UNIVERSITÉ DE GENÈVESection de PhysiqueDépartement de Physique de la MatièreQuantique

FACULTÉ DES SCIENCESProfesseur T. GiamarchiProfesseur C. Kollath

Numerical study of thermal effects inlow dimensional quantum spin

systems

THÈSE

présentée à la Faculté des Sciences de l’Université de Genèvepour obtenir le grade de docteur ès Sciences, mention Physique

par

Emanuele Coirade

Côme (Italie)

Thèse n◦ Thesis-number

GENÈVEAtelier d’impression ReproMail

2016

Contents

1 Introduction 11.1 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Spin-1/2 Chains and Ladders 52.1 Homogeneous spin-1/2 chains . . . . . . . . . . . . . . . . . . . . 52.2 Inhomogeneous spin-1/2 chains: the isotropically dimerized case . . 62.3 Spin-1/2 two-leg ladders . . . . . . . . . . . . . . . . . . . . . . . 92.4 Spin chain mapping and boson representation for the ladder model . 10

2.4.1 Beyond spin chain mapping: treatment around hc1 . . . . . 112.4.2 Beyond spin chain mapping: treatment around hc2 . . . . . 12

2.5 Experimental realizations . . . . . . . . . . . . . . . . . . . . . . . 13

3 Methods 153.1 DMRG & MPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1.1 Basic ideas of DMRG & MPS . . . . . . . . . . . . . . . . 163.1.2 Ground state search . . . . . . . . . . . . . . . . . . . . . . 193.1.3 Time evolution . . . . . . . . . . . . . . . . . . . . . . . . 213.1.4 Finite temperature . . . . . . . . . . . . . . . . . . . . . . 223.1.5 Dynamical correlations at finite temperature . . . . . . . . . 23

3.2 Tomonaga-Luttinger Liquid (TLL) . . . . . . . . . . . . . . . . . . 253.2.1 Bosonization of the spin-1/2 chain . . . . . . . . . . . . . . 263.2.2 Evaluation of TLL parameters . . . . . . . . . . . . . . . . 27

4 NMR Relaxation Rate 314.1 Spin-lattice relaxation rate 1/T1 . . . . . . . . . . . . . . . . . . . 31

i

CONTENTS

4.2 What we compute, why and how . . . . . . . . . . . . . . . . . . . 334.3 Results for 1/T1 as a function of the temperature . . . . . . . . . . 35

4.3.1 Heisenberg model . . . . . . . . . . . . . . . . . . . . . . 364.3.2 XXZ model . . . . . . . . . . . . . . . . . . . . . . . . . . 404.3.3 Dimerized model . . . . . . . . . . . . . . . . . . . . . . . 41

4.4 Summary & conclusions of the chapter . . . . . . . . . . . . . . . . 43

5 Dynamical Correlations at Finite Temperature of a Dimerized Spin-1/2Chain 455.1 What we compute: definitions . . . . . . . . . . . . . . . . . . . . 465.2 Results for dynamical correlations at finite temperature . . . . . . . 47

5.2.1 Isotropic gapped system h = 0 . . . . . . . . . . . . . . . . 485.2.2 Gapped regime at finite h . . . . . . . . . . . . . . . . . . . 505.2.3 Tomonaga-Luttinger liquid regime . . . . . . . . . . . . . . 525.2.4 High field gapped phase . . . . . . . . . . . . . . . . . . . 55

5.3 Band-narrowing effects . . . . . . . . . . . . . . . . . . . . . . . . 565.4 Summary & conclusions of the chapter . . . . . . . . . . . . . . . . 58

6 Grüneisen Parameter and Quantum Phase Transitions in Spin-1/2 Lad-ders 636.1 The Grüneisen parameter: definition and computation . . . . . . . . 63

6.1.1 Computation of Γmag via spin-chain mapping . . . . . . . . 646.1.2 Strong coupling expansion & refined results . . . . . . . . . 67

6.2 Analytical results against experiments and DMRG . . . . . . . . . . 696.3 Summary & conclusions of the chapter . . . . . . . . . . . . . . . . 71

7 General conclusions & Perspectives 75

A Technical aspects of chapter 4 77A.1 Equality between the time integrals of onsite S+− and S−+ correla-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77A.2 Derivation of the result in Eq. (4.11) . . . . . . . . . . . . . . . . . 78

B Extrapolation Method for Spin-Lattice Relaxation Rate 81

C Consistency Test Using the XX Model 83

D Dynamics of a single excitation in the strong dimerization limit 87D.1 h = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87D.2 S+− correlations for h > hc2 . . . . . . . . . . . . . . . . . . . . . 88

ii

CHAPTER 1

Introduction

Quantum spin systems are an extremely fascinating subject of study because of theirrich physics. Depending on their microscopic characteristics (dimensionality, geom-etry, local spin S, type of interaction) and on the external environment (temperature,applied magnetic field, pressure), one can detect many interesting phenomena. Vari-ous combinations of these ingredients have been investigated theoretically [1, 2] fordecades, leading to several important results and observations. Among them one cancite for example the discovery of the spinon nature of the excitation spectrum of thespin-1/2 antiferromagnetic chain [3], the different nature of integer and half-integerspin chains (which are gapped or gapless, respectively) [4, 5], or the connectionsbetween high temperature supeconductivity and quantum magnetism in a 2D squarelattice [6].

More specifically, low-dimensional quantum magnets like spin chains or laddersare particularly intriguing because quantum fluctuations are extreme and no orderedstate is usually possible. In one dimension, the interaction between the excitationsleads to various exotic states, ranging from gapped phases to phases possessingquasi-long-range magnetic order known as Tomonaga-Luttinger liquids (TLL) [7].Thanks to recent advancements in chemistry, it is now possible to grow pure, singlecrystalline samples of low-dimensional spin systems, characterised by small cou-pling constants (in the order of ∼ 10 K) [8, 9]. On the one hand, this allows oneto experimentally manipulate the sample and to drive it through different quantumphases, for example via the application of a magnetic field [10, 11] or by exertingpressure on it [12]. On the other hand, the energy scale of the exchange couplinggets closer to temperatures accessible by experiments, so that thermal effects start to

1

1. INTRODUCTION

play a fundamental role in the description of the system.In order to provide a complete picture of the different types of order that can be

present, the nature of the different phases and also what happens at the quantum crit-ical points (QCP), a set of probes sensitive to the correlation functions of the system,which can also be evaluated both experimentally and theoretically, is needed. Onthe experimental side, fortunately, a set of suitable probes such as neutron scatter-ing [13], electronic spin resonance [14], Raman scattering [15] and Nuclear Mag-netic Resonance (NMR) [16, 17] exists. On the theoretical side, it is important tohave reliable numerical and analytical methods to access the desired quantities andto allow for a direct comparison with experiments. Field theory methods such asbosonization [7] provide up to now a very good description of the correlation func-tions of such systems [10] for temperatures very low compared to the characteristicenergy scale. Via Bethe ansatz it is possible to compute for integrable models theneutron spectra at zero temperature [18–20]. Density matrix renormalization group(DMRG) techniques [21] provide a quantitative description of the thermodynam-ics, and of dynamical properties at zero temperature of low-dimensional spin sys-tems [10, 11, 22]. However, a direct method of computation representing a bridgebetween the very low temperature regime and the high temperature one was miss-ing. Recent developments in DMRG/MPS techniques [23, 24] have opened the pathfor the computation of real time dynamics at finite temperature [25] and here wewill make intensive use of this new feature. This technique offers some advantageswhen compared to other numerical methods. It can access directly the real time dy-namics avoiding the delicate problem of the analytic continuation from imaginarytime which characterizes quantum Monte-Carlo (QMC) [26], and it is capable totreat systems of much bigger size if compared for instance to exact diagonalizationmethods [27].

In this thesis we show how MPS methods can be used for a quantitative computa-tion of dynamical correlation functions in one dimensional spin-1/2 systems at finitetemperature, especially in regimes where the temperature plays a crucial role. Byputting together the analytical results in the low temperature limit, and the numericalresults at finite T, one can now get a full description of quantities like the NMR re-laxation time or the spectra obtained via inelastic neutron scattering measurements.We also perform an analytical study of the thermal effects close to criticality in spin-1/2 two-leg ladder systems. A more detailed outline of the thesis is given in thefollowing.

1.1 Outline of the thesisThis thesis is strongly based on Refs. [28–30], but it also contains a broader discus-sion about some technical aspects and methods. It is organized as follows:

• In chapter 2 we introduce the three spin-1/2 models that will be investigated:

2

1.1 Outline of the thesis

the XXZ chain, the isotropically dimerized chain and the two-leg ladder. Theircorresponding Hamiltonians and their phase diagrams are discussed. In thecase of the ladder we show how the model can be mapped onto a purely 1Dchain in some specific regimes, and other subsequent approximations whichallow a direct analytical solution for the quantities we are interested in. Ourresults for the dimerized chain and for the ladder are computed with explicitreference to existing compounds, which are presented in the last part of thechapter.

• In chapter 3 we provide a description of two theoretical techniques, one nu-merical and one analytical, which are particularly suited for the treatment oflow dimensional physics: DMRG (with its MPS reformulation) and the TLLtheory. Starting from the generic MPS representation of a quantum state, wediscuss how the search for a ground state of a given Hamiltonian is performedand how dynamics and finite temperature can be introduced in this framework.We then combine these two ingredients to tackle the computation of dynam-ical correlation functions at finite temperature. In the second part we presentthe TLL theory focusing on its applications to the case of the spin chain, andwe propose a new method to determine numerically the TLL parameters.

• Chapter 4 is dedicated to the numerical computation of the NMR spin-latticerelaxation rate (often called 1/T1) for the chain models. After having definedthe quantity and discussed its connection with spin-spin correlation functions,we show how MPS results for finite temperature dynamics can offer very gooddescription of it, also in regimes where analytical treatments are particularlydifficult.

• In chapter 5 we present the MPS results for dynamical correlation functionsat finite temperature of a dimerized spin-1/2 chain at different magnetic fieldsand temperatures. We discuss the main features of the spectra through differ-ent quantum phases, and try to interpret some of these structures in the strongdimerization limit or using the TLL theory. We also investigate how the tem-perature acts on the spectra and in particular on the dispersion of a single tripletexcitation.

• In chapter 6 we deal with the computation of the so called Grüneisen param-eter for the ladder model. After having defined this quantity and highlightedits importance for the detection and the investigation of quantum phase transi-tions, we compute it analytically close to criticality using the strong couplingexpansion and an effective free fermion theory. Our results are compared toexperimental measurements and DMRG simulations.

• In chapter 7 we give a general overview on our results and discuss furtherperspectives.

3

1. INTRODUCTION

• In appendix A we give additional technical details about some calculations andapproximations performed in chapter 4.

• In appendix B we present the extrapolation method adopted in chapter 4 andhow we associate an errobar to the extrapolated results.

• In appendix C we benchmark our MPS results for dynamical correlations atfinite temperature against exact results available for the XX model.

• In appendix D we describe, in the case of the dimerized chain in the gappedphases and in the strong dimerization limit, how some computed spectra canbe interpreted by simply considering the dynamics of a single excitation.

4

CHAPTER 2

Spin-1/2 Chains and Ladders

In this chapter we present in detail the three spin-1/2 models which we will study inthis work: we start from the XXZ chain, then we move to the isotropically dimerizedchain, and finally we consider the two-leg ladder model. For all of them we remindbriefly their main physical features and for the last one we introduce the spin chainmapping and the boson representation, which provides a simpler interpretation ofthe physics of the system in specific limits. Finally we present more in detail someinteresting experimental realizations of the dimerized chain and of the two-leg ladderto which our calculations refer directly.

2.1 Homogeneous spin-1/2 chainsWe start by considering the very well known XXZ Hamiltonian. The chain is sub-jected to a magnetic field h is applied along the z direction. The Hamiltonian is givenby

H = J∑j

[1

2

(S+j S−j+1 + h.c.

)+ ∆Szj S

zj+1

]− h

∑j

Szj , (2.1)

where Sαj = 12σ

αj is a spin operator for a spin 1/2 on site j, α = x, y, z denotes

its direction, and σαj the Pauli matrices. S±j = Sxj ± iSyj are the spin rising andlowering operators. The parameter J gives the spin coupling strength, ∆ is dimen-sionless and measures the anisotropy. The g factor, the Bohr magneton and ~ havebeen absorbed into h and J , which both have here the dimensions of an energy. For

5

2. SPIN-1/2 CHAINS AND LADDERS

Figure 2.1: Schematic phase diagram at zero temperature of the XXZ model as a func-tion of the magnetic field h and of the anisotropy ∆. The phase denoted with ’Ferro’is the ferromagnetic one in which spins are polarized along the z direction. XY de-notes a massless phase characterized by dominant in-plane antiferromagnetic correla-tions which is a TLL [7]. ’Néel’ denotes an antiferromagnetically ordered Ising phasealong z. Dashed lines separate the different phases. The behavior of the boundary as afunction of h around the point ∆ = 1 reflects the Berezinski-Kosterlitz-Thouless (BKT)behavior of the gap at the transition. The figure is taken from Ref. [29] and inspired byFig. 1.5 in [2].

the isotropic case ∆ = 1, the model corresponds to the Heisenberg (or XXX) Hamil-tonian, isotropic in the three directions. For ∆ = 0 we have the XX model whichcan be mapped via a Jordan-Wigner transformation [7] onto a free-fermion modelwith a fixed chemical potential and possesses exact solution. The phase diagram atzero temperature of the XXZ model as a function of the magnetic field and of theanisotropy is given in Fig. 2.1. The boundary between the XY and ferromagneticphases is described by the relation h = J(1 + ∆). The boundary between the XYand Néel phases is given by the triplet gap, which is a function of ∆ [2]. In this workwe will limit ourselves to the case 0 ≤ ∆ ≤ 1.

2.2 Inhomogeneous spin-1/2 chains: the isotropicallydimerized case

The second model we consider in this work is the dimerized Heisenberg chain sub-jected to a magnetic field h along the z direction, which is described by the Hamil-tonian

H =∑j

(J + (−1)jδJ

)Sj · Sj+1 − h

∑j

Szj . (2.2)

where Sj denotes the vector of the spin at site j. The spin coupling strength isalternated with values Js = J + δJ (strong bonds) and Jw = J − δJ (weak bonds).Also in this case the g factor, the Bohr magneton and ~ have been absorbed into h and

6

2.2 Inhomogeneous spin-1/2 chains: the isotropically dimerized case

J-δJ J-δJJ+δJ

j+1 j+2jj-1

Figure 2.2: Pictorial representation of the dimerized Heisenberg chain: a spin 1/2 islocated on each black square, the strength of the coupling (between nearest neighborspins only) is alternated with values Js = J + δJ and Jw = J − δJ .

(J±δJ), which all have here the dimensions of an energy. A pictorial representationof this system is given in Fig. 2.2.

The phase diagram of this model as a function of the magnetic field and at lowtemperature is reported in Fig. 2.3, lower part. At zero magnetic field (h = 0) themodel has a non-trivial spin-0 ground state (quantum disordered phase) with a gapof order Js to the first excitation which is a band of spin-1 excitations (triplons) [7].The role of the magnetic field along z is to progressively reduce this gap, up to thefirst critical field hc1 when the gap closes. At h > hc1 a quantum critical phase ariseswith gapless excitations (TLL phase). If we further increase the magnetic field, thespins of the chain progressively polarize and above the second critical magnetic fieldhc2 the ground state is fully polarized with a gapped spectrum [7].

A very intuitive explanation of this behavior can be provided by considering thelimit of strong dimerization. As we will see in the end of this chapter, we will stickto a choice of parameters that will put us exactly in that limit. When J ∼ δJ ←→Js � Jw the strong bonds are essentially decoupled from each other. In this singlestrong bond picture, each of them has four eigentstates: the singlet state

|s〉 =|↑↓〉 − |↓↑〉√

2(2.3)

with energy Es = −3Js/4, total spin S = 0 and z−projection Sz = 0, and the threetriplet states ∣∣t+⟩ = |↑↑〉 ,

∣∣t0⟩ =|↑↓〉+ |↓↑〉√

2,∣∣t−⟩ = |↓↓〉 (2.4)

with S = 1; Sz = 1, 0,−1, and energies E+ = Js/4 − h, E0 = Js/4, E− =Js/4+h, respectively. The ground state is |s〉 below the critical value of the magneticfield hc1 and |t+〉 above. The dependence of the energies on h is shown in Fig. 2.3.A small but finite Jw delocalizes triplets and creates bands of excitations with a finitebandwidth for each triplet branch. This leads, as anticipated, to three distinct phasesin the dimerized system of Eq. (2.2) depending on the magnetic field:

1. Quantum disordered phase, characterized by a spin-singlet ground state anda gapped spectrum. This phase appears for magnetic fields ranging from 0 to

7

2. SPIN-1/2 CHAINS AND LADDERS

Figure 2.3: Sketch of the energy spectrum of excitations for the dimerized chain underthe application of a magnetic field h in the limit of large dimerization (J ≈ δJ). Thediagram can be very well understood in the single strong bond picture. At h = 0 there isan energy gap of the order of Js between the singlet (ground state) and the three tripletstates. The magnetic field h splits the triplets and brings down the excitation energy ofthe state

∣∣t+⟩. Due to the presence of the weak bonds the triplets can be delocalizedand, thus, have a dispersion in energy of the order of Jw (the boundaries of which arerepresented by the dotted lines). At sufficiently high magnetic field, the energy of thelowest triplon band is close to the energy of the singlet state. This leads in the extendedsystem to a quantum critical phase for h > hc1 with gapless excitations. The systemremains in this phase up to the point h = hc2 for which the triplon band is totally filledand a fully polarized phase arises. Picture adapted from Fig. 2 in Ref. [10]

.

hc1. The magnetic field progressively splits the triplets energy, bringing downthe |t+〉 one, and closes the gap.

2. TLL phase, characterized by a gapless excitation spectrum. It occurs betweenthe critical fields hc1 and hc2. The magnetization per site increases from 0 to0.5 for h running from hc1 to hc2. The low energy physics can be describedby the TLL theory.

3. Fully polarized phase, characterized by a fully polarized ground state and agapped excitation spectrum. This phase appears above hc2.

8

2.3 Spin-1/2 two-leg ladders

𝑺𝑗−1,1 𝑺𝑗,1 𝑺𝑗+1,1

𝑺𝑗−1,2 𝑺𝑗,2 𝑺𝑗+1,2

J⊥ J∥

J∥

Figure 2.4: Pictorial representation of the two-leg spin-1/2 ladder: a spin 1/2 is locatedon each black square, the strength of the coupling (between nearest neighbor spins only)is J⊥ on the rungs and Jw on the legs. J⊥ is considerably bigger than J‖ in this work.

2.3 Spin-1/2 two-leg laddersFinally we consider also the spin-1/2 two-leg ladder model, in a magnetic field halong the z direction, described by the Hamiltonian

H = H⊥ +H‖, (2.5)

whereH⊥ = J⊥

∑j

Sj,1 · Sj,2 − h∑j

(Szj,1 + Szj,2

), (2.6)

and

H‖ = J‖∑j

2∑l=1

Sj,l · Sj+1,l. (2.7)

Here Sl,k is the spin-1/2 (vector) located on the l-th leg and on the j-th rung. J⊥ isthe coupling along the rung, J‖ is the coupling along the legs. Again the g factor,the Bohr magneton and ~ have been absorbed into h, J⊥ and J‖ which all have herethe dimensions of an energy. A sketch of the model is given in Fig. 2.4. This modelshares many features with the dimerized chain presented in the previous section andfor the phase diagram we can easily refer to Fig. 2.3 with the mapping J⊥ ←→ Jsand J‖ ←→ Jw, and J⊥ � J‖. Therefore we have again

1. A Quantum disordered phase, characterized by a spin-singlet ground state oneach rung and a spectrum with a gap of the order J⊥. This phase appears formagnetic fields ranging from 0 to hc1.

2. A TLL phase, characterized by a gapless excitation spectrum, occurring be-tween the two critical fields hc1 and hc2. The low energy physics can bedescribed by the TLL theory.

9

2. SPIN-1/2 CHAINS AND LADDERS

3. A Fully polarized phase, characterized by full polarization and a gapped exci-tation spectrum, occurring above hc2.

2.4 Spin chain mapping and boson representation forthe ladder model

When J⊥ � J‖, the ladder problem can be reduced to a simpler spin chain problem,especially in the vicinity of the two critical fields hc1 and hc2. In these regions states|s〉 and |t+〉 are close to each other in energy, while the other two

∣∣t0⟩ and |t−〉represent highly excited states. The essence of the spin-chain mapping [31–34] is toproject out

∣∣t0⟩ and |t−〉 bands from the Hilbert space of the model, since they arealmost completely unoccupied. One can thus map each strong bond to a pseudo-spin1/2 according to the relations

˜|↓〉 = |s〉 =1√2

(|↑↓〉 − ↓↑〉) ,

˜|↑〉 =∣∣t+⟩ = |↑↑〉 . (2.8)

The transformation that expresses the original spin operators Si (i = 1, 2) of eachrung in terms of the new pseudo-spin 1/2 operators S is given by:

S±1,2 =: ∓ 1√2S±, Sz1,2 =:

1

4

(1 + 2Sz

). (2.9)

The ladder Hamiltonian in Eqs. (2.5-2.7) becomes after this transformation an XXZchain Hamiltonian, with anisotropy 1/2:

HXXZ = J‖∑j

[1

2

(S+j S−j+1 + S−j S

+j+1

)+

1

2Szj S

zj+1

]−

−(h− J⊥ −

J‖

2

)∑j

Szj + L

(−J⊥

4+J‖

8− h

2

). (2.10)

L is the length of the ladder. This new Hamiltonian can be solved analytically usingBethe ansatz, see for instance Refs. [18,19]. Close to the first critical field and at lowtemperature, we are in a regime of extreme diluteness of excitations (very few triplets“+” in a sea of singlets). One can get an idea of the physics in this limit adopting aneffective theory which consists in considering the excitations as non interacting. Wethen apply to Hamiltonian (2.10) a Jordan-Wigner transformation given by{

S+j = (−1)jc†je

−iπ∑

l<j c†l cl ,

Szj = c†jcj − 12 ,

(2.11)

10

2.4 Spin chain mapping and boson representation for the ladder model

and we then neglect the term quadratic in density. This leads to the following freefermion Hamiltonian

Hff = −J‖

2

∑j

(c†j+1cj + c†jcj+1

)− (h− J⊥)

∑j

c†jcj −3

4LJ⊥, (2.12)

which can be exactly solved for all quantities of interest.This spin chain mapping represent a part of a more general strong coupling ex-

pansion of the model. In order to go beyond the previous results, it becomes neces-sary to say more about this approach. As anticipated, the four-dimensional Hilbertspace on each rung is spanned by the four states |s〉, |t+〉,

∣∣t0⟩ and |t−〉, and we in-troduce the four corresponding operators s†j , t

†j,+, t†j,0 and t†j,− creating on the rung

j a singlet or a triplet. By imposing an hard-core constraint on each rung j (only oneof the four state can exist on a single rung)

s†jsj + t†j,+tj,+ + t†j,0tj,0 + t†j,−tj,− = 1, (2.13)

we can rewrite the Hamiltonian of the ladder model in terms of boson operators. H⊥is quadratic in these operators, while H‖ is quartic and it has a much more complexstructure [35]. This procedure is the starting point also for the simple spin chainmapping discussed just above. To go back to Eq. (2.10) what we need to do is toneglect completely the possibility of having higher excitations

∣∣t0⟩ and |t−〉. Thiscan be achieved by suppressing all terms containing operators tj,0, tj,−, t†j,0 and t†j,−,and by simplifying the hard-core constraint Eq. (2.13) as

s†jsj + t†j,+tj,+ = 1. (2.14)

If we now collect the remaining term and make an additional transformation back tothe spin operators of the form{

Szl = t†l,+tl,+ −12 ,

S+l = t†l,+sl,

(2.15)

the structure of Eq. (2.10) is restored.

2.4.1 Beyond spin chain mapping: treatment around hc1

Let’s try to refine the previous picture a little bit and we start by the vicinity of hc1.We start as always from a landscape where only few triplets |t+〉 are present in asea of singlets |s〉, but we want to add very few excitations

∣∣t0⟩ and |t−〉 createdby temperature. Following this idea we rearrange the ladder Hamiltonian written interms of rung operators by re-expressing s†jsj in terms of the densities of the othertriplets (exploiting the hard-core constraint), by setting elsewhere sj ≈ s†j ≈ 1, and

11

2. SPIN-1/2 CHAINS AND LADDERS

by neglecting all terms cubic or quartic in triplet operators. After some algebra weare left with an Hamiltonian containing three species of non-interacting fermions(i.e. the three triplets) which we write in a more compact form as

H1 =∑

l=+,0,−

∑j

[J‖

2

(c†l,jcl,j+1 + h.c.

)− hlc†l,jcl,j

], (2.16)

with the evident association c†l,j = t†l,j for all j and for l = +, 0,− and:

h+ = h− J⊥,h0 = −J⊥, (2.17)

h− = −h− J⊥.

According to our expectations, the magnetic field for the triplets∣∣t0⟩ is fixed and

prevents the creation of these excitations at zero temperature. The same happensfor the remaining triplets |t−〉 since we are in the vicinity of the first critical field.This Hamiltonian can be solved exactly for all the quantities we are interested in, forvalues of the field around the first critical one.

2.4.2 Beyond spin chain mapping: treatment around hc2

Close to hc2, the scene is completely dominated by triplets |t+〉 and there are onlyfew singlets. As before, we take into account the possibility of having in additionvery few higher excitations

∣∣t0⟩ and |t−〉 thermally activated. This time we rearrangethe ladder Hamiltonian in terms of rung operators by re-expressing t†+,jt+,j in termsof the densities of the other triplets and of the singlets (exploiting again the hard-coreconstraint), by setting elsewhere t+,j ≈ t†+,j ≈ 1, and by neglecting all terms cubicor quartic in sj , t0,j and t−,j operators. After some algebra we are left once morewith an Hamiltonian containing three species of non-interacting fermions (i.e. the“0” and “-” triplets, and singlets) of the form:

H2 =J‖

2

∑j

[s†jsj+1 + h.c.

]−(J⊥ + J‖ − h

)∑j

s†jsj+

+J‖

2

∑j

[t†j,0tj+1,0 + h.c.

]−(J‖ − h

)∑j

t†j,0tj,0−

− 2(J‖ − h

)∑j

t†j,−tj,− +

(J⊥4

+J‖

2− h)L. (2.18)

This Hamiltonian can be solved exactly for all the quantities we are interested in, forvalues of the field around the second critical one.

12

2.5 Experimental realizations

Copper Nitrogen Hydrogen Oxygen

Figure 2.5: Chemical structure of the copper nitrate[Cu (NO3)2 · 2.5D2O

]. Red

spheres are Cu2+ ions which are the spin carriers. Solid, red thick lines represent thestrong coupling Js, Solid, red thin lines represent the weak coupling Jw, while dashedgreen lines stand for the intra-chain interaction, which is weak and therefore not consid-ered in this work. The sketch has been realised according to the crystal structure detailedin Ref. [39].

2.5 Experimental realizations

Results that will be presented in this work for the dimerized chain and for the laddermodel are computed within choices of parameters referring to existing compounds.

As for the dimerized chain we explicitly refer to the quantum magnet coppernitrate [Cu (NO3)2 · 2.5D2O] experimentally investigated via neutron scattering inRefs. [36–38]. This compound shows a dimerized structure with coupling constantsJs = J + δJ = 5.28 kBK and Jw = J − δJ = 1.474 kBK, and critical fieldsexperimentally measured hc1 = 4.39 kBK and hc2 = 6.73 kBK. A sketch of theatomic structure of the compound is shown in Fig. 2.5.

As for the ladder model we consider two organic compounds belonging to theso called “Hpip” family, namely [(C5H12N)2 CuBr4], which is often called BPCB,and its chlorine counterpart [(C5H12N)2 CuCl4]. The first of these materials was pre-sented originally in Ref. [40] and it has been intensively investigated in the followingyears both theoretically and experimentally, see for instance Ref. [10] and referencestherein. The atomic structure of the bromine compound is sketched in Fig. (2.6).It is characterized by coupling constants J⊥ = 12.92 kBK and J‖ = 3.33 kBK,and critical fields hc1 = 9.69 kBK and hc2 = 19.95 kBK. However, the values ofthe couplings of this compound make it difficult to access experimentally the uppercritical field. More recently a new member of this family of compounds have been

13

2. SPIN-1/2 CHAINS AND LADDERS

Figure 2.6: Chemical structure of BPCB compound. Blue spheres are Cu2+ ions whichare the spin carriers. Solid thick blue lines and dashed thick blue lines stand for theinteraction path J⊥ and J‖, respectively. Picture adapted from Fig.1(a) in Ref. [41].

synthetized, by replacing Br atoms with Cl ones [42]. This substitution induces a re-duction of the exchange parameters and therefore of the critical fields, making bothof them more easily accessible in experiments. For the chlorine compound we thushave J⊥ = 3.42 kBK and J‖ = 1.34 kBK, and critical fields hc1 = 2.39 kBK andhc2 = 6.06 kBK.

14

CHAPTER 3

Methods

In this chapter we present the methods that will be used in this work for the the-oretical investigation of the spin systems already introduced. In particular we willfocus on the DMRG, especially in its more recent matrix product states (MPS) re-formulation, and on the TLL theory: these techniques are particularly suited for thetreatment of one-dimensional or quasi-one-dimensional systems.

We start from a general overview on DMRG, which represents a very power-ful tool to describe strongly correlated quantum lattice systems, especially in onedimension. This techinque was originally introduced by S. R. White in the begin-ning of the 90’s [43, 44]. Along the years many extensions and generalizations havebeen proposed and successfully implemented, offering the possibility to access staticproperties at finite temperature [45–47] and dynamical properties at zero tempera-ture [48–50]. These two results have opened the path to the computation of dynam-ical correlations at finite temperature. In this work we will make extensive use ofthese recent algorithms, following what is proposed in Refs. [23, 24].

Finally we introduce an analytical low-energy description for the gapless regimeof our spin systems, the TLL theory [7]. This theory has proven to be extremely suc-cessful in describing quantitatively the low-energy physics of many one-dimensionalsystems. It will be used in this work to benchmark some of the numerical results andto help in giving a physical interpretation of them. We will see the importance in thistheory of two parameters, the TLL parameters u and K, and we propose a methodto compute them numerically via DMRG.

The combination of numerical and analytical results (DMRG/MPS+TLL theory)allows today to obtain a complete, detailed picture of the model for all energy and

15

3. METHODS

temperature regimes.

3.1 DMRG & MPSIn this section we first give a short review of the basic ideas of the method, focusingon its application to one-dimensional, finite systems at zero temperature. Then wediscuss the implementation of the real time evolution (useful for the simulation of thedynamics) and then of the imaginary time evolution (allowing for the computation ofthermodynamic quantities). Finally, by combining these two procedures as explainedin Refs. [23, 24], we describe how to compute finite temperature dynamics. Wepresent the algorithm we use in this work and we also discuss how to get momentum-frequency correlations from our results.

3.1.1 Basic ideas of DMRG & MPSThe most difficult problem while dealing with numerics for quantum many-bodysystem is the exponential growth of the Hilbert space of the system with its size.If we take for instance a spin-1/2 chain of L sites, the corresponding Hilbert spacewould be of size 2L. Luckily, the first assumption on which DMRG relies is that onecan always find a reduced Hilbert space, often extremely small if compared to the fullone, which contains the relevant physics and which can be parametrized efficientlyby the MPS formalism. In fact, quantum many-body states can be represented, asone would guess from the name of the formalism, as products of matrices. Alsothe operators we use to measure the relevant quantities can be re-interpreted in thismatrix language, in this case we speak about matrix product operators (MPO). Theground state search is then performed, as we will see, as a variational procedure inthe MPS space.

Before entering into details, let us introduce one of the most useful results oflinear algebra: the so called singular value decomposition (SVD). SVD guaranteesthat for any arbitrary rectangular matrix M of dimensions m × n there exists a de-composition such that

M = USV †. (3.1)

S is diagonal with non-negative elements Saa = sa called singular values and ithas dimensions (min(m,n) ×min(m,n)). Singular values are organized such thatsa ≤ sa′ if a > a′. The number r of singular values which are non-zero gives therank of M.U has dimensionsm×min(m,n) andU†U = I, while V † has dimensionsmin(m,n)× n and V V † = I. It is particularly important for the following to stressthat it is possible to optimally approximate a matrix M of rank r by a matrix M’ ofrank r′ < r properly chosen. This approximation consists in setting to zero all butthe first r′ singular values of S to zero (and to shrink the column dimension of U andthe row dimension of V † accordingly).

16

3.1 DMRG & MPS

The basic idea of the DMRG techinque consists in splitting the Hilbert space intwo blocks which we call left (L) and right (R). Any pure state on a bipartite latticeLR can be written as

|Ψ〉 =∑α,β

Cαβ∣∣φLα⟩ ∣∣φRβ ⟩ , (3.2)

where∣∣φL,R⟩ are in the set of the orthonormal basis of the two blocks, of dimensions

NL and NR, in which we have divided the system. The coefficients Cαβ can be readas the elements of a matrix C. As previously said, the size of each block basis growsexponentially with the lattice size. In order to make the computation accessible oneneeds at some point to reduce the size of the bases keeping only N < NL, NR statesto approximate each of them.

The SVD procedure represents the basis of a very useful representation of quan-tum states of a bipartite system LR, the Schmidt decomposition. It also provides apowerful optimization of the truncated bases. Let us now reconsider the state |Ψ〉 inEq. (3.2) and let’s perform an SVD of the matrix C. We get

|Ψ〉 =∑α,β

min(NL,NR)∑r=1

UαrSrrV∗βr

∣∣φLα⟩ ∣∣φRβ ⟩ =

=

min(NL,NR)∑r=1

(∑α

Uαr∣∣φLα⟩

)sr

∑β

V ∗βr∣∣φRβ ⟩

=

=

min(NL,NR)∑r=1

sr∣∣rL⟩ ∣∣rR⟩ , (3.3)

where sr = Srr are the elements on the diagonal of the matrix S. If this last sum isperformed over only the N ≤ min(NL, NR) positive and nonzero singular values,what we get is the Schmidt decomposition of our quantum state. This decompositionis related to the reduced density matrices by

ρL = TrR |Ψ〉 〈Ψ| =∑r

s2r

∣∣rL⟩ ⟨rL∣∣ , (3.4)

ρR = TrL |Ψ〉 〈Ψ| =∑r

s2r

∣∣rR⟩ ⟨rR∣∣ , (3.5)

where∣∣rL,R⟩ are eigenvectors of ρL,R, and s2

r are the eigenvalues. An approximatedescription of the state |Ψ〉 can be obtained by truncating its Schmidt decompositionretaining only the states

∣∣rL⟩ and∣∣rR⟩ corresponding to the D largest values sr:

|Ψ′〉 ≈D∑r=1

sr∣∣rL⟩ ∣∣rR⟩ , (3.6)

17

3. METHODS

with s1 ≥ s2 ≥ s3 ≥ · · · ≥ 0. If normalization is desired, the retained singularvalues need to be rescaled. The accuracy of the approximation can be evaluated bythe truncation error εD defined as the difference in norm between the exact state andthe approximated one using the optimized basis:

εD =∑r>N

s2r = ‖|Ψ〉 − |Ψ′〉‖2 (3.7)

For a givenD this error is minimized by the truncation procedure adopted in Eq. (3.6)and it clearly depends on the distribution of the sr.

Let’s go back now to our chain of L sites with d-dimensional local state spaces{σi} on each site i = 1, . . . , L. It can be shown that any representative state of aspin-1/2 chain (and more in general of any quantum state on a lattice) can be writtenas:

|Ψ〉 =∑

σ1,...,σL

cσ1...σL|σ1 . . . σL〉 . (3.8)

Let’s see now how it can be represented exactly in a matrix product form. Additionaldetails can be found in Ref. [21], here we follow the same procedure described there.We start by reshaping the vector cσ1...σL

with dimension dL into a matrix M withdimensions

(d× dL−1

):

cσ1...σL−→ Mσ1,(σ2...σL) (3.9)

An SVD of M gives

Mσ1,(σ2...σL) =

r1∑`1

Uσ1,`1S`1,`1(V †)`1,(σ2...σL) =

r1∑`1

Uσ1,`1c`1σ2...σL, (3.10)

where we have reshaped back into a vector the product between matrices S andV †, and r1 ≤ d. Now we can decompose the matrix U into a collection of d rowvectors Lσ1 with elements Lσ1

`1, and reshape at the same time c`1σ2...σL

into a matrixM(`1σ2),(σ3...σL) of dimensions (r1d× dL−2). We then get

cσ1...σL=

r1∑`1

Lσ1

`1M(`1σ2),(σ3...σL). (3.11)

Now this new matrix M can undergo a new SVD following the same idea as beforeto give

cσ1...σL=

r1∑`1

r2∑`2

Lσ1

`1Lσ2

`1,`2M(`2σ3),(σ4...σL), (3.12)

where M(`2σ3),(σ4...σL) has now dimensions (r2d × dL−3) and r2 ≤ r1d ≤ d2. Ifwe continue to SVD until the end of the chain, what we get is

cσ1...σL=

∑`1,...`L−1

Lσ1

`1Lσ2

`1`2. . .LσL−1

`L−2,`L−1LσL

`L−1= Lσ1Lσ2 . . .LσL−1LσL ,

(3.13)

18

3.1 DMRG & MPS

where the last compact form implies the matrix product induced by the sums overthe `i indexes. The quantum state can be now represented in the matrix product stateform as:

|Ψ〉 =∑

σ1...σL

Lσ1Lσ2 . . .LσL−1LσL |σ1 . . . σL〉 . (3.14)

These matrices L are such that∑σiLσi†Lσ = I and they are called left-normalized,

and the corresponding MPS left-canonical. The MPS formalism is quite versatile,therefore one can easily get for example right-canonical MPS, if one starts the seriesof SVDs from the right, or even mixed-canonical if one starts from both ends of thechain. In this last case the decomposition gives:

|Ψ〉 =∑

σ1...σL

Lσ1 . . .LσlSRσl+1 . . .RσL |σ1 . . . σL〉 , (3.15)

with singular values at the bond (l + 1, l). Within this notation the Schmidt decom-position presented above into two blocks (left block from 1 to l, right block froml + 1 to L) is easy to reintroduce. If we reconsider Eq. (3.3) one immediately notesthe parallelism ∣∣rR⟩ =

∑σ1...σl

(Lσ1 . . .Lσl) |σ1 . . . σl〉 (3.16)

∣∣rL⟩ =∑

σl+1...σL

(Rσl+1 . . .RσL) |σl+1 . . . σL〉 (3.17)

From a simple dimensional analysis it turns out immediately that in practical calcu-lations such decompositions cannot be performed exactly given the limited computa-tional resources. Luckily it turns out that in most cases the state can be neverthelessdescribed accurately using matrices of reduced dimensions. If we consider for in-stance the mixed-canonical representation we have just introduced, it is possible tocut the spectra of the reduced density operators and to optimally approximate (in the2-norm) the state following the same philosophy of Eq. (3.6). This argument can begeneralized from the approximation incurred by a single truncation to that caused byL−1 truncations, one at each bond. If we make no assumptions on the normalizationof the matrices, we can write a general MPS for open boundary conditions as:

|Ψ〉 =∑

σ1,...,σL

Mσ1Mσ2 . . .MσL−1MσL |σ1 . . . σL〉 . (3.18)

3.1.2 Ground state search

Let us now move to the presentation of the standard algorithm which allows one tocompute the ground state |Ψ0〉 for a given Hamiltonian H (expressed as an MPO)

19

3. METHODS

ℳ𝜎1 𝑀𝑖𝑛.

ℳ𝜎2 …ℳ𝜎𝐿−1 ℳ𝜎𝐿 𝐿𝑎𝑛𝑐𝑧𝑜𝑠 ℳ1𝜎1ℳ𝜎2… ℳ𝜎𝐿−1 ℳ𝜎𝐿

↙ 𝑆𝑉𝐷 ↙

ℳ1𝜎1ℳ𝜎2 𝑀𝑖𝑛.

…ℳ𝜎𝐿−1 ℳ𝜎𝐿 𝐿𝑎𝑛𝑐𝑧𝑜𝑠 ℳ1𝜎1ℳ1𝜎2… ℳ𝜎𝐿−1 ℳ𝜎𝐿

↙ 𝑆𝑉𝐷 ↙

( … 𝑡𝑜 𝑡ℎ𝑒 𝑟𝑖𝑔ℎ𝑡 𝑒𝑛𝑑 𝑜𝑓 𝑡ℎ𝑒 𝑐ℎ𝑎𝑖𝑛 … )

ℳ1𝜎1ℳ1𝜎2 …ℳ1

𝜎𝐿−1ℳ𝜎𝐿 𝑀𝑖𝑛.

𝐿𝑎𝑛𝑐𝑧𝑜𝑠 ℳ1𝜎1ℳ1𝜎2…ℳ1

𝜎𝐿−1ℳ1𝜎𝐿

ℳ1𝜎1ℳ1𝜎2…ℳ1

𝜎𝐿−1ℳ2𝜎𝐿 𝐿𝑎𝑛𝑐𝑧𝑜𝑠 ℳ1𝜎1ℳ1𝜎2 …ℳ1

𝜎𝐿−1ℳ1𝜎𝐿 𝑀𝑖𝑛.

↘ 𝑆𝑉𝐷 ↘

ℳ1𝜎1ℳ1𝜎2…ℳ2

𝜎𝐿−1ℳ2𝜎𝐿 𝐿𝑎𝑛𝑐𝑧𝑜𝑠 ℳ1𝜎1ℳ1𝜎2 …ℳ1

𝜎𝐿−1

𝑀𝑖𝑛.

ℳ2𝜎𝐿

↘ 𝑆𝑉𝐷 ↘

( … 𝑡𝑜 𝑡ℎ𝑒 𝑙𝑒𝑓𝑡 𝑒𝑛𝑑 𝑜𝑓 𝑡ℎ𝑒 𝑐ℎ𝑎𝑖𝑛 … )

1)

2)

3)

4)

5)

6) Repeat points from 1) to 5) using the updated matrices.

Sw

eepin

g p

rocedu

re

Figure 3.1: Pictorial representation of the finite size DMRG algorithm: starting fromthe initial MPS, we optimize the first matrix in order to minimize the energy (point 1), wethen move to the next site and do the same thing (point 2), and so on until we reach theend of the chain (point 3). Then the direction is reversed (points 4-5) and the procedurecontinued until one reaches the other end of the chain. This sweeping procedure stopswhen convergence onto the ground state is reached. In the figure indexes 1, 2 are thereto distinguish explicitly the original matrix from the optimized one.

with open boundary conditions. In order to find the optimal approximation to it, wehave to find the optimal MPS |Ψ〉 = |Ψ0〉 of some dimension D that minimizes

E =〈Ψ| H |Ψ〉〈Ψ|Ψ〉

. (3.19)

The most efficient way of doing it is a variational search in the MPS space. Thealgorithm starts from some initial guess |Ψ〉. It minimizes the energy on the first siteby solving the standard eigenvalue problem (exploiting Lanczos algorithm1) for therepresentative matrixMσ1 , taking its initial value as a starting point. Then it movesto the second site, usually via SVD to mantain the desired normalization structure,

1The Lanzos algorithm [51] is an adaptation of the power method, particularly suited for numericalcomputation, for finding the extremal eigenvalues of a matrix A (in our case the minimal). The powermethod guarantees that, starting from a random vector |ψ〉 and applying A several times to it such that|ψ〉n+1 = A|ψ〉n, in the large n limit xn/‖xn‖ approaches the normed eigenvector corresponding tothe desired eigenvalue.

20

3.1 DMRG & MPS

and performs the same optimization as before. This is continued until the end ofthe chain is reached. Then the direction of the procedure is reversed. Dimensions ofthe resulting matrices are kept limited according to the finite computational resourcesand to the requirements on the precision, and optimal truncations are performed. Thissweep through the system is continued until the quantity 〈Ψ| H2 |Ψ〉 − 〈Ψ| H |Ψ〉2is reasonably close to 0, then we can take |Ψ〉 as our desired ground state |Ψ0〉. Apictorial representation of the procedure can be found in Fig. 3.1. To speed up theprocedure and to avoid the possibility of getting stuck in a non-global minimum, it isimportant to choose as a starting point a state which is not totally random, but prettyclose to the correct ground state.

DMRG offers also the possibility to compute physical quantities and ground stateproperties in the thermodynamic limit. The algorithm used in this work is based oniTEBD (infinite time-evolving block decimation) [52] and it relies as well on theMPS formalism. We will speak about infinite size MPS in the following. The algo-rithm starts from the identification of the unit cell of the lattice and the preparation ofa random initial state for this unit cell. At each step of the algorithm the system’s sizeis increased by introducing two additional sites. The procedure relies on a transla-tionally invariant variational approximation to the ground state wavefunction and it isbased on imaginary time evolution (see the two following sections). At each step wediscard the smallest singular values and their associated singular vectors once matrixdimensions exceed D and build updated optimal bases. This procedure continuesuntil the system is large enough to guarantee that the ground state is substantiallyinsensitive to the addition of new sites.

3.1.3 Time evolutionThe time evolution [48–50] in the framework of the MPS reformulation of DMRG[21] is implemented such that an effective reduced Hilbert space is chosen at eachtime step to describe the relevant physics.

One of the most famous and used implementation relies on the Trotter decompo-sition of the time evolution operators. Let’s start by assuming that the Hamiltonianswe have to deal with all consist of nearest-neighbor interactions (which is what wehave for the models studied in this work). This means that H can be written asH =

∑j hj , where hj contains the interaction term between site j and site j + 1

(bond j). In order to perform and monitor the evolution we need also to discretizetime: t = Nδt. Let’s consider a single time step δt and simply start with a first-orderTrotter decomposition

eiHδt = eiHevenδteiHoddδt +O(δt2). (3.20)

Heven/odd contain all terms acting on even/odd bonds. The error is given by the factthat bond Hamiltonians do not commute with each other in general. On the otherhand all time evolutions on odd and even bonds, respectively, do commute, and we

21

3. METHODS

can therefore group them together as in Eq. (3.20). Higher-order decompositionexist, for which the error per timestep is reduced. In this work we will use a fourth-order Trotter decomposition:

eiHδt =

5∏j=1

eiHoddδtj/2eiHevenδtjeiHoddδtj/2 +O(δt5), (3.21)

where δt1 = δt2 = δt4 = δt5 = 14− 3√4

δt and δt3 = δt − 4δt1.The time evolution is performed on a given state by applying for each time step δt

the MPOs of the bonds following the order associated to the chosen decomposition,and by truncating the resulting MPS according to the availability of computationalresources. This error related to the compression adds to the one due to the choiceof the time step. After each step we may evaluate a generic observable P in timein the standard way, 〈Ψ(t)| P |Ψ(t)〉, or we can even more interestingly computedynamical correlations:⟨

P (t)Q⟩

= 〈Ψ| eiHtP e−iHtQ |Ψ〉 . (3.22)

More details about the computation of correlations will be given in section 3.1.5.

3.1.4 Finite temperatureIn order to use the MPS representation also for finite temperature calculations weneed to introduce the density matrix of the physical state: ρβ = e−βH/Zβ , whereZβ = Tr(e−βH) and the inverse temperature β = 1/(kBT ). This density matrix canbe obtained in the form of an MPS purification which is a pure state in an enlargedHilbert space [21, 24, 45–47]:

ρβ −→|ρβ〉 =∑σi,σ′i

Mσ1σ′1

1 . . .MσLσ′L

L |σ1 . . . σL〉 ⊗ |σ′1 . . . σ′L〉 (3.23)

|ρβ〉 ∈ H ⊗Haux ,

where |σi〉 and |σ′i〉 represent orthonormal site basis states respectively forH and forthe auxiliary Hilbert spaceHaux. |ρβ〉 is constructed such that

Traux |ρβ〉 〈ρβ | = ρβ . (3.24)

We choose the auxiliary space such thatHaux = H and Traux denotes the trace overthis space. An MPS representation of |ρβ〉 is obtained by applying an imaginarytime evolution to the state maximizing the entanglement between the local physicaland the local auxiliary sites, |ρ0〉, and following the same procedure discussed in theprevious section:

|ρβ〉 = e−βH/2 |ρ0〉 , (3.25)

22

3.1 DMRG & MPS

where

|ρ0〉 ∝L⊗i=1

(∑σi

|σi〉 ⊗ |σi〉aux

), (3.26)

with σ denoting the state non equal to σ. This maximally entangled state correspondsto the physical infinite temperature state ρ0 ∝ I, i.e. if one traces out the auxiliarydegrees of freedom one obtains the identiy. Further, in each term the state |σi〉aux ischosen such that the total magnetization is conserved in the following calculations,which enlightens considerably the numerical effort. Finally, in order to avoid anoverflow error, the state |ρβ〉 needs to be normalized at each step of the imaginarytime evolution. The expectation value of an MPO O acting on the physical system isdirectly related to its thermodynamic average by the relations

⟨O⟩T

=Tr[Oe−βH

]Tr[e−βH

] = 〈ρβ | O |ρβ〉 (3.27)

3.1.5 Dynamical correlations at finite temperatureResults of sections 3.1.3 and 3.1.4 can be joined together in order to obtain dynamicalcorrelations at finite temperature. Let’s consider two spin operators such as Sλj andSµl , where (Sλ)† = Sµ and j, l are the sites of application of the operators. Theobject we want to compute is then⟨

Sλj (t)Sµl (0)⟩T

= Sλµ(x, t) = Tr(ρβS

λj (t)Sµl (0)

), (3.28)

where x = j − l and j, l are site indexes. Using the cyclic property of the trace, andexpliciting the time dependence of the operator Sλ and of the density matrix, onecan rewrite Eq. (3.28) as follows:⟨

Sλj (t)Sµl (0)⟩T

=1

ZβTr([e−βH/2

]Sλj

[e−iHtSµl e

−βH/2eiHt]). (3.29)

The square brackets indicate which parts of this expression are approximated as anMPS [24]. The bracketing is not unique and several different approaches exist. How-ever, we found this one tested in Ref. [23] to be often the most efficient for the hereconsidered correlations. The approximation of the bracketed operators is calculatedusing an imaginary and real time evolution, and the application of the local opera-tors. The adopted scheme for computation is sketched and described in Fig. 3.4. Ineach step of the real or imaginary time evolution, the evolved state is approximatedby an MPS with bond dimensions as small as possible for a given constraint on theprecision. Typical values for maximum bond dimensions used here are up to 1000.Depending on the magnitude of the singular values after each decomposition, wekeep those which are bigger than a minimal truncation ε. This has been chosen of

23

3. METHODS

Type equation here.

Id

𝑒−𝛽𝐻 /2 𝑆𝑙𝜇𝑒−𝛽𝐻 /2

𝑒−𝑖𝐻 𝑡𝑆𝑙𝜇𝑒−𝛽𝐻 /2𝑒𝑖𝐻 𝑡

Tr 𝑒−𝛽𝐻 /2 𝑆𝑗𝜇𝑒−𝑖𝐻 𝑡𝑆𝑙

𝜇𝑒−𝛽𝐻 /2𝑒𝑖𝐻 𝑡

i-tMPS

tMPS

a) b)

c)

d)

Figure 3.2: Schematical representation of the algorithm used for the computation ofdynamical correlations at finite temperature. The initial state at finite temperature isprepared via imaginary time evolution a). A copy is created. The operator Sµl is appliedon this copy b) and then a (double) real time evolution c) is performed. At each real timestep the second operator is measured by sandwiching it through the two resulting states(d). The final result gives us the desired correlation. In the picture i-tMPS indicates theimaginary time evolution, tMPS the real time evolution. After Fig. 2 in Ref. [24].

the order of 10−20 for imaginary time evolution and 10−10 for real time evolution. Inthe real time evolution we also monitor the total amount of information discarded af-ter each truncation, which we call maximal truncated weight. The real time evolutionis automatically stopped when this value goes beyond 10−6 (10−5 for simulations athigh temperature).

Finite temperature correlations in space and time are relevant for us because oftheir connection, as we will see in chapter 4, to the NMR relaxation rate 1/T1, but inthis work we also study dynamical correlations in momentum and frequency space.Spectral functions of the form

Sλµ(q, ω) =

∫ +∞

−∞dx

∫ +∞

−∞dt ei(ωt−qx)Sλµ(x, t) (3.30)

can be directly related to the inelastic neutron scattering (INS) cross-section (seechapter 5) and more in general they provide precious information about the spectrumof the system. Since our simulations are performed on a one-dimensional latticeand the time evolution is performed within discrete time steps, our double Fouriertransform needs to be discretized. In this work we will always make use of thefollowing formula for the transformation, which is inspired by Eq. (C8) in Ref. [10]:

Sλµ(q, ω) ≈ δtNt∑

n=−Nt+1

L−12∑

x=−L−12

ei(ωmtn−qkx)Sλµ(x, tn). (3.31)

24

3.2 Tomonaga-Luttinger Liquid (TLL)

Here Nt is the total number of time steps of amplitude δt made in the simulation,ωm = πm

Ntδt, for m ranging from−Nt+ 1 to Nt, qk = 2πk

L with k = 0, . . . , L−1; Lis the size of the system and x has now become a discrete quantity expressed in unitsof the lattice spacing. This definition in (3.31) implies that the operator Sµ is appliedat t = 0 on the central site of the chain, which has in this case an odd number ofsites. Moreover, to perform the sum in (3.31) we need also results at negative times.For all the models we consider here, simple considerations on the symmetries of thelattice and on the properties of the spin operators allow to conclude that

Sλµ(x,−t) =[Sλµ(x, t)

]∗. (3.32)

We therefore get results at negative times for free by computing those at positivetimes.

The finite calculation time tmax = Ntδt and the finiteness of the lattice induceartificial oscillations in our results in momentum and frequency space. To eliminatethese oscillations we apply a gaussian filter to the space-time correlations beforeFourier-transforming, again following what is suggested in Ref. [10] and referencesthere in.

Sλµ(x, tn) −→ f(x, tn) · Sλµ(x, tn), (3.33)

wheref(x, tn) = e−(4x/L)2

e−(2tn/tmax)2

. (3.34)

This filter is everywhere non-negative and brings smoothly to zero the correlations atlarge distance and time. The application of this filter consists in other words in con-volving momentum-frequency correlations with a gaussian function, which guaran-tees the non-negativity of the results. This procedure reduces further the momentum-frequency resolution, but minimizes numerical artifacts.

3.2 Tomonaga-Luttinger Liquid (TLL)The Tomonaga-Luttinger Liquid (TLL) theory, proposed first by Tomonaga in 1950,reformulated by Luttinger in 1963 and refined by Mattis and Lieb in 1965, is thekeystone of the analytical description of many gapless one-dimensional systems. Inthese systems the role of interactions is crucial and the excitations are generally col-lective [7]. This is in contrast to what one has in higher dimensions where elementaryexcitations are quasiparticles and many fermionic systems belong to the universalityclass of the so called Fermi liquids [53].

Within TLL theory collective excitations can be seen as free bosonic excitationswith linear spectrum. In this context the physics of the system, and in particular thedynamics of the low-energy excitations, can be described by the Hamiltonian [7]

HTLL =1

2π

∫dx[uK (∇θ(x))

2+u

K(∇φ(x))

2], (3.35)

25

3. METHODS

where we have set ~ = 1, and θ(x) and φ(x) are bosonic fields characterized by thecanonical commutation relations

[φ(x),∇θ(x′)] = iπδ(x− x′). (3.36)

Here x is a continuous variable indicating the position. Local operators of the un-derlying model can be written in terms of these two fields (bosonization). The twoparameters u and K appearing in Hamilltonian (3.35) are non-universal and dependon the underlying model. u is the velocity of propagation of the bosonic excitations,i.e. the sound velocity. K is dimensionless and it usually called Luttinger parame-ter. Once these two parameters are determined, all the time-space correlations canbe determined asymptotically by the field theory associated to Hamiltonian (3.35) upto non-universal amplitudes.

In this section we briefly discuss the TLL predictions for the low energy physicsof spin-1/2 chains, and we explain also our procedure for the computation of the twoparameters of the theory.

3.2.1 Bosonization of the spin-1/2 chainThe gapless regime of the spin-1/2 XXZ model, Eq. (2.1), is a paradigmatic examplefor the application of the TLL theory [7]. The Hamiltonian for m = 0, being m themagnetization per site, can be rewritten in terms of the bosonic fields as

H = HTLL −Jz

2(πα)2

∫dx cos(4φ(x)), (3.37)

Here HTLL is the quadratic TLL Hamiltonian (3.35) with

uK = vF = J sin(kF ), (3.38)

u

K= vF

[1 +

2J∆

πvF(1− cos(2kF ))

]. (3.39)

Where J and ∆ refer to Hamiltonian (2.1) and we have set the lattice spacing to 1.Local spin operators can be expressed via the bosonic fields φ and θ according to therelations

Sz(x) = − 1

π∇φ(x) +

(−1)x

παcos(2φ(x)), (3.40)

S+(x) =e−iθ(x)

√2πα

[(−1)x + cos(2φ(x))] . (3.41)

where α is a small cutoff of the order of the lattice spacing. A finite magnetizationm can be absorbed by a simple shift of the field φ of the form φ→ φ− πmx. Thesedefinitions can be used for the computation of the two-point time-ordered spin-spin

26

3.2 Tomonaga-Luttinger Liquid (TLL)

correlation functions for the bosonized XXZ Hamiltonian in the massless regime andin imaginary time [7]:

〈Sz(r)Sz(0, 0)〉 = m2 +K

2π2

y2α − x2

(x2 + y2α)2

+ C2 cos(π(1 + 2m)x)

(1

r

)2K

,

(3.42)⟨S+(r)S−(0, 0)

⟩= C3 cos(2πmx)

(1

r

)2K+ 12K

+ C4 cos(πx)

(1

r

) 12K

. (3.43)

Here r = (x, uτ + α Sign(τ)) = (x, yα), being τ the imaginary time, while C2, C3

and C4 are non-universal constants. As it can be seen from Eqs. (3.42) - (3.43), Szz

correlations develop low-energy modes at q ∼ 0 and q ∼ π(1± 2m), while for S+−

they are located at q ∼ ±2πm and q ∼ π.

3.2.2 Evaluation of TLL parametersEquations (3.38) and (3.39) are valid only for the XXZ model. Having a procedureable to compute the values of the TLL parameters u and K for various other sys-tems is of crucial importance, since from these two values, as we saw above, onecan obtain a lot of information about the behavior of the correlations. In this workwe propose a new scheme for the numerical evaluation of these two important pa-rameters. This procedure has been adopted in Ref. [29] and it can be summarized asfollows: the ratio K/u can be evaluated from the static TLL susceptibility, while theproduct u ·K is related to the variation of the ground state energy with respect to aflux threaded onto the system. Then, the two values of u and K trivially follow byrecombination of the two previous results.

The ratio K/u is determined from the static TLL susceptibility of the systemaccording to the relations [7]

κ−1 = Ld2E0

dM2and κ =

K

uπ

=⇒ K

u

π

Ld2E0

dM2

, (3.44)

where L is the size of the system, E0 is the ground state energy, M is the totalmagnetization. The second derivative has to be discretized since M in a spin-1/2system can only vary by integer steps (thus ∆M = 1), thus:

K

u(M) =

π(∆M)2

L [E0(M + ∆M) + E0(M −∆M)− 2E0(M)]

=π

L [E0(M + 1) + E0(M − 1)− 2E0(M)]. (3.45)

27

3. METHODS

E0 can be evaluated at fixed values of magnetization via standard finite-size DMRG.The magnetization, defined at the beginning of the simulation by the initial distri-bution of spins, is thus set as a conserved quantum number. At this point it is alsopossible to determine the values of the magnetic field corresponding to each possiblevalue of the magnetization of the system. These values of h will become useful forthe next step of the procedure, and we know that they can obtained from the curveE0(M) according to the relation:

h(M0) =dE

dM

∣∣∣∣M0

=E0(M + 1

2∆M)− E0(M − 12∆M)

∆M, (3.46)

where ∆M = 2 in this case.The product u ·K can be determined by studying the variation of the ground state

energy of the system in response to a variation of a flux through the system [7,54–56].To be more precise, for a fixed value M of the magnetization (and therefore for thecorresponding value of h previously obtained),

uK(M) = πLd2E0(Φ,M)

dΦ2

∣∣∣∣Φ=0

, (3.47)

The flux is represented by twisted periodic boundary conditions, Ψ(L) = Ψ(0) ·eiΦ.This condition can be transferred into the Hamiltonian via the following transforma-tion:

S+j S−j+1 −→ S+

j S−j+1 · e

iΦL ,

S−j S+j+1 −→ S−j S

+j+1 · e

−iΦL , (3.48)

which distributes homogeneously the total flux Φ along the chain. For each fixedvalue of M we evaluate E0(Φ)|M via an infinite size MPS algorithm for symmetricvalues of Φ around zero at finite magnetic field. The resulting ground state energyE0(Φ) close to Φ = 0 can be approximated by a parabola and we fit the pointswith a second degree polynomial of the form P (Φ) = aMΦ2 + bMΦ + cM , whereaM , bM and cM are the fit parameters. According to Eq. (3.47), the fit parameter aMis related to the product

uK(M) = 2πLaM . (3.49)

In Fig.3.3 a curve E0(Φ) for the dimerized model at h = 4.209Jw is reported asan example with the corresponding fit using a 2nd degree polynomial (bM ≈ 0and therefore not reported). This procedure is adopted in chapters 4 and 5. In thefollowing we show as an example our results for u(m) and K(m) (being m themagnetization per site) for the dimerized chain compound presented in chapter 2.The curve for the parameter u tends correctly to 0 for m → 0 and m → 0.5. Thecurve for K which should tend to 1 for those values of m shows on the contrary adownturn in both limits. We attribute these errors to extremely difficult derivation

28

3.2 Tomonaga-Luttinger Liquid (TLL)

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8

−1.5813125

−1.581312

−1.5813115

−1.581311

−1.5813105

−1.58131

−1.5813095

Flux Φ

Energy

per

bon

d[unitsofJw]

h = 4.209 · Jw

Infinite size MPS

Fit: E(Φ) = 5.781·10−6· Φ

2 - 1.5813127

Figure 3.3: Numerical result using infinite size MPS for the ground state energy as afunction of the flux Φ for the dimerized model at h = 4.209Jw (blue stars). The pointsare fitted using a 2nd degree polynomial (red curve).

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

Magnetization per site m

TLL

par

amet

er ’

u’ [

K]

0 0.1 0.2 0.3 0.4 0.50.7

0.75

0.8

0.85

0.9

0.95

1

Magnetization per site m

TLL

par

amet

er ’

K’

(uni

tless

)

Original dataInterpolated data

Figure 3.4: Numerical results for the TLL parameters u (left) and K (right), as a func-tion of the magnetization per site m, for the dimerized chain copper nitrate compound.Results for the parameter K have been interpolated close to m = 0 and m = 0.5 usingthe constraints K(m = 0) = K(m = 0.5) = 1 to reduce the numerical error. In thoseregions K is difficult to obtain by recombining the product uK and the ratio K/u, sinceuK → 0 while K/u→∞.

29

3. METHODS

of K in those regions. Since K =√uK · Ku , and since the two quantities under

the square root sign tend respectively to zero and to infinity for m going to 0 or 0.5,any small numerical error can heavily influence the result. We decided thereforeto discard the last points at the two ends of the curve, and to interpolate for thecorresponding values of m adding the two additional constraints K(0) = K(0.5) =1.

30

CHAPTER 4

NMR Relaxation Rate

In this chapter we focus on our DMRG results for the so called spin-lattice relaxationrate 1/T1. Its inverse, the spin-lattice relaxation time T1, is one of the importanttime-scales of NMR measurements. After a brief introduction about this quantityand its physical meaning, we will discuss how to connect it with what we computenumerically. Then, we also presents some available analytical expressions which wewill use to benchmark our data. Finally, we present and comment our results for thetwo models considered (XXZ and dimerized spin-1/2 chains).

The content of this chapter is based on Ref. [29]. Figures and parts of the text aretaken from there. Recently, a similar analysis was carried out for the DTN compoundby Capponi, Dupont and Laflorencie [57].

4.1 Spin-lattice relaxation rate 1/T1

In NMR experiments the nuclear spins of the sample, previously polarized by an ap-plied magnetic field, are perturbed using an electromagnetic pulse. The term relax-ation describes how the signal re-emitted by the sample deteriorates with time. Thisdeterioration is usually studied by considering two processes, each of them charac-terized by its own specific time constant. During the first process, called spin-latticerelaxation, responsible of the loss of signal’s intensity and characterized by the timeconstant T1, the component of the magnetization along the direction of the appliedmagnetic field (denoted here by z) reaches thermodynamic equilibrium with its sur-roundings (the lattice) after the perturbation [16, 17]. The evolution of the nuclear

31

4. NMR RELAXATION RATE

magnetization along z is:

Mz(t) = Mz,eq

(1− e−t/T1

). (4.1)

The second process, called spin-spin relaxation and characterized by another timeconstant T2, is related to the broadening of the NMR signal and to the decay of themagnetization component perpendicular to the applied field.

In a solid the ratio 1/T1 (spin-lattice relaxation rate) can be related directly tothe spin-spin correlations of the electronic system via the Redfield equations [17]:

1

T1=γ2n

2

[A2⊥ (Sxx(ω0) + Syy(ω0)) +A2

‖Szz(ω0)

]= (4.2)

=γ2n

2

[A2⊥

2

(S+−(ω0) + S−+(ω0)

)+A2

‖Szz(ω0)

]. (4.3)

where γn is the nuclear gyromagnetic ratio of the measured nuclear spin, A⊥ andA‖ are the longitudinal and transverse components of the hyperfine tensor, Sαα(ω0)with α = x, y, z are the local spin-spin correlation functions at the nuclear Larmorfrequency ω0, which can be obtained via a Fourier transform of the onsite correla-tions in real time:

Sαα(ω0) =

∫ +∞

−∞dt eiω0t

⟨Sαj (t)Sαj (0)

⟩T. (4.4)

Also,

Sxx(ω0) + Syy(ω0) =1

2

[S+−(ω0) + S−+(ω0)

](4.5)

In Eq. (4.4), the expectation value 〈. . .〉T denotes the thermal and quantum averagedefined as

〈· · · 〉T =Tr[e−βH · · · ]

Tr[e−βH ]. (4.6)

Note that in formula (4.2) it is implicitly assumed that the hyperfine coupling term isessentially q independent. This covers a large number of cases, for example the onesin which the relaxation is measured on the site carrying the electronic spin. Thereare also interesting cases for which the q dependence of the hyperfine term can filtersome modes, for example the modes at q = π if the relaxation is measured mid-point between two neighboring sites. This leads to different formulas and interestingproperties [58–60]. Note that techniques similar to the ones used here but computingthe finite temperature, space and time dependent spin correlations allow to treat thisproblem as well. We leave this more complicated case for further studies, and focushere to the generic case for which the local spin-spin correlation is sufficient.

Although the principle of what is measured by the ratio 1/T1 is simple, the theo-retical determination is far from trivial. Very often various schemes of approximation

32

4.2 What we compute, why and how

of the exact formula are used. The first approximation consists in assuming that theNMR frequency in Eqs. (4.2)-(4.3) is low enough (usually in the hundred of MHzrange) compared to the temperature and that it can safely be set to zero [16,17]. Thesecond approximation is usually to reduce the local correlation function, which is asum over all momenta, to a sum taken around special momentum values (e.g. theq ∼ 0 values and values around either the antiferromagnetic wavevector q ∼ π/a,where a is the lattice spacing, or a corresponding incommensurate one when themagnetization is finite). This last approximation is reasonable when the character-istic scale of excitations (typically the temperature) is low enough compared to themagnetic exchange, so that the excitations around these wavevectors can be well sep-arated. Finally, to compute the correlations, some continuous approximations suchas bosonization, exploiting the above points, are usually employed. This set of ap-proximations has allowed a connection between NMR measurements and theoreticalpredictions for quantum chains and ladder systems [10].

We have already mentioned that recent developments in chemistry have provideda successful set of magnetic systems, which can be driven from zero magnetizationto full saturation by the application of experimentally reachable magnetic fields. Inthis materials the exchange energy scale (. 10K) is now much closer to the typi-cal measurement temperatures [8, 9]. This invalidates partly, or pushes to very lowtemperatures the approximations discussed above. A direct method to compute theNMR relaxation time without having to resort to these approximations is thereforeneeded. We will see in this chapter that numerical techniques such as time dependentMPS (tMPS) methods at finite temperature, are suitable candidates for this task.

4.2 What we compute, why and how

In the previous section we learned that the ratio 1/T1 can be obtained once one has atool compute onsite spin-spin correlations at finite frequency and temperature. Nu-merically we can access onsite correlations at finite temperature in real time. Let usnow then reconsider the Fourier transform in Eq. (4.4). The time integral over infinitetime is only valid theoretically, since neither in the experiment nor in the simulationone could expect doing the sum over an infinite interval of time. In practice, twotime scales compete. One is the typical time t ∼ 1/ω0 above which one can expectthe oscillations coming from the frequency ω0 to become strong and regularize theintegral. The second time scale hidden in the correlation itself is the decay of thecorrelation linked to the finite temperature. Since typical NMR frequencies are ofthe order ω0 ' 20 MHz while the typical lowest temperatures at which such exper-iments are done are of the order of 40 mK ' 790 MHz, for all practical purposeswe can expect that the decay due to the temperature regularizes the integral. Wewill thus in the following give the expression by taking the usual limit ω0 → 0 andkeeping in Eq. (4.4) a finite integration domain up to a maximum time t0.

33

4. NMR RELAXATION RATE

Using the approximations discussed above, one obtains that

Sλµ(ω0 → 0) '∫ +t0

−t0dt⟨Sλj (t)Sµj (0)

⟩T

= 2

∫ +t0

0

dt Re⟨Sλj (t)Sµj (0)

⟩T.

(4.7)

where we have exploited the fact that Sλµ(−t) = (Sλµ(t))∗, and (λ, µ) can be(±,∓) or (z, z). Since we have set ω0 = 0 inside the integral in the above ex-pression, the two time integrals of +− and −+ correlations also become identical(additional details can be found in Appendix A). Note of course that this statementis not true for the correlations themselves at finite time. We can thus compute theone that is the most convenient numerically depending on the specific case.

Since in this work we focus on the parameter dependence of generic Hamiltoni-ans we will omit the factors γ2

nA2⊥ and γ2

nA2‖, which depend on the specific material.

For a specific material they have to be considered and in general both terms mightbe important. However, in this work we are not focusing on a particular compoundand we have chosen to consider for each example only one of the terms separately.We thus compute numerically(

1

T1

)±∓

= 2

∫ +t0

0

dt Re⟨S±j (t)S∓j (0)

⟩T, (4.8)(

1

T1

)zz

= 2

∫ +t0

0

dt Re[⟨Szj (t)Szj (0)−m2

⟩T

]. (4.9)

Note that with the definitions in Eqs. (4.8)-(4.9) the units of 1/T1 become time andnot one over time as for the original definition in Eq. (4.1). We evaluate the real partof the correlations inside the integrals within tMPS at discrete time steps of amplitudeδt. The convergence with the time step of the evolution is assured. The amplitude ofthe time step is chosen to be small enough to guarantee good approximation of theintegral evaluated via trapezoidal integration:

2

∫ +tmax

0

dt Re⟨Sλj (t)Sµj (0)

⟩T≈

≈N∑l=1

δt[Re⟨Sλj ((l − 1)δt)Sµj (0)

⟩T

+ Re⟨Sλj ((lδt)Sµj (0)

⟩T

], (4.10)

whereN is the total number of time steps at which the correlations are evaluated andδt as previously said is the amplitude of a single time step. j = L/2, being L the sizeof the chain, for all cases investigated. Depending on the available computationalresources and on the constraints on the desired precision, runs are stopped after acertain tmax = Nδt. In many cases this tmax is large enough such that correlations

34

4.3 Results for 1/T1 as a function of the temperature

are practically zero for larger times. In other cases it is not possible due to thenumerical complexity to reach such a large tmax. In order to have an idea of the valueof the extended integral, we extrapolate its value for tmax → +∞ and associateto it an error bar. In Appendix B the details of the extrapolation method and thedetermination of the error bars are given.

It is also important to remind that the NMR relaxation rate 1/T1 in one dimensioncan be computed in the low energy TLL representation [7]. This calculation is validwhen the temperature is low enough compared to the typical spin energy scales. Inthat case, if one is not at the isotropic point (K > 1/2), neglecting the subdominanttemperature corrections and the zz contribution in Eq. (4.2) (small compared to the+− term), one finds [10]

1

T1' lim

ω0→0− 2

βω0ImχR+−(x=0, ω0) '

'4Ax cos

(π

4K

)u

(2πkBT

u

) 12K−1

B

(1

4K, 1− 1

2K

), (4.11)

where u and K are the TLL parameters associated to the model, and Ax is the am-plitude coefficient relating the microscopic spin operator Si on the lattice with theoperators in the continuous field theory. These coefficients have been computedboth analytically and numerically in various contexts ranging from chains to lad-ders [10, 32, 61–63]. χR+−(x = 0, ω0) is the retarded, onsite S+− correlation func-tion at the frequency ω0 (for the q resolved susceptibility see Refs. [7, 64, 65]). Notealso that in this formula ~ and the lattice spacing has been set to one, thus omitted.More details about the calculation can be found in Appendix A.

Eq. (4.11) has provided a quantitative estimation of the NMR in ladder systemsfor which the relaxation time could be measured [41,66]. It will thus provide us botha benchmark for the numerical evaluation of the relaxation time at low temperaturein the gapless phase, as well as an estimation of the deviation from these ideal lowenergy properties.

4.3 Results for 1/T1 as a function of the temperatureIn the following subsections we present our numerical results for the Heisenbergmodel, the XXZ model and the dimerized model. To test the accuracy of our numer-ical results, we performed some calculations for the XX model and compared themwith exact analytical results. More details can be found in Appendix C.

The numerical results shown in this section are obtained with the followingchoices of parameters:

• For the Heisenberg model and the XXZ model we chose L = 100 and cor-relations are evaluated on the site j = 50. For the imaginary time evolution

35

4. NMR RELAXATION RATE

we adopted a minimal truncation of εβ = 10−20, 400 as maximum numberof retained states, and steps of amplitude δβ = 0.01 J−1. For the real timeevolution we adopted a minimal truncation of εt = 10−10, 800 as maximumnumber of retained states, a maximum truncated weight of 10−6 and real timesteps of amplitude δt = 0.05~/J . In all cases we could reach tmax = 30~/J .

• For the dimerized model we considered a chain of L = 130 spins and cor-relations are evaluated on the site j = 65. For the imaginary time evolutionwe chose a minimal truncation of εβ = 10−20, 500 as maximum numberof retained states, and steps of amplitude δβ = 0.01474 J−1

w . For the realtime evolution we adopted a minimal truncation of εt = 10−10, 500 as max-imum number of retained states for temperatures kBT < 0.68 Jw, 800 for0.68 Jw < kBT < 1.36 Jw and 2000 for higher temperatures. The maximumtruncated weight is 10−6 in most cases, 10−5 for the highest temperatures.The amplitude of the real time steps is δt ≈ 0.0737 ~/Jw and a. The tmaxreached still decreases from 59 ~/Jw for the lowest temperatures, to 15 ~/Jwfor the highest ones, according to the requested precision.

The system sizes were chosen such that the perturbations do not yet reach the bound-ary of the system for times up to tmax. The resulting finite system size effects aresmall compared to the uncertainties introduced by the finite cut off of the time inte-gral and are therefore neglected.

4.3.1 Heisenberg model

Let us start by considering the Heisenberg model, i.e. the XXZ model in Eq. (2.1)with isotropic coupling ∆ = 1, of spins 1/2 under a magnetic field, applied alongthe z direction. A pictorial representation of the phase diagram as a function ofh is easily derived from Fig. 2.1 and given in Fig. 4.1. At low magnetic field theground state of the system is a gapless TLL, whereas above the critical magneticfield (hc = 2J) a gapped ferromagnetic phase develops. In order to explore thedifferent phases and the quantum critical point we focused on the fields h = 0 andh = J in the gapless phase, h = hc = 2J at the quantum critical point and h = 5Jin the gapped phase. In Fig. (4.2) the results for the 1/T1 relaxation rate of the S+−

correlations are shown at the magnetic field values h = 0, h = J in the gaplessphase, and h = 2J at the quantum critical point.

The behavior of the relaxation time at zero magnetic field has been investigatedpreviously both by analytical [67, 68] and numerical methods such as the quantumMonte-Carlo using the maximum entropy method in order to continue to the realtime axis [69, 70] and tDMRG methods [71]. In the asymptotic low-temperaturelimit one obtains [64, 65] 1/T1 = const if logarithmic corrections ln1/2(1/(kBT ))are neglected. Since the behavior at h = 0 has been studied in detail in Refs. [69,71],we here show the h = 0 case for (1/T1)+− mainly for comparison. At temperatures

36

4.3 Results for 1/T1 as a function of the temperature

Figure 4.1: Phase diagram for the Heisenberg model (∆ = 1) as a function of themagnetic field h, as derived from Fig. 2.1. For low magnetic field, the ground state isa TLL and the phase is gapless. In contrast, above the critical magnetic field hc = 2Ja ferromagnetic phase, which exhibits a gap in its excitation spectrum, arises. Picturetaken from Ref. [29].

above kBT/J & 0.5 an almost linear increase of the relaxation time with increasingtemperature can be seen. At low temperature (1/T1)+− shows an almost constantbehavior. At temperatures below kBT/J . 0.2 it even increases again while lower-ing the temperature. This behavior is consistent with the logarithmic corrections andhas been analyzed in [69]. The rise at larger temperatures has been treated in [71]and has been found compatible with an exponential increase with a scale of the orderof the magnetic exchange J .

In the TLL region (h = J), as described in Eq. (4.11), the low temperaturebehavior of the relaxation rate should be approximately described by an algebraicdecay with the exponent 1

2K − 1, which is fully determined by the TLL parameterK [32, 41]. At larger temperature a breakdown of this low energy prediction is ex-pected. In Fig. 4.2 our numerical results for (1/T1)+− are quantitatively comparedto the TLL predictions. Up to temperatures of about kBT/J ≈ 0.2 the numericalpoints agree within the error bars with the prediction made in Eq. (4.11). This com-parison is achieved using previously extracted values for the Tomonaga-Luttingerparameter K = 0.66(1), the amplitude Ax = 0.119(1) from Refs. [62, 63], andu = 1.298(5) J/~ (lattice spacing equal to 1) extracted from separate calculationswhich we performed using standard finite-size DMRG methods following what wasdiscussed in Section 3.2.2. Thus, all parameters in Eq. (4.11) are fully determined.For temperatures larger than kBT/J ≈ 0.2 the numerical results are much higherthan the decaying TLL prediction. This is to be expected and clearer in the q spacefor which the local correlation can be seen as a sum over all q points. The TLL for-mula only contains the part coming from one of the low energy q points (q aroundπ in the absence of magnetic field). At higher temperatures other q points start tocontribute significantly to the sum. The numerical results even seem to show a slightmaximum around kBT/J ≈ 0.5 and then remain more or less constant in value upto the shown maximal temperature.

At the quantum critical point h = 2J , an algebraic divergence of the 1/T1

with the temperature is also expected in the low T limit. It is predicted to behaveas ∝ (kBT/J)−0.5 as obtained in Refs. [72–74]. This behavior has been experi-

37

4. NMR RELAXATION RATE

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5

kBT/J

J(1/T

1) +

−/h

MPS, h = hc = 2J

Fit: a ·(

kBTJ

)

−0.5

MPS, h = J

Analytics: 0.57(1) ·(

kBTJ

)

−0.24(1)

MPS, h = 0

Figure 4.2: (1/T1)+− as defined in Eq. (4.8), multiplied by J/~ to have a dimensionlessquantity, as a function of kBT/J for an XXX model under magnetic field. Dots witherrorbars are MPS results, solid lines are the analytical predictions and/or fits. a =0.71(2) (fit parameter). Picture taken from Ref. [29].

mentally observed for example in the Heisenberg chain compound copper pyrazinein Ref. [75] and discussed in Ref. [76]. In this situation the prefactor is not eas-ily extracted and therefore we fit the expected algebraic behavior J(1/T1)+−/~ =a(kBT/J)−0.5 with a free fit parameter a. We obtain very good agreement of ournumerical results with the fit using the value a = 0.71(2) in the entire regime oftemperatures up to kBT/J ≈ 2, as shown in Fig. 4.2. This means that our resultsare in agreement with the predictions of the quantum critical regime extending upto these temperatures. In addition in Fig. 4.3 we offer a comparison between ourmagnetization data computed at finite temperature, and the scaling function close to(or on) a field-induced quantum critical point, which have been calculated and usedin the literature [72,76,77]. At h = 2J = hc, from Eqs. (2) and (3) in [76], we knowthat the magnetization per site should behave as

m(T ) = mS −(

2kBT

J

)d/2M(δhc/kBT ) = (4.12)

= 0.5− 0.24312

√kBT

J(4.13)

where mS = 0.5 is the magnetization per site at saturation, d = 1 is the dimension

38

4.3 Results for 1/T1 as a function of the temperature

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

kBT/J

Magnetizationper

site

m

MPS

m(T ) = 0.5 − 0.24132√

kBTJ

Figure 4.3: Magnetization per site as a function of kBT/J for the Heisenberg modelat the critical field h = 2J . Blue stars are MPS results (error bars are smaller than thesymbol size), red circles are the analytical predictions. Picture taken from Ref. [29].

of the system, δhc = 0 is the distance from the critical field and

M(δhc/kBT ) =1

π

∫ ∞0

dx1

ex2−(δhc/kBT ) + 1. (4.14)

We observe a good agreement between numerical results and the analytical predic-tion, which is supposed to be valid in the low temperature limit. We see numericaldata approaching the analytics as the temperature is lowered.

In Fig. 4.4 the results for the 1/T1 relaxation rate for the Szz correlations at afield of h = 5J are reported. The system at this magnetic field exhibits a gappedenergy spectrum, and we denote the gap by ∆g = h − 2J = 3J . Due to thegapped energy spectrum an exponential decay with temperature is expected and in-deed observed numerically. In order to validate the exponential form for a differentparameter regime we consider also the XX model at the same magnetic field whichcan be mapped onto free fermions. The corresponding gap in the energy spectrumis given by ∆g = h − J = 4J . We computed the longitudinal part of the 1/T1 an-alytically (the density-density correlation for the corresponding free fermions), andtried successfully a fit with an exponential of −∆g/kBT . The analytical calcula-tion was performed up to a finite time tmax = 30~/J , the errorbars come from theextrapolation procedure for tmax →∞.

39

4. NMR RELAXATION RATE

0 1 2 3 4 5−20

−15

−10

−5

0

J/kBT

ln(J(1/T

1) z

z/h)

MPS, XXX, h = 5J

Fit: a−∆g

kBT, ∆g = 3J

Analytics, XX, h = 5J

Fit: c −∆g

kBT, ∆g = 4J

Figure 4.4: Logarithm of J(1/T1)zz/~ (unitless quantity) from Eq. (4.9) as a functionof J/kBT . We considered the Heisenberg and the XX model, with h = 5J . Blue dotswith errorbars are MPS results, red dots with errorbars (due to extrapolation of the resultat infinite tmax) are analytical results. Solid lines are the fits according to the expectedbehavior. a = 0.5(1) and b = 1.0(1) are the fit parameters. Picture taken from Ref. [29].

4.3.2 XXZ model

In order to explore more in detail the behavior of the relaxation time in the TLLphase, we move to the spin-1/2 XXZ model, Eq. (2.1), in absence of magnetic field(h = 0). For anisotropies 0 ≤ ∆ < 1 the ground state of this model is a TLLphase. As we discussed for the Heisenberg model at h = J , we expect that atlow temperature the relaxation rate corresponding to the S+− correlations shows analgebraic divergence as given in Eq. (4.11). We consider the cases ∆ = 0, 0.5, 0.7as shown in Fig. 4.5. The corresponding values of the Luttinger liquid parametersK and u are calculated using the Bethe ansatz formulas given e.g. in Ref. [7], whilethe amplitudes Ax are taken from Ref. [62]. Their rounded values are summarizedin table 4.1.

The agreement of the TLL prediction of the algebraic divergences and our numer-ical results is extremely good at low temperatures. For larger values of the anisotropythe divergence becomes weaker until for ∆ = 1 one leaves the TLL region and noalgebraic divergence is seen. As expected the predictions for the Luttinger liquidbehavior disagree above a certain temperature of the order of kBT/J ≈ 0.2. Abovethis scale our numerical results show an upturn and the different curves even cross.Our results clearly show the importance for systems with small exchange constantsto be able to go beyond the asymptotic expressions in order to make comparison with

40

4.3 Results for 1/T1 as a function of the temperature

∆ K u Ax0 1 1 0.14710.5 0.75 1.299 0.1340.7 0.6695 1.4103 0.1297

Table 4.1: Values for the three relevant parameters u, K, and Ax for different values ofthe anisotropy ∆ in the XXZ model. u has the units of J/~ (lattice spacing equal to 1here). Table taken from Ref. [29].

0 0.5 1 1.5 20

1

2

3

4

kBT/J

J(1/T1) +

−/h

MPS, ∆ = 0

Analytics: 0.8704 · (kBT/J)−0.5

MPS, ∆ = 0.5

Analytics: 0.6368 · (kBT/J)−1/3

MPS, ∆ = 0.7

Analytics: 0.581 · (kBT/J)−0.2532

MPS, ∆=1

Figure 4.5: (1/T1)+− as defined in Eq. (4.8), multiplied by J/~ to have a dimensionlessquantity, as a function of kBT/J for the XXZ model at different anisotropies. Dots witherror bars are MPS results, solid lines are analytical predictions. Picture taken fromRef. [29].

the experiments.

4.3.3 Dimerized modelAs a final example, we consider the isotropically dimerized spin-1/2 chain in pres-ence of a magnetic field along the z direction as defined in Eq. (2.2). This modelcan describe very well some interesting compounds like for example the copper ni-trate [Cu(NO3)2 · 2.5D2O] studied in Refs. [36, 38]. It is useful to remind that forthis compound the coupling parameters are determined as J/kB ≈ 3.377 K andδJ/kB ≈ 1.903 K and we will focus on these strongly dimerized parameters in thefollowing. Also we remind that the ground state of the system at h = 0 has zeromagnetization. A gap of ∆g ∼ 4.4 kBK separates the ground state from the first

41

4. NMR RELAXATION RATE

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

kBT/Jw

Jw(1/T

1) −

+/h

MPS

Fit: a · e−

∆gkBT

0.5 1 1.5 2−4

−2

0

Jw/kBTln(J

w(1/T1) −

+/h)

Figure 4.6: (1/T1)−+ as defined in Eq. (4.8) multiplied by Jw/~ to obtain a dimension-less quantity, plotted as a function of kBT/Jw for the isotropically dimerized spin-1/2chain at h = 0. Dots with error bars are MPS results, solid lines are combination of a fitwith the analytical prediction. The fit parameter is a ≈ 4.7(3). In the inset, logarithmicrepresentation of the same quantities when positive. Picture taken from Ref. [29].

excited state. In a magnetic field, the system shows a first quantum critical point ata magnetic field hc1. At this point the system undergoes a transition from a gappedphase to a gapless, TLL phase. Here we focus on two cases: the gapped phase forh = 0 and the TLL phase at h ≈ 1.01 ·∆g & hc1.

We calculate the relaxation time for the onsite correlation S−+ (at h = 0,〈S−+〉 = 〈S+−〉 = 2〈Szz〉). Due to the presence of a gap ∆g in the absence ofa magnetic field, the temperature dependence of the relaxation rate at low temper-atures is expected to be exponentially activated. i.e. ∝ e−∆g/kBT . In Fig. 4.6 weshow that our results agree very well with this exponential activation.

At larger temperatures kBT/Jw > 1 a saturation effect seems to set in. Inthe inset, lower temperature points have been cut because of the difficulties in theextrapolation which led to negative (though pretty close to 0) extrapolated values, asshown in the main panel of Fig. 4.6.

In contrast for the case h ∼ 3.02Jw the low energy physics can be described bythe TLL theory. Thus, the expected behavior of the relaxation rate as a function ofthe temperature is an algebraic divergence at low T of the form ∝ (kBT/Jw)

12K−1.

The TLL parameter K which enters in this formula has been determined by sepa-rate calculations of the compressibility using MPS and the flux dependence of theenergy using infinite-size MPS calculations as explained in section 3.2.2, givingK ≈ 0.81(3). The numerically obtained relaxation time is shown in Fig. 4.7 and

42

4.4 Summary & conclusions of the chapter

0 0.2 0.4 0.6 0.8 1 1.20

0.5

1

1.5

2

2.5

kBT/Jw

Jw(1/T

1) −

+/h

MPS

Fit: a · T−b

Analytics+fit: c · T−0.38(2)+ d

Analytics+fit: g · T−0.5 + l

Figure 4.7: (1/T1)−+ as defined in Eq. (4.7) multiplied by Jw/~ to obtain a dimension-less quantity, as a function of kBT/Jw, for the isotropically dimerized spin-1/2 chainat h ≈ 3.02Jw. Dots with errobars are MPS results, solid lines are fits or combinationof fit and analytics. The fit parameters are a ≈ 0.62(1), b ≈ 0.63(2), c ≈ 1.42(2),d ≈ −0.84(1), g ≈ 0.87(1), and l ≈ −0.25(1). Picture taken from Ref. [29].

compared to the TLL predictions.The black line represents the fit using the separately determined exponent 1

2K−1.A constant offset has been added since the behavior is not entirely dominated by thedivergence. Deviations between the comparison of the analytical prediction and thenumerical calculation are seen. We attribute these deviations to the proximity of thequantum critical point. In this regime, the TLL behavior is valid only for very lowtemperatures kBT ≤ h − hc1. From our numerical results only the lowest tem-perature point lies within this region. To verify the influence of the quantum criticalpoint, a fit using the critical power law shifted by a constant offset is performed. Thisfit leads for the intermediate temperature points to good results (see the green curve).A fit in which also the exponent is a fit parameter leads to an even larger exponent ofb ≈ 0.63 (red curve).

4.4 Summary & conclusions of the chapter

Exploiting MPS techinques, we computed in this chapter the spin-lattice relaxationrate 1/T1 for a wide range of temperatures, for different Hamiltonians and for dif-ferent quantum phases. In particular, we have considered the XX, Heisenberg, andXXZ Hamiltonians, plus the isotropically dimerized case. For the non-dimerized

43

4. NMR RELAXATION RATE

cases we have performed a detailed study of the gapless phase, the gapped phase andalso of the quantum critical point. Numerical results were in very good agreementwith analytical results available in the low-temperature limit. We have shown the de-viation from the low-T law at finite temperature and we have swept through quantumcritical points, situations in which theoretical results are more difficult to obtain. Ourcalculations prove that the MPS method can be successfully used to obtain the NMRrelaxation rate in regimes in which the field theoretical asymptotic results would notbe applicable. The overlap between the regimes in which the numerical methodsare applicable and the regimes covered by the field theoretical asymptotic methodsallows essentially a full description of the NMR behavior for the accessible regimeof temperatures. Having a method which can quantitatively compute the NMR re-laxation time for a given microscopic Hamiltonian, rather than simple asymptoticexpressions, should allow to test that the Hamiltonian which is supposed to describethe real system does not miss an important term, and to fix the various coefficientsby comparing the computed temperature dependence with the experimentally mea-sured one. This is similar in spirit to what was achieved by the comparison of thecomputed neutron spectra with the measured ones for DIMPY [78].

One interesting perspective is the investigation of the behavior of the relaxationmechanism of the spin excitations close to the quantum critical point. Indeed thenature of the relaxation mechanism is potentially different depending on whether oneconsiders the Szz term or the S±∓ ones. For 3D systems a self energy analysis ofthe transverse part of 1/T1 was suggesting [74] a behavior 1/T1 ∝ e−3∆g/kBT dueto the necessity of making three magnon excitations to be able to scatter a magnonand get a finite lifetime while the Szz part leads, as shown in the present paper, to1/T1 ∝ e−∆g/kBT . Our numerical results which are able to correctly determinethe exponential decay in the controlled cases of the longitudinal excitations are thuspotentially able to address this issue and potentially make contact on the experimentson that point [79]. Such a study clearly going beyond the scope of the present work,is thus left for future projects.

We have seen that the present method works efficiently if the systems are one-dimensional. An important challenge on the theoretical level is to extend the presentanalysis to the case of higher dimensional systems. A first natural extension wouldbe the application to quasi-one-dimensional systems like for example spin-1/2 two-leg ladders. In this direction some existing experiments measuring NMR in "Hpip"compounds (see chapter 2) could be of potential interest [66, 80]. More in general,although other methods such as quantum Monte-Carlo exist, the dynamical corre-lations in real time are still a challenge for which the MPS methods could bringuseful contributions. Indeed the (numerically) rather complete knowledge of theone-dimensional correlation functions allow to incorporate them into approximationschemes such as RPA to capture large part of the higher dimensional physics. An-other route is to solve clusters of one dimensional structures, which allows to at leastincorporate part of the transverse fluctuations.

44

CHAPTER 5

Dynamical Correlations at Finite Temperature of aDimerized Spin-1/2 Chain

As discussed previously, DMRG/MPS methods has allowed to obtain the frequencyand momentum resolved spin-spin correlation functions for several one-dimensionaland quasi-one-dimensional systems, see for example [10]. For integrable modelsBethe ansatz gives nowadays the possibility to access momentum and frequency re-solved correlations [18–20]. However, up to very recently, these predictions werelimited to zero temperature and only finite temperature calculations of the thermo-dynamics were accessible. In addition, the analytical treatment of temperature ef-fects beyond the bosonization limit turns out to be particularly difficult. For gappedsystems some semi-classical approximations [81] or form factor expansions [82] canbe made, but no complete treatment existed. Recently, full dynamical calculationsat finite temperatures in the framework of DMRG/MPS methods [23, 24] have beenperformed in simple cases, opening the path to the study of temperature effects onspectral functions for more complicated models.

In this chapter we use our MPS algorithm for the dynamics at finite temperatureto make an analysis of the properties of the spin-1/2 dimerized chain under magneticfield, focusing also on the thermal effects. The model we refer to and the corre-sponding real material are presented in chapter 2 (see in particular section 2.5 andFig. 2.5). This dimerized system shows in absence of magnetic field a ground statethat is a spin liquid with a finite spin gap. Application of the magnetic field closesthe gap and transforms the problem into a TLL [7] in a similar way than for laddersystems [73].

In the following we first give a brief introduction about what we exactly compute

45

5. DYNAMICAL CORRELATIONS AT FINITE TEMPERATURE OF ADIMERIZED SPIN-1/2 CHAIN

and its relation with inelastic neutron scattering measurements. We then move tothe results for dynamical correlations for different values of temperature and fieldand we give an interpretation of the different structures seen. Finally we show howtemperature affects the characteristics of the spectra. The content of this chapter isbased on Ref. [30]. Figures and parts of the text are taken from there.

5.1 What we compute: definitions

As already stated in the title of this chapter, we are interested in dynamical spin-spincorrelation functions at finite temperature for the dimerized spin-1/2 chain. Numeri-cally we compute objects of the form⟨

Sλj (tn)Sµl (0)⟩T

= Sλµ(d, tn), (5.1)

where (λ, µ) can be (±,∓) or (z, z), tn = nδt is the discretized time at which thecorrelation is evaluated (n is an integer), j, l are the site indexes and d = j − l. Thechain has a size of L = 130. Also here, the expectation value 〈. . . 〉T denotes thatthe correlation is evaluated at finite temperature T. Since the unit cell of the originallattice corresponds to d = 2 (see Fig. 2.2 for clarity), we choose to focus on objectsof the form: ⟨

Sλj′,r1(tn)Sµl′,r2(0)⟩T

= Sλµr1r2(d′, tn) , (5.2)

where j′ and l′ are now the strong bond indexes. In addition we have attachedindexes r1 and r2 indicating the positions of the spins on each strong bond: 1 forleft, 2 for right. With this notation we have for each couple (λ, µ) 4 possible speciesof correlations which we will name 11, 12, 21 and 22 following the choice for r1, r2

indexes. d′ = j′ − l′ ranges from − L−12 to L−1

2 , with L = L/2. In the (z, z) casewe substract m2 (m is the magnetization per site). In this work we consider only 11correlations (L = 130 ⇒ 65 points), since they contain already the most interestingphysics of the model and of the phases considered. 21 correlations can be obtainedin the same run.

Our data then undergo a double Fourier transform to move to momentum andfrequency space. In order to reduce artificial oscillations and other numerical arti-facts, due to finite system length and finite tmax reached by simulations, we apply agaussian filter to the correlations before Fourier transforming, see Eqs. (3.33)-(3.34).

For the double Fourier transform we use the same conventions adopted in Ref. [10]and detailed already in Eq. (3.31) which we recall here for clarity:

Sλµ11 (q, ω) ≈ δtNt∑

n=−Nt+1

L−12∑

d′=− L−12

ei(ωmtn−qkd′)Sλµ11 (d′, tn).

46

5.2 Results for dynamical correlations at finite temperature

Nt is the total number of time steps made for the real time evolution, ωm = πmNtδt

for m integer ranging from −Nt + 1 to Nt, and qk = 2πkL

for k = 0, 1, . . . , L − 1.For us L = 65. It is clear from the equation above that we also need results atnegative times, which we can easily obtain by exploiting Eq. (3.32). Finally, weextract the real part of these filtered correlations, Re

(Sλµ11 (q, ω)

), which contains

the most important information.Our results for correlations are directly connected to Inelastic Neutron Scattering

(INS) measurements as explained for instance in [13]. Zero temperature dynamicalcorrelations allowed already successful comparisons against experimental results atlow temperature [10,78]. For the dimerized system neutron scattering measurementshas been performed in Refs. [37,83,84]. To relate precisely our results to the neutronabsorption data for a given material, one needs incorporate the specific details ofthe compound (position of atoms, etc.) and to sum over polarizations (since mostexperiments are done with unpolarized neutrons). These extra elements, althoughimportant, are in general complicating the analysis. In this paper we thus directlyfocus on the spin-spin correlations themselves for each polarization to analyze theeffects of a finite temperature.

5.2 Results for dynamical correlations at finite tem-perature

We present in this section our results for dynamical spin-spin correlation functionsat finite temperature. We consider the three correlations S+−

11 , S−+11 and Szz11 . From

S±∓ correlations one can easily access Sxx and Syy . Since we are interested inthe physics of the various phases of the system (see Fig. 2.3) we consider severalmagnetic fields. In particular we examine in details, and for various temperatures:

• h = 0. This value of the field puts us in the gapped regime. For this field thesystem is isotropic.

• h = 2.868Jw < hc1, being hc1 ∼ 2.976Jw. Although we are still in thegapped regime, the gap is very small since the field is very close to the criticalone.

• h = 3.716Jw. This magnetic field puts the system well inside the masslessregion. In this regime the low energy part of the spectrum is expected to bedescribed by a TLL.

• h = 4.674Jw > hc2, being hc2 ∼ 4.566Jw. In this case we are in the fullypolarized regime and the system is also gapped (at zero temperature).

We remind that Jw = J−δJ = 1.474 kBK is the amplitude of the coupling betweenthe strong bonds (Js = J + δJ = 5.28 kBK) according to Hamiltonian (2.2) and

47

5. DYNAMICAL CORRELATIONS AT FINITE TEMPERATURE OF ADIMERIZED SPIN-1/2 CHAIN

the model presented in section 2.5. For each one of the regimes above we examinethe physics encoded in the spectra and the effects of the temperature. In particu-lar in this section we examine in details the results for the temperatures 120 mK ≈0.082 Jw/kB , 500 mK ≈ 0.339 Jw/kB and 2 K ≈ 1.356 Jw/kB . We have alsoresults available also for 300 mK ≈ 0.204 Jw/kB , 800 mK = 0.534 Jw/kB and4 K = 2.714 Jw/kB . For all these temperatures the system is still fully in thequantum regime. Results obtained for the lowest temperature are substantially indis-tinguishable from those at T = 0. For the imaginary time evolution in temperaturewe chose 500 as retained states maximum, a minimal truncation εβ = 10−20 and animaginary-time steps of amplitude δβ = 0.02948 J−1

w for the two lowest tempera-tures or δβ = 0.01474 J−1

w for the others.On the technical side, because of the increasing complexity of the simulation

of the real time evolution for higher temperatures and for magnetic fields insideor very close to the critical region, the maximum time reached is often reduced,causing a worse resolution in frequency for some of the plots that are shown below.Nevertheless, our resolution is still sufficient to see interesting effects on the structureof the correlations. In the following we also give for each case the choice we madefor the parameters of the simulations.

5.2.1 Isotropic gapped system h = 0

In absence of magnetic field the dimerized system is gapped and perfectly isotropicin the three directions x, y, z. In the limit of large gap one has a single triplet exci-tation that proceed by magnetic exchange from strong rung to strong rung (occupiedby singlets). The spectrum can be very well approximated by the one of an isolatedparticle moving with a tight binding Hamiltonian with a hopping matrix element−Jw/4 = −(J − δJ)/4 (triplet ↔ particle, singlet ↔ hole). More details aboutthis result can be found in Appendix D. One thus expects for the spectral function anarrow cosine shape of amplitude ∼ Jw/2. We show for example the 〈SzSz〉 corre-lations at kBT = 0.339Jw in Fig. 5.1. The cosine shape is well observed with anamplitude which is reasonably compatible with the value Jw/2 but also shows theproper modification due to the fact that the gap is not infinite on the strong bonds.The broadening is very small at low temperature. Comparing Fig. 5.1 and Fig. 5.2,which reports 〈SzSz〉 correlations at kBT = 1.356Jw, we notice a significant re-duction of the intensity as well as an increase on broadening. In addition intensityappears at low energy. This is to be expected: at finite temperature the trace on allpossible states contains now a more significant fraction of

∣∣t0⟩ = (|↑↓〉+ |↓↑〉)/√

2states. These states are transformed into singlet states (see below) upon neutronscattering [10].

For 〈SzSz〉 correlations we chose in all cases for the real time evolution a min-imal truncation of εt = 10−10 and a real time step δt = 0.0737 J−1

w . As alreadyanticipated, the computational time needed to perform the simulation increases dra-

48

5.2 Results for dynamical correlations at finite temperature

0 π 2πq · a

ω/Jw

ℜ{Szz11 (q, ω)} · Jw

0

1

2

3

4

5

6

7

8Cosine with amplitude Jw/2

0

5

10

Figure 5.1: Case kBT = 0.339Jw, h = 0, 〈SzSz〉 correlations. A cosine band atfinite energy shows the gapped nature of the system. The blue curve is the cosine onegets by approximating the spectrum of the system with the one of a single triplet in a seaof singlets (see text and Appendix D for details) with corresponding amplitude of Jw/2.a is the lattice spacing. Picture taken from Ref. [30].

0 π 2πq · a

ω/Jw

ℜ{Szz11 (q, ω)} · Jw

0

1

2

3

4

5

6

7

8

0

1

2

Figure 5.2: Case kBT = 1.356Jw, h = 0. 〈SzSz〉 correlations. The cosine bandat finite energy is still clearly visible but with a reduction in intensity and a strongerbroadening, especially at q close to 0 and 2π/a. New structures appear at low energy ifcompared to Fig. 5.1. a is the lattice spacing. Picture taken from Ref. [30].

49

5. DYNAMICAL CORRELATIONS AT FINITE TEMPERATURE OF ADIMERIZED SPIN-1/2 CHAIN

kBT [Jw] Ret. St. Max. Max. Trunc. W. tmax [J−1w ] Comp. Time

0.082 500 10−6 44.22 ∼1 day0.204 500 10−6 44.22 ∼1 day0.339 500 10−6 44.22 ∼2.5 days0.534 500 10−6 44.22 ∼8.5 days1.356 1000 10−5 16.7299 ∼9 days2.714 2500 10−5 10.318 ∼20 days

Table 5.1: Choice of parameters, tmax reached in the evolution and computational time(Comp. Time) needed for 〈SzSz〉 correlations and no magnetic fields. As temperatureapproaches values comparable to Jw the computational cost increases dramatically, andthe precision is lost much earlier despite the choice of a much higher number of retainedstates (Ret. St. Max.) and of weaker constraint on the maximum truncated weight (Max.Trunc. W.).

matically with temperature. This can be easily seen from table 5.1, where the otherparameters of the simulation, the tmax reached and the computational time neededare displayed for all the available temperatures.

The computational time grows from about 1 day for the lower temperature, to about20 days for the higher one for which the maximum time reached is much smaller (onethird). Stronger thermal fluctuations make the numerical description of the behaviorof the system more and more complicated.

5.2.2 Gapped regime at finite h

At h = 2.868Jw < hc1 the gap is almost closed. The three correlations are shown inFig. 5.3 and Fig. 5.4 for the two temperatures kBT = 0.339Jw and kBT = 1.356Jw.

They differ from each other in a way that can be understood qualitatively in thesingle strong dimer picture, an analysis that becomes even quantitatively correct ifthe dimerization is large. At h = 0 the single dimer is a singlet |s〉. The application ofS+ to the first site of the dimer induces a transition |s〉 → |t+〉, and the magnetic fieldbrings down the energy of |t+〉 (to the value of the energy of |s〉 for h = hc1). Thisexplains why the cosine band for 〈S−S+〉 correlations drops to almost zero energy inthe left plot of Fig. 5.3. Following the same line of reasoning, the application of S−

to the first site of the dimer induces a transition |s〉 → |t−〉 which gains energy as hgrows. This explains why the cosine band for 〈S+S−〉 now climbs to higher energyin the central plot of Fig. 5.3. Finally, the application of Sz to the first site of thedimer induces a transition |s〉 →

∣∣t0⟩ which is non sensitive in terms of energy to theapplication of h. This explains why the cosine band for 〈SzSz〉 remains at the sameenergy in the right plot of Fig. 5.3. Already at kBT = 0.339Jw some temperature

50

5.2 Results for dynamical correlations at finite temperature

0 π 2πq · a

ω/Jw

ℜ{S−+11 (q, ω)} · Jw

0

2

4

6

8

10

02468

0 π 2πq · a

ω/Jw

ℜ{S+−

11 (q, ω)} · Jw

0

2

4

6

8

10

0246

0 π 2πq · a

ω/Jw

ℜ{Szz11(q, ω)} · Jw

0

2

4

6

8

10

0

2

4

Figure 5.3: Case T = 0.339Jw, h = 2.868Jw.⟨S−S+

⟩(left),

⟨S+S−

⟩(center)

and 〈SzSz〉 (right) correlations. The gap of the system is almost closed by the magneticfield. Cosine bands displace depending on the correlation considered and temperatureeffects (broadening, lower energy structures) can be already seen. a is the lattice spacing.Picture taken from Ref. [30].

0 π 2πq · a

ω/Jw

ℜ{S−+11 (q, ω)} · Jw

0

2

4

6

8

10

012

0 π 2πq · a

ω/Jw

ℜ{S+−

11 (q, ω)} · Jw

0

2

4

6

8

10

012

0 π 2πq · a

ω/Jw

ℜ{Szz11(q, ω)} · Jw

0

2

4

6

8

10

00.511.5

Figure 5.4: Case T = 1.356Jw, h = 2.868Jw.⟨S−S+

⟩(left),

⟨S+S−

⟩(center)

and 〈SzSz〉 (right) correlations. Temperature effects are enhanced, low energy featureslook more pronounced. For

⟨S−S+

⟩correlations the cosine band is substantially lost

especially close to q = 0 and q = 2π/a. a is the lattice spacing. Picture taken fromRef. [30].

51

5. DYNAMICAL CORRELATIONS AT FINITE TEMPERATURE OF ADIMERIZED SPIN-1/2 CHAIN

Corr. Ret. St. Max. Max. Trunc. W. tmax [J−1w ] Comp. Time

〈S−S+〉 700 10−6 44.22 ∼21.5 days〈S+S−〉 2000 10−6 23.2155 ∼67 days〈SzSz〉 1000 10−6 44.22 ∼58 days

Table 5.2: Choice of parameters (retained state maximum and maximum truncatedweight), tmax reached in the evolution and computational time (Comp. Time) neededfor the three correlations at kBT = 0.339Jw and h = 2.868Jw.

Corr. Ret. St. Max. Max. Trunc. W. tmax [J−1w ] Comp. Time

〈S−S+〉 1600 10−5 19.3094 ∼68.5 days〈S+S−〉 2200 10−5 12.529 ∼29 days〈SzSz〉 1800 10−5 17.3195 ∼52 days

Table 5.3: Choice of parameters (retained state maximum and maximum truncatedweight), tmax reached in the evolution and computational time (Comp. Time) neededfor the three correlations at kBT = 1.356Jw and h = 2.868Jw.

effects are clearly seen in the diffusion on top of the cosine bands, and in the risingof lower energy structures. At a much higher temperature (Fig. 5.4) these effects areenhanced, and the maxima of intensity of the signal are much lower.

For the three correlations we chose in all cases shown in the plots for the real timeevolution a minimal truncation of εt = 10−10 and a real time step δt = 0.0737 J−1

w .The other parameters of the simulation, the tmax reached and the computational timeneeded are listed for each correlation in tables 5.2 and 5.3.

In this case the computational time is in general much higher than the case h = 0and one can hardly reach the maximum value tmax = 30 (kBK)−1 = 44.22 J−1

w .This was expected since the gap is almost closed and small thermal fluctuations caneasily create excitations in the system making the simulation more complicated. Itis also interesting to notice the different efforts needed depending on the correlationwe want to compute: for instance 〈S+S−〉 is by far the most difficult target, and thiswill be the case also for the other fields considered in this chapter (in the TLL phaseand in the fully polarized phase). This can be again explained by considering thephysics of the system: a spin flip from "up" to "down" in these regions of the phasediagram represents a very strong excitation if compared to the opposite one.

5.2.3 Tomonaga-Luttinger liquid regimeAt h = 3.716Jw, in the middle of the TLL region and at finite magnetization, struc-tures become much more complex. At low temperature and very low energy the

52

5.2 Results for dynamical correlations at finite temperature

0 π 2πq · a

ω/Jw

ℜ{S−+11 (q, ω)} · Jw

0

2

4

6

8

10

051015

0 π 2πq · a

ω/Jw

ℜ{S+−

11 (q, ω)} · Jw

0

2

4

6

8

10

0510

0 π 2πq · a

ω/Jw

ℜ{Szz11(q, ω)} · Jw

0

2

4

6

8

10

0123

Figure 5.5: Case T = 0.082Jw, h = 3.716Jw.⟨S−S+

⟩(left),

⟨S+S−

⟩(center)

and 〈SzSz〉 (right) correlations. Blue arrows indicate the positions of the zero energypoints according to the low energy TLL description (mapping to an homogeneous spinchain). The agreement between the numerical results and the theoretical expectations onthe position of the zero energy points is excellent. a is the lattice spacing. Picture takenfrom Ref. [30].

structures can be understood if we consider a mapping to an homogeneous spin chain(single strong dimer↔ spin). We know from [7, 73] that at finite magnetization andat low enough temperature, for an homogeneous spin-1/2 chain, 〈SzSz〉 correlationshave low energy modes at q = 0, 2π and at q = π(1 ± 2m), while XY correlationsdevelop incommensurability at q = 2πm and q = 2π(1−m), while the q = π pointstays commensurate. Here with m we indicate the magnetization per site of the ho-mogeneous chain. Taking into account this and considering that we are computingcorrelations of “11” species (see above), we find in our framework that for all thethree correlations 〈S−S+〉, 〈S+S−〉 and 〈SzSz〉 zero energy points should sit onmomenta q · a = 0, 4πmd, 2π(1 − 2md), 2π. Here md is the magnetization persite of the dimerized chain. The theoretical expectation are in very good agreementwith the numerical results as shown in Fig. 5.5. Our results are obtained with the fol-lowing choice of parameters: real time step of δt = 0.0737 J−1

w , minimal truncationεt = 10−10, maximum truncated weight of 10−6 (10−5 for the highest temperature).The maximum number of retained states kept grows as temperature increases from700 to 1000 for 〈S−S+〉 correlations, from 1500 to 2400 for 〈S+S−〉 correlations,and from 700 to 1700 for 〈SzSz〉 correlations.

In Figs. 5.6 and 5.7, we see how the thermal effects affect the structures. Thesignal becomes more diffuse and the intensity is redistributed. This is particularlyevident if one compares Fig. 5.6 and Fig. 5.7, where we adopted the same color scaleto highlight this effect.

53

5. DYNAMICAL CORRELATIONS AT FINITE TEMPERATURE OF ADIMERIZED SPIN-1/2 CHAIN

0 π 2πq · a

ω/Jw

ℜ{S−+11 (q, ω)} · Jw

0

2

4

6

8

10

0123

0 π 2πq · a

ω/Jw

ℜ{S+−

11 (q, ω)} · Jw

0

2

4

6

8

10

0123

0 π 2πq · a

ω/Jw

ℜ{Szz11(q, ω)} · Jw

0

2

4

6

8

10

00.511.5

Figure 5.6: Case T = 0.339Jw, h = 3.716Jw.⟨S−S+

⟩(left),

⟨S+S−

⟩(center)

and 〈SzSz〉 (right) correlations. The temperature starts to play a role with respect tothe previous picture, deforming the structures and redistributing the intensities. a is thelattice spacing. Picture taken from Ref. [30].

0 π 2πq · a

ω/Jw

ℜ{S−+11 (q, ω)} · Jw

0

2

4

6

8

10

0123

0 π 2πq · a

ω/Jw

ℜ{S+−

11 (q, ω)} · Jw

0

2

4

6

8

10

0123

0 π 2πq · a

ω/Jw

ℜ{Szz11(q, ω)} · Jw

0

2

4

6

8

10

00.511.5

Figure 5.7: Case T = 1.356Jw, h = 3.716Jw.⟨S−S+

⟩(left),

⟨S+S−

⟩(center) and

〈SzSz〉 (right) correlations. The higher temperature modifies more heavily the struc-tures, especially at very low energy. To highlight this, the same colorscale of the previouspicture has been used here. a is the lattice spacing. Picture taken from Ref. [30].

54

5.2 Results for dynamical correlations at finite temperature

0 π 2πq · a

ω/Jw

ℜ{S−+11 (q, ω)} · Jw

0

2

4

6

8

10

0123

0 π 2πq · a

ω/Jw

ℜ{S+−

11 (q, ω)} · Jw

0

2

4

6

8

10

0246

0 π 2πq · a

ω/Jw

ℜ{Szz11(q, ω)} · Jw

0

2

4

6

8

10

0

0.5

1

Figure 5.8: Case T = 0.339Jw, h = 4.674Jw.⟨S−S+

⟩(left),

⟨S+S−

⟩(center)

and 〈SzSz〉 (right) correlations. At relatively low temperature⟨S−S+

⟩correlations are

zero because all spins are up. In the central plot two cosines both with amplitude Jw/2are superposed to the structures we associate to the |s〉 (lower cosine) and to the |t0〉(upper cosine) excitations, according to the single strong dimer treatment. a is the latticespacing. Picture taken from Ref. [30].

5.2.4 High field gapped phase

At h = 4.674Jw, the system is again gapped and fully polarized, we therefore expect〈S−S+〉 and 〈SzSz〉 correlations to be zero. Again, temperature acts on the corre-lations by creating weak structures due to thermal excitations in the 〈S−S+〉 and〈SzSz〉 cases already at the lower temperature. This because the gap at this specificvalue of the field is pretty small. By comparing Fig. 5.8 and Fig. 5.9 the diffusion ofthe signal and the redistribution of the relative intensities can be seen for all the threecorrelations.

Our results are obtained with the following choice of parameters: real time stepof δt = 0.0737 J−1

w , minimal truncation εt = 10−10, maximum truncated weightof 10−6 (10−5 for the higher temperature). The maximum number of retained stateskept grows as temperature increases from 700 to 1400 for 〈S−S+〉 correlations, from1600 to 2400 for 〈S+S−〉 correlations, and from 800 to 1000 for 〈SzSz〉 correla-tions.

Results for 〈S+S−〉 correlations can again be very well understood qualitativelyin the single strong dimer picture: at zero temperature the system is made by acollection of |t+〉 on each strong bond, the application of S− on a site induces atransition on that strong bond to a state which is a combination of a singlet |s〉 andof a triplet |t0〉. The two cosine bands of amplitude ∼ Jw/2 are representative ofthese two excitations: the lower one for the singlet, which shows a very small gapcompatible with the value h − hc2, the upper one for the |t0〉. More details on thisresult can be found in Appendix D. At finite temperature, thermal excitations can be

55

5. DYNAMICAL CORRELATIONS AT FINITE TEMPERATURE OF ADIMERIZED SPIN-1/2 CHAIN

0 π 2πq · a

ω/Jw

ℜ{S−+11 (q, ω)} · Jw

0

2

4

6

8

10

0

1

2

0 π 2πq · a

ω/Jw

ℜ{S+−

11 (q, ω)} · Jw

0

2

4

6

8

10

0123

0 π 2πq · a

ω/Jw

ℜ{Szz11(q, ω)} · Jw

0

2

4

6

8

10

00.511.5

Figure 5.9: Case T = 1.356Jw, h = 4.674Jw.⟨S−S+

⟩(left),

⟨S+S−

⟩(center) and

〈SzSz〉 (right) correlations. Some signal is now present in⟨S−S+

⟩correlations at low

energy and q ∼ 0 or 2π/a because thermal fluctuations can make some spin flips in thesystem. The other two correlations get broadened and there is a different redistributionof relative intensity. a is the lattice spacing. Picture taken from Ref. [30].

generated in the initial state. The applications of S− to one of these excitations (forexample a singlet) can create a high-energy |t−〉 which propagates on the chain andgenerates the highest faint structure.

5.3 Band-narrowing effectsOne interesting effect of the temperature on the correlation functions, in addition tothe diffusion effects and variation of intensity mentioned above, is how the dispersionof a triplet |t+〉 excitation can be modified. Indeed, as seen in the previous section,for h < hc1 this excitation has essentially a cosine-like form. At zero temperaturethe dispersion depends essentially on the weak exchanges between the strong bonds(the amplitude of the cosine is Jw/2 in the limit Js � Jw). At finite temperature wesee that the shape more or less persists but the amplitude can now be renormalizedby the temperature. How the temperature affects such a dispersion is not easy to ana-lyze analytically. For ladders, estimations based on bond-operators description havebeen performed in Ref. [85] in the gapped regime. Similar analyzes are in principledirectly applicable to the dimerized systems as well. However, these approximationscannot easily take into account the broadening due to the temperature. The thermaleffect is thus mostly to shift an otherwise perfectly sharp spectrum

δ(ω − E(k))→ δ(ω − E(k, T )) (5.3)

In that approximate solution one question is therefore how the temperature canchange the amplitude of the dispersion E(k, T ) for k varying in the Brillouin zone.As we will see in our numerical results such an approximation is quite limited in

56

5.3 Band-narrowing effects

0 2 4 6

3

3.5

4

Locationintensity

peaks[ω/Jw]

kBT/Jw

S−+ Correlations

q = 0

q = π/(2a)

q = π/a

2 2.5 3 3.5 4 4.5 5 5.50

0.5

1 q = 0

2 2.5 3 3.5 4 4.5 5 5.50

0.5

1

Renorm

alizedintensities

q = π/(2a)

2 2.5 3 3.5 4 4.5 5 5.50

0.5

1

ω/Jw

q = π/a

Figure 5.10: Position of intensity maxima at q = 0, π2a, πa

as a function of kBT/Jwfor

⟨S−S+

⟩correlations in the case h = 0. The squeezing of the cosine band is clearly

seen. As for the plots on the right, curves from T = 0.082Jw to T = 0.543Jw are allon top of each other (blue curve). See text for a detailed legend of the plots on the right.a is the lattice spacing. Picture taken from Ref. [30].

describing the real correlation functions. Indeed the broadening of the bands due tothe temperature is quite large and more importantly asymetric. It is thus impossibleto reduce the analysis of the temperature effects to a simple shift of the position ofthe maximum as (5.3) would imply. A full description of the spectrum is thereforeneeded.

In this section we thus use our calculations at finite temperature to perform suchan analysis for the dimerized system. We consider three values of magnetic fieldsinside of the first gapped region, namely h = 0, h = 1.695Jw and h = 2.868Jw,several values of the temperature between 0.082Jw and 6.78Jw, and the three corre-lations of the previous section. Figures from 5.10 to 5.16 report some of our results.Each figure is composed by four plots, one on the left and three on the right. On theleft, for each given temperature kBT/Jw and for each fixed values of momentumq = 0 (bottom of the band), q = π/(2a) (intermediate point) and q = π/a (top ofthe band), a being the lattice spacing, we select the energy corresponding to the max-ima in intensity of the cosine band related to the |s〉 → |t+〉 excitation. The error barof each point is essentially the finite resolution in energy. On the right we plot thecuts at fixed momentum (from top to bottom q = 0, q = π/(2a) and q = π/a), andwe limit ourselves in energy to the vicinity of the maximum in intensity. In each ofthese plots on the right, to each color corresponds a different temperature, in particu-lar: black↔ 0.082Jw, purple↔ 0.204Jw, light blue↔ 0.339Jw, blue↔ 0.543Jw,red↔ 1.356Jw, green↔ 2.712Jw, yellow↔ 3.39Jw, orange↔ 6.78Jw. In eachcurve values have been divided by the maximum in order to renormalize all maximato one.

For h = 0 (Figs. 5.10 and 5.11) we report only 〈S−S+〉 and 〈SzSz〉 corre-

57

5. DYNAMICAL CORRELATIONS AT FINITE TEMPERATURE OF ADIMERIZED SPIN-1/2 CHAIN

0 2 4 6

2.8

3

3.2

3.4

3.6

3.8

4

4.2

Loca

tionintensity

pea

ks[ω/J

w]

kBT/Jw

Szz Correlations

q = 0q = π/(2a)q = π/a

2 2.5 3 3.5 4 4.5 5 5.50

0.5

1 q = 0

2 2.5 3 3.5 4 4.5 5 5.50

0.5

1

Renorm

alizedintensity

q = π/(2a)

2 2.5 3 3.5 4 4.5 5 5.50

0.5

1

ω/Jw

q = π/a

Figure 5.11: Position of intensity maxima at q = 0, π2, π as a function of kBT/Jw for

〈SzSz〉 correlations in the case h = 0. The squeezing of the cosine band is clearly seen.As for the plots on the right, curves from T = 0.082Jw to T = 0.543Jw are all on topof each other (blue curve). See text for a detailed legend of the plots on the right. a isthe lattice spacing. Picture taken from Ref. [30].

lations since the curves for 〈S−S+〉 and 〈S+S−〉 are identical at all temperatures.In both figures, despite the big error bars at high temperatures, the amplitude of thecosine band, as defined by the position of the maxima, is clearly reduced as the tem-perature increases. However one also sees from the plots on the right that the smalldisplacement of the maxima is accompanied by a broad asymmetric broadening ofthe spectra. This broadening, which is much larger than the shift of the maximumitself, contributes largely to the flattening of the spectrum with temperature.

Similar effects are observed at the other fields. For h = 1.695Jw (Figs. 5.12 and5.13) the system is still in the gapped regime, and we report again only 〈S−S+〉 and〈SzSz〉 correlations since the curves for 〈S−S+〉 and 〈S+S−〉 are again identical atall temperatures. We observe as for the previous case the squeezing of the band asthe temperature increases and the broadening of the structures due to temperature.

For h = 2.868Jw (Figs. 5.14→ 5.16) the system is extremely close to the firstquantum critical point, although it is still in the gapped regime at zero temperature.The squeezing of the band as the temperature increases is once more clear for thethree correlations, together with the broadening of the structure.

5.4 Summary & conclusions of the chapterExploiting the same MPS technique used already in the previous chapter, we havecomputed dynamical correlation functions of a dimerized system of spins 1/2 as afunction of frequency, momentum and temperature. We have analyzed the magneticfield and the temperature effects on such correlations. At zero field, section 5.2.1,gapped phase, the system is isotropic and the three correlations show the same struc-

58

5.4 Summary & conclusions of the chapter

0 2 4 61

1.5

2

2.5

Loca

tionintensity

pea

ks[ω/Jw]

kBT/Jw

S−+ Correlations

q = 0

q = π/(2a)

q = π/a

0.5 1 1.5 2 2.5 3 3.50

0.5

1 q = 0

0.5 1 1.5 2 2.5 3 3.50

0.5

1

Renorm

alizedintensities

q = π/(2a)

0.5 1 1.5 2 2.5 3 3.50

0.5

1

ω/Jw

q = π/a

Figure 5.12: Position of intensity maxima at q = 0, π2, π as a function of kBT/Jw for⟨

S−S+⟩

correlations in the case h = 1.695Jw. The squeezing of the cosine band isclearly seen. As for the plots on the right, curves for T = 0.082Jw and T = 0.204Jware on top of each other (purple curve). See text for a detailed legend of the plots on theright. a is the lattice spacing. Picture taken from Ref. [30].

0 2 4 6

3

3.5

4

4.5

Locationintensity

peaks[ω/J

w]

kBT/Jw

Szz Correlations

q = 0q = π/(2a)q = π/a

2 2.5 3 3.5 4 4.5 5 5.50

0.5

1 q = 0

2 2.5 3 3.5 4 4.5 5 5.50

0.5

1

Renorm

alizedintensity

q = π/(2a)

2 2.5 3 3.5 4 4.5 5 5.50

0.5

1

ω/Jw

q = π/a

Figure 5.13: Position of intensity maxima at q = 0, π2, π as a function of kBT/Jw

for 〈SzSz〉 correlations in the case h = 1.695Jw. The squeezing of the cosine band isclearly seen. As for the plots on the right, curves for T = 0.082Jw and T = 0.204Jware on top of each other (purple curve). See text for a detailed legend of the plots on theright. a is the lattice spacing. Picture taken from Ref. [30].

59

5. DYNAMICAL CORRELATIONS AT FINITE TEMPERATURE OF ADIMERIZED SPIN-1/2 CHAIN

0 2 4 60

0.5

1

1.5

Loca

tionintensity

pea

ks[ω/Jw]

kBT/Jw

S−+ Correlations

q = 0

q = π/(2a)

q = π/a

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1 q = 0

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

Renorm

alizedintensities

q = π/(2a)

0 0.5 1 1.5 2 2.50

0.5

1

ω/Jw

q = π/a

Figure 5.14: Position of intensity maxima at q = 0, π2, π as a function of kBT/Jw for⟨

S−S+⟩

correlations in the case h = 2.868Jw. The squeezing of the cosine band isclearly seen and a non-trivial behavior is observed at low T. See text for a detailed legendof the plots on the right. a is the lattice spacing. Picture taken from Ref. [30].

0 2 4 65.5

6

6.5

7

Locationintensity

peaks[ω/J

w]

kBT/Jw

S+− Correlations

q = 0q = π/(2a)q = π/a

4.5 5 5.5 6 6.5 7 7.5 80

0.5

1 q = 0

4.5 5 5.5 6 6.5 7 7.5 80

0.5

1

Renorm

alizedintensity

q = π/(2a)

4.5 5 5.5 6 6.5 7 7.5 80

0.5

1

ω/Jw

q = π/a

Figure 5.15: Position of intensity maxima at q = 0, π2, π as a function of kBT/Jw for⟨

S+S−⟩

correlations in the case h = 2.868Jw. The squeezing of the cosine band isclearly seen and a non-trivial behavior is observed at low T. See text for a detailed legendof the plots on the right. a is the lattice spacing. Picture taken from Ref. [30].

60

5.4 Summary & conclusions of the chapter

0 2 4 6

3

3.5

4

4.5

Loca

tionintensity

pea

ks[ω/J

w]

kBT/Jw

Szz Correlations

q = 0

q = π/(2a)

q = π/a

2 2.5 3 3.5 4 4.5 5 5.50

0.5

1 q = 0

2 2.5 3 3.5 4 4.5 5 5.50

0.5

1

Renorm

alizedintensity

q = π/(2a)

2 2.5 3 3.5 4 4.5 5 5.50

0.5

1

ω/Jw

q = π/a

Figure 5.16: Position of intensity maxima at q = 0, π2, π as a function of kBT/Jw

for 〈SzSz〉 correlations in the case h = 2.868Jw. The squeezing of the cosine band isclearly seen and a non-trivial behavior is observed at low T. See text for a detailed legendof the plots on the right. a is the lattice spacing. Picture taken from Ref. [30].

ture. The spectrum can be very well approximated in the single strong bond pictureby the one of an isolated particle (excitation) described by a tight binding Hamilto-nian. At finite field, but still in the gapped phase (h ≤ hc1), section 5.2.2, the threecorrelations differentiate from each other. The behavior can be again understoodqualitatively in the strong bond picture by associating to each correlation the corre-sponding triplet excitation induced by the application of the first operator. At finitefield, but this time in the TLL phase (hc1 < h < hc2), section 5.2.3, the low temper-ature structures can be explained according to the TLL theory. At high field, section5.2.4 (h > hc2, gapped), the structures can be again interpreted in the single strongbond picture by considering a single excitation described by a tight binding Hamil-tonian. In all the cases analysed here we have shown how the temperature broadensthe spectrum and redistributes intensity among the structures. Temperature is alsoresponsible of a dramatic increase of the computational time and of the numericaleffort needed to simulation of the real time evolution.

In addition we have studied the effects of the temperature on the dispersion ofa single triplet excitation in the gapped regime. We have seen that the temperatureleads to a general narrowing of the bands for the triplon and that the broadening ofthe bands is much larger than the shift of the maxima. In most of the cases, thisbroadening is asymmetric with respect to the maximum peak and it largely con-tributes to flatten and to redistribute the intensities. Asymmetric broadenings werealso computed for integrable models by form factor expansion in Ref. [82], or for thelow temperature properties of ladders in the strong rung limit in Ref. [86].

Our calculations are potentially directly relevant to neutron scattering experi-ments that have been done on compounds such as the copper nitrate alternating chain[Cu (NO3)2 · 2.5D2O] [84] investigated in Refs. [36–38,83]. In particular, our results

61

5. DYNAMICAL CORRELATIONS AT FINITE TEMPERATURE OF ADIMERIZED SPIN-1/2 CHAIN

can deal with the thermal effects on the spectra without the need to resort to uncon-trolled approximations. To be able to properly perform the comparison with neutronscattering experiments it is of course necessary to take into account the structure ofthe system (for example the non-linear nature of the chain) as well as the propermixture of correlation functions induced by unpolarized neutron measurements. Asin the case of chapter 4, a natural extension would be to move to finite temperaturedynamics of quasi-one-dimensional spin systems, like for instance spin-1/2 two-legladders as BPCB [10] or DIMPY [11]. Another interesting direction would be totake into account the effect of the inter-chain (or inter-ladder) coupling by means ofmean field or RPA treatments.

62

CHAPTER 6

Grüneisen Parameter and Quantum PhaseTransitions in Spin-1/2 Ladders

In this chapter we focus on our analytical results for the so called Grüneisen pa-rameter for the spin-1/2 two-leg ladder model presented in chapter 2. We havealready seen in section 2.5 that this model is appropriate to describe two mem-bers of the so called “Hpip” family: (C5H12N)2 CuBr4 (bromine compound) and(C5H12N)2 CuCl4 (chlorine compound). The Grüneisen parameter is a very inter-esting quantity to focus on if one wants to study quantum phase transitions. Aftera brief introduction in which we define the magnetic Grüneisen parameter and de-scribe its peculiarities, we present the derivation of our analytical expressions for thisquantity. Our results will be then compared to numerical and experimental resultstaken from Ref. [28].

6.1 The Grüneisen parameter: definition and compu-tation

The magnetic Grüneisen parameter Γmag links the temperature derivative of the mag-netization to the magnetic heat capacity, according to the relation

Γmag = −∂m/∂Tcv,mag

. (6.1)

Γmag represents a very sensitive tool to detect and study quantum phase transitions,since it exhibits in the limit of T → 0 a divergence in the vicinity of a quantum

63

6. GRÜNEISEN PARAMETER AND QUANTUM PHASE TRANSITIONS INSPIN-1/2 LADDERS

critical point (QCP). This makes the difference with respect to ∂m/∂T and the mag-netic heat capacity, which can indicate a phase transition at finite temperature butapproach 0 in the limit of zero temperature. In addition, the Grüneisen parameterexhibits a change of sign while sweeping through the QCP [87]. In the case of thespin-1/2 ladder models presented already in chapter 2, one can immediately identifythe two QCPs by studying the behavior of Γmag as a function of the magnetic field.

To compute the Grüneisen parameter close to criticality we make intensive useof the results already discussed in chapter 2.

6.1.1 Computation of Γmag via spin-chain mapping

The first thing we want to do is to compute analytically the Grüneisen parameter forthe ladder model close to criticality, exploiting the spin-chain mapping of the ladderalready discussed in section 2.4 which transforms our full ladder Hamiltonian intoan XXZ one. By applying the Jordan-Wigner transformation (2.11) we then moveto an Hamiltonian expressed in fermionic creation and annihilation operators [7].Fermions in this picture are the very few excitations present in the system (triplets"+" close to hc1 and singlets close to hc2). Under these conditions we can get an ideaof the physics of the system by adopting an effective theory in which we considerthe excitations as non interacting. We therefore neglect quadratic terms in fermionicdensity and transform the Hamiltonian into an XX one (2.12), which is diagonal inmomentum space:

HXX,k =∑k

(−J‖ cos k + J⊥ − h

)c†kck −

3

4LJ⊥

=∑k

εkc†kck −

3

4LJ⊥, (6.2)

Here we adopted the following definition of the Fourier transform:

cj =1√L

∑k

eikjck ,

where k = −π(L−2)L , . . . , − 2π

L , 0, 2πL , . . . , π (since j = 0, 1, . . . , L− 1).

If we want to perform comparisons with experiments it is important to stress that thefirst critical field of our effective model of Eq. (6.2) does not match the first criticalfield hlad

c1 of the two real compounds. It can be easily seen from that equation thathc1 = J⊥ − J‖, since the energy of the first excitation for k = 0 is J⊥ − J‖ − h(and that hc2 = J⊥ + J‖). On the other hand hc1 < hlad

c1 for both compounds. Weneed therefore to rescale h such that hc1 matches hlad

c1 . For each specific value of thefield hexp used in experiments, the corresponding value of h that will be used in our

64

6.1 The Grüneisen parameter: definition and computation

calculation is obtained from the mapping

h =

(hexp − hlad

c1

)(hc2 − hc1)

hladc2 − hlad

c1

+ hc1, (6.3)

which is of course valid only close to criticality.Let us now go back to Eq. (6.2). The free energy per spin is given by

f(β, h) = −3

4J⊥ −

1

2πβ

∫ π

−πdk ln(1 + e−βεk),

where β is the inverse temperature. Once we know f , we can get (implicit) expres-sion for all quantities relevant for us, in particular the magnetization m, its tempera-ture derivative ∂m

∂T and the magnetic specific heat cv,mag:

m(β, hz) = −∂f∂h

=1

π

∫ π

0

dk1

1 + e−β(J‖ cos k−J⊥+h),

∂m

∂T(β, h) = −β2 ∂m

∂β= −β

2

4π

∫ π

0

dkJ‖ cos k − J⊥ + h

cosh2

(β

2

(J‖ cos k − J⊥ + h

)) , (6.4)

cv,mag(β, h) = −β2 ∂2

∂β2(βf) =

β2

4π

∫ π

0

dk

(J‖ cos k − J⊥ + h

)2cosh2

(β

2

(J‖ cos k − J⊥ + h

)) .(6.5)

Integrals (6.4) and (6.5) can be solved numerically, and their ratio can give us thedesired Grüneisen parameter. We will solve them close to the first critical field, andthe h employed there has been already rescaled according to (6.3).

At low temperature only the lowest energy states are occupied, therefore we canexpand the dispersion relation around k = 0 up to the second order to simplify a bitthe integrals:

εk ' −J‖ +J‖

2k2 + J⊥ − h.

Taking the β →∞ limit, all the relevant thermodynamic quantities can be expressedin terms of known integrals of the form

fν(α) =1

Γ(ν)

∫ ∞0

xν−1

ex−α + 1dx for ν > 0. (6.6)

In this specific case α = β(J‖ + h − J⊥) = βδh, where δh is the distance fromthe critical field. For α � 1 (which means T � δh) integrals of this kind can be

65

6. GRÜNEISEN PARAMETER AND QUANTUM PHASE TRANSITIONS INSPIN-1/2 LADDERS

expanded asymptotically1 to give:

m(T → 0, δh) =1

π

√2δh

J‖

[1− π2

24

(T

δh

)2

− 7π4

384

(T

δh

)4

+ . . .

], (6.7)

cv,mag(T → 0, δh) =πT

6

√2

J‖δh

[1 +

21π2

40

(T

δh

)2

+ . . .

]. (6.8)

Using results in Eqs. (6.7)-(6.8) we can derive an expression for Γmag:

Γ(T → 0, δh) =1

2δh+

7π2

40

T 2

δh3+O

(T 4

δh5

). (6.9)

The first term in this expansion agrees with the universal scaling relation proposed inRef. [88] for the zero temperature limit. The expected behavior close to the criticalpoint is given in its most general form by:

Γ(T → 0, δh) = −Gh1

δh. (6.10)

The coefficient Gh is a combination of critical exponents and of the dimensionality:

Gh =ξ(d− y0z)

y0. (6.11)

In this case d = 1 is the dimensionality, y0 = 1 characterizes the power-law behaviorof the specific heat for a Fermi liquid, z = 2 is the dynamical critical exponent,ξ = 1

2 is the exponent related to the correlation length [89]. Putting all together weget correctly Gh = − 1

2 , as obtained in (6.9).Let us now try to evaluate the behavior of Γmag at the critical field as a function of

temperature, in the zero temperature limit. Magnetization, its temperature derivativeand the heat capacity can be expressed again in terms of integrals of the form givenin Eq. (6.6), but this time with α = 0. Such integrals can be exactly solved2 and the

1For α� 1 and ν > 0:

fν(α) =1

Γ(ν)

∫ ∞0

dxxν−1

ex−α + 1= αν

∑l

d2l

Γ(ν + 1− 2l)α−2l,

wheren∑l=0

(−1)lπ2l

(2l + 1)!d2n−2l = δ0,n.

From this last result one immediately sees that: d0 = 1, d2 = π2

6, d4 = 7π4

360.

2A couple of useful mathematical relations:∫ ∞0

x−1/2

ex + 1dx = (1−

√2)√π ζ(1/2). (6.12)∫ ∞

0

√x

ex + 1dx =

(2−√

2)√π

4ζ(3/2). (6.13)

66

6.1 The Grüneisen parameter: definition and computation

results lead to the following scaling relation:

Γ(T, δh = 0) =

√2

6

ζ(1/2)

ζ(3/2)

1

T∼ −0.527 · T−1, (6.14)

where ζ(x) is the Riemann zeta function. The temperature dependence agrees withthe universal scaling relation at criticality proposed in Ref. [88]. The expected be-havior is given in its most general form by:

Γ(T, δh = 0) = −GTT−1/(νz). (6.15)

GT is defined in Ref. [88] as a combination of some universal and non-universalparameters and it is therefore difficult to make a one-to-one comparison. We can atleast check the agreement for the exponent: ν = 1

2 and z = 2, thus − 1νz = −1 as

expected.

6.1.2 Strong coupling expansion & refined resultsThe results we obtained up to now can be refined in order to take into account moreproperly the temperature effects and to give a qualitative interpretation of the physicsbehind them.

Let us recall what discussed already in chapter 2. Starting from the general strongcoupling expansion of the ladder model, we introduced the four operators s†j , t

†j,+,

t†j,0 and t†j,− creating a singlet or one of the three triplets on the site j and we rewrotethe original Hamiltonian in terms of these new operators.

Results of the previous subsection, namely Eq. (6.4) and Eq. (6.5), can be ob-tained by neglecting at this point all terms containing triplets "0" or "-" terms andusing an hard-core constraint for singlets and triplets "+".

In order to improve a bit our approximation, what we can do around the firstcritical field is to make use of the full hard-core constraint, Eq. (2.13), to re-expressthe density of singlet in terms of the densities of the triplets, to neglect only termscubic or quartic in triplet operators and to set elsewhere single singlet operators equalto unity. What we got is Eq. (2.16):

H1 =∑

l=+,0,−

∑j

[J‖

2

(c†l,jcl,j+1 + h.c.

)− hlc†l,jcl,j

],

whereas c†l,j = t†l,j and being

h+ = h− J⊥,h0 = −J⊥,h− = −h− J⊥.

67

6. GRÜNEISEN PARAMETER AND QUANTUM PHASE TRANSITIONS INSPIN-1/2 LADDERS

To determine the Grüneisen ratio around hc1 we need again the temperature deriva-tive of the magnetization and the specific heat of the system described byH1. Thanksto its structure, these two quantities can be computed separately for each of the threespecies of fermions and then summed up in order to get the desired result.

∂m

∂T(β, h) =

β2

4π

∑l=+,−

∫ π

0

εl(k)

cosh2

(βεl(k)

2

) dk , (6.16)

cv,mag (β, hz) =β2

4π

∑l=+,0,−

∫ π

0

ε2l (k)

cosh2

(βεl(k)

2

) dk , (6.17)

whereas

ε+(k) = −J‖ cos(k)− h+ J⊥ ,

ε0(k) = J‖ cos(k)− J⊥ ,ε−(k) = J‖ cos(k)− h− J⊥ .

Integrals in (6.16) and (6.17) can be evaluated numerically. Again, if we want tomake proper comparison with experiments, we need to map the first critical field ofthe true compound onto the first critical field of the triplets "+". This can be achievedfollowing the same strategy of Eq. (6.3).

Calculation close to the second critical field are performed in a similar fashion,but this time we use the full hard-core constraint to re-express the density of thetriplets "+", we neglect cubic or quartic terms in singlets and triplets "0" and "-",and we set elsewhere single triplet "+" operators equal to unity. What we got isEq. (2.18):

H2 =J‖

2

∑j

[s†jsj+1 + h.c.

]−(J⊥ + J‖ − h

)∑j

s†jsj+

+J‖

2

∑j

[t†j,0tj+1,0 + h.c.

]−(J‖ − h

)∑j

t†j,0tj,0−

− 2(J‖ − h

)∑j

t†j,−tj,− +

(J⊥4

+J‖

2− h)L.

Interesting quantities to compute the Grüneisen ratio can be computed quite easilyas before

∂m

∂T(β, h) =

β2(h− J‖

)cosh2

(β(h− J‖

)) +β2

4π

∑l=s,0

∫ π

0

εl(k)

cosh2

(βεl(k)

2

) dk , (6.18)

68

6.2 Analytical results against experiments and DMRG

cv (β, h) =β2(h− J‖

)2cosh2

(β(h− J‖

)) +β2

4π

∑l=s,0

∫ π

0

ε2l (k)

cosh2

(βεl(k)

2

) dk , (6.19)

whereas

εs(k) = J‖ cos(k)− h+ J⊥ + J‖ ,

ε0(k) = J‖ cos(k)− h+ J‖ . (6.20)

Also in this case we notice that the critical value of h for singlets is different fromthe second critical field hlad

c2 of both ladder models. We need therefore to adopt aproper mapping following again the same strategy adopted in Eq. (6.3).

6.2 Analytical results against experiments and DMRGExploiting the results we have just obtained we can compute the Grüneisen parameterΓmag as a function of magnetic field (at fixed temperature) and temperature (at fixedmagnetic field). Let’s start from the first case, and let’s focus mainly on the chlorinecompound.

In Fig. 6.1 we compute the Grüneisen parameter as a function of the magneticfield, for 4 different temperatures, analytically. Dashed lines are obtained via the freefermion effective theory coming from spin-chain mapping of the ladder, as explainedin subsection 6.1.1. Solid lines are obtained exploiting the refined approximationpresented in subsection 6.1.2. The difference between the two approaches becomesmore and more important as the temperature increases, and this is expected since thesolid lines take into account higher excitations thermally activated. In all cases thecurves present correctly a change of sign around the critical field, and a divergenceseems to set in around that point as the temperature is lowered. By comparing solidlines of Fig. 6.1 and the experimental points of Fig. 3 in Ref. [28], a good agreementis observed. With respect to the bromine compound, the chlorine one is characterizedby a smaller ratio J‖/J⊥: this reduces the gap to the higher energy triplets makingthe role of temperature more important.

Let’s now move to the discussion of the temperature dependence of the Grüneisenratio for fields very close to the critical ones. In Fig. 6.2 experimental results for|Γmag| (dots with corresponding errorbars) for different temperatures and for bothcompounds are reported. In order to collect the results in one figure we renormal-ize by J⊥. The fact that the two compounds possess different energy scales allowus to cover a wide range of energies (temperatures). To make the picture clearerand to emphasize occurring power-law behaviors we chose a log-log scale. Reddots correspond to the bromine compound at h ∼ hc1, green dots to the chlorinecompound at h ∼ hc1, blue dots to the chlorine compound at h ∼ hc2. hc2 forthe bromine compound is too high and could not be reached by experiments. Solid

69

6. GRÜNEISEN PARAMETER AND QUANTUM PHASE TRANSITIONS INSPIN-1/2 LADDERS

1 1.5 2 2.5 3 3.5 4 4.5 5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Magnetic field [T]

Γ mag

[1

/T]

T=320 mK, free fermions (3 species)T=420 mK, free fermions (3 species)T=520 mK, free fermions (3 species)T=620 mK, free fermions (3 species)T=320 mK, free fermions (1 species)T=420 mK, free fermions (1 species)T=520 mK, free fermions (1 species)T=620 mK, free fermions (1 species)

Figure 6.1: Analytical results for Γmag(h) for the HpipCuCl4 compound. To obtain thedashed lines, the XXZ Hamiltonian coming from the spin-chain mapping of the ladderis approximated in the limit of extreme diluteness of excitations by an XX one (effectivetheory). Solid lines are obtained from the strong coupling expansion, taking into accountthe presence of higher energy excitations which can only be created by temperature.

green and light blue lines are DMRG results for the ladder model at the experimen-tal field obtained by P. Bouillot. The red solid line corresponds to analytical resultswithin the XX approximation for the bromine compound, close to hc1. The dottedblack line represents the behavior in temperature at the critical field of the Grüneisenratio, with a prefactor obtained by fitting the experimental data with a power lawaT−1. The correct exponent (-1) is the one computed already in Eq. (6.14). We de-termined a = 0.46. The very good collapse of the results on the top left of Fig. 6.2,obtained for the two compounds and for different theoretical models at low temper-atures, proves the validity of the universal behavior (dotted black line). At highertemperatures (right part of Fig. 6.2) the microscopic structure of the material be-comes important and deviations from the scaling occur. We observe clearly differentbehavior for the two QCPs for the chlorine compound. This is due to the energies ofthe higher triplet excitations which shift with the magnetic field. Here the agreementbetween experimental data with errorbars and the DMRG calculations for the lad-der system (solid lines) is extremely good. The trend we can observe for the lowestavailable temperatures seems to indicate once more a clear tendency to a collapseonto the universal critical behavior.

In the case of the chlorine compound, deviations from the critical behavior atexperimental temperatures are quite remarkable especially for h ∼ hc1. DMRGsimulations for the full ladder model can capture these effects. In the following wewill adopt our approximation of subsection 6.1.2 and try to compute the same curves.In Fig. 6.3 analytical results within our approximation, for the chlorine compound

70

6.3 Summary & conclusions of the chapter

log 𝑘𝐵𝑇/J⊥

log

Γ 𝑚𝑎𝑔∙J⊥/𝑔𝜇𝐵

Figure 6.2: |Γ(T )| renormalized by J⊥ in order to deal with unitless quantities, inlog-log scale. Results for HpipCuBr4 are available only at h ∼ hc1. Red dots are exper-imental data and the red line is the analytics within the XX approximation. Results forHpipCuCl4 are reported close to both critical fields (green and blue dots are experimen-tal data, solid lines are DMRG results for the full ladder model). The black dotted lineis the critical behavior at criticality Γ ∝ T−1 with prefactor determined with a fit of theexperimental points. Picture taken from Ref. [28].

at both critical fields, are compared to the DMRG ones for the full ladder model.DMRG results already showed a very good agreement with experimental data inFig. 6.2. The new approximation is able to capture also from a quantitative pointof view the deviations from the critical behavior at the experimental temperatures.The presence of higher energy excitations (triplets "0" and "-") seems to be the mainresponsible for the bending of the curve at h ∼ hc1. The quality of the resultscoming from this approximation is confirmed once more if we go back to the brominecompound and compare against DMRG results, as it can be seen in Fig. 6.4

6.3 Summary & conclusions of the chapter

In this chapter we have focused on two compounds whose microscopic structure canbe described by a two-leg spin-1/2 ladder Hamiltonian: by applying and increasing amagnetic field to the system, two phase transitions take place at respectively h = hc1(from gapped spin liquid to Tomonaga-Luttinger liquid) and h = hc2 > hc1 (fromTomonaga-Luttinger liquid to gapped, fully polarized).

71

6. GRÜNEISEN PARAMETER AND QUANTUM PHASE TRANSITIONS INSPIN-1/2 LADDERS

−1 −0.8 −0.6 −0.4−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

log(kBT/J⊥)

log(|Γmag|J

⊥/g

µB)

DMRG (ladder), h=4.375 TFree fermions (3 species), h=4.375 TDMRG (ladder), h=1.75 TFree fermions (3 species), h=1.75 T

Γmag

=0.46⋅ T−1

Figure 6.3: |Γ(T )| renormalized by J⊥ in order to deal with unitless quantities, inlog-log scale, computed analytically and numerically for the HpipCuCl4 at both criticalfields. Solid lines are analytical results obtained within the refined approximation (freefermions, 3 species). Crosses are DMRG data for the full ladder model obtained byP. Bouillot. The black dotted line is the critical behavior at criticality Γmag ∼ 0.46T−1

with prefactor previously determined with a fit of the experimental points.

By studying the field dependence of the Grüneisen parameter we have shownanalytically the development of a divergence as the temperature is lowered, and achange of sign, around the criticality. By studying the temperature dependence, wehave shown how at low enough temperature and at criticality, results for both com-pounds collapse onto what is the expected asymptotic behavior. Close to criticalities,free fermions effective theory are able to describe in a satisfactory way the physics ofthe ladder models only at very low temperature. By keeping more terms in the strongcoupling expansion of the model, including the possibility of having higher energyexcitations thermally activated, we have managed to describe well the deviationsfrom the critical behavior at intermediate temperatures.

The study performed in this chapter has shown once more the model characterof the two "Hpip" compounds. The Grüneisen parameter can be successfully usedas a tool to detect and characterize quantum phase transitions in spin systems. Theseresult open the path to the investigation of more complicated systems, for examplethe disordered ones. Samples with randomly alternated bromine and chlorine bondscan be now investigated to understand the effects of disorder and Bose glasses in

72

6.3 Summary & conclusions of the chapter

−1 −0.8 −0.6 −0.4−1.5

−1

−0.5

0

0.5

1

log(kBT/J⊥)

log(|Γmag|J

⊥/gµB)

DMRG (ladder), h=14.4996 TFree fermions (3 species), h=14.4996 TDMRG (ladder), h=7 TFree fermions (3 species), h=7 T

Γmag

=0.46⋅ T−1

Figure 6.4: |Γ(T )| renormalized by J⊥ in order to deal with unitless quantities, inlog-log scale, computed analytically and numerically for the HpipCuBr4 at both criticalfields. Solid lines are analytical results obtained within the refined approximation (freefermions, 3 species). Crosses are DMRG data for the full ladder model. The black dottedline is the critical behavior at criticality Γmag ∼ 0.46T−1, the prefactor is the same usedin Fig. 6.3.

low-dimensional antiferromagnets.

73

6. GRÜNEISEN PARAMETER AND QUANTUM PHASE TRANSITIONS INSPIN-1/2 LADDERS

74

CHAPTER 7

General conclusions & Perspectives

In this work we have performed a study, mainly numerical, of the thermal effectsin low-dimensional spin systems. This thesis is composed by three main projects,corresponding to chapters 4-5-6. In the following we will briefly summarize theresults:

1. In chapter 4, using an MPS technique for real time dynamics at finite temper-ature, we have performed a numerical quantitative study of the temperaturedependence of the NMR 1/T1 relaxation rate for spin-1/2 chains (XXZ anddimerized). We were able to reproduce in all cases the expected behavior inthe low energy (temperature) limit and at criticality, we have shown interestingdeviations from these analytical results at intermediate temperatures. We com-puted the TLL parameters numerically using an infinite size MPS algorithm.

2. By means of the same numerical method, in chapter 5 we have investigated theexcitation spectra of the dimerized spin-1/2 chain, for different temperaturesand applied magnetic field. We have characterized and explained qualitativelythe structures seen in the three phases of the model by resorting to the singlestrong bond picture (gapped phases) or to the TLL theory (gapless phase). Wehave shown how temperature comes into play by asymmetrically broadeningthe spectra and by redistributing the intensity of the structures. We also dis-cussed the band narrowing effect for the triplon excitations, and its interplaywith the broadening of the spectrum.

3. In chapter 6, by combining analytical results, DMRG and experimental mea-surements performed on spin-1/2 two-leg ladder models under magnetic field,

75

7. GENERAL CONCLUSIONS & PERSPECTIVES

we demonstrated the validity of the Grüneisen parameter as a tool to detectand study quantum phase transitions. We discussed its field and temperaturedependence at criticality via several approximations of the true model (spin-chain mapping, strong coupling expansion). We have noticed once more thecrucial role of the temperature in experiments.

In the first two parts we have successfully demonstrated how recent developments inDMRG/MPS techinques allow for the simulation of the dynamics of one-dimensionalspin systems at finite temperature. This offers new opportunities for direct compar-isons for instance with NMR and neutron scattering experiments, since now by com-bining analytical and numerical methods it is possible to cover the whole range oftemperatures experimentally accessible with quantitative predictions. We have seenthat for temperatures lower than the energy scale of the system (which is typicallythe coupling J), the algorithm is extremely powerful: real time evolution at finitetemperature can be performed efficiently and with accuracy up to long times. Dif-ferently, in regions where temperature is comparable to J or higher, it is not easy toachieve a satisfactory resolution in frequency combined with a good precision. Inthis regime the number of retained states needed to perform efficiently the time evo-lution grows extremely fast, causing a substantial increase of the computational time(which is proportional to the cube of the number of retained states) also on powerfulmachines. This of course leaves space for future optimizations of the algorithm.

In this work we have computed the dynamics at finite temperature of purely 1Dsystems. A natural extension would be the exploration of quasi-1D systems, like forexample spin-1/2 two-leg ladders for which, as we have seen in chapters 2 and 6,excellent experimental realizations exist. This extension is already quite challengingsince it would need an enlargement of the local Hilbert space, and we also know thatthe computational time grows exponentially fast with the increase of the transversedirection. These preliminary considerations show the complexity of the scenario forthe purely 2D problem. Nevertheless, in the last years, several attempts have beenmade in this direction for the computation of static and thermodynamic quantities intwo dimensions [90]: the 2D DMRG [91], a natural extension of the MPS methodknown as Projected Entangled Pair State (PEPS) [92] or the Multiscale EntanglementRenormalization Ansatz (MERA) [93].

76

APPENDIX A

Technical aspects of chapter 4

A.1 Equality between the time integrals of onsite S+−

and S−+ correlations

In this section we show that∫ +∞

−∞dt⟨S+j (t)S−j (0)

⟩=

∫ +∞

−∞dt⟨S−j (t)S+

j (0)⟩. (A.1)

Let’s start by the Lehmann representations of the two correlations inside the integrals⟨S+j (t)S−j (0)

⟩=∑m,n

e−βEnei(En−Em)t 〈n|S+j |m〉 〈m|S

−j |n〉 , (A.2)

⟨S−j (t)S+

j (0)⟩

=∑m,n

e−βEnei(En−Em)t 〈n|S−j |m〉 〈m|S+j |n〉 =

inverting m and n... =∑m,n

e−βEmei(Em−En)t 〈n|S+j |m〉 〈m|S

−j |n〉 . (A.3)

Comparing Eqs. (A.2)-(A.3) we see that the time integral of each of the two correla-tion reduces to sums of integrals of complex exponentials. Since∫ +∞

−∞dt ei(En−Em)t =

∫ +∞

−∞dt ei(Em−En)t = 2πδ(En − Em), (A.4)

77

A. TECHNICAL ASPECTS OF CHAPTER 4

one clearly sees that the equality (A.1) is true. Now, in this work we are not per-forming integrals up to infinite time but to a finite time t0, therefore we need to showthat

e−βEn

∫ +t0

−t0dt ei(En−Em)t = e−βEm

∫ +t0

−t0dt ei(Em−En)t. (A.5)

=⇒ 2e−βEnsin((En − Em)t0)

En − Em= 2e−βEm

sin((Em − En)t0)

Em − En= (A.6)

= 2e−βEmsin((En − Em)t0)

En − Em(A.7)

In the limit t0 → ∞, the function sin((En−Em)t0)En−Em

tends to δ(En − Em). For largebut finite values of t0 this sinc function selects values of En − Em . 1/t0. Infirst approximation one sees that the two members of the identity to prove differ bymultiplicative factors of order e−β/t0 ∼ 1 in all cases considered in this work.

A.2 Derivation of the result in Eq. (4.11)We start from the result for the time ordered, onsite, +− correlations in real time.It can be obtained directly from Eq. (C.55) in Ref. [7] with the obvious substitution2K → 1

2K and K → 14K (K is the dimensionless TLL parameter):

χT+−(x=0, t) =−(παβu

)1/2K

[sinh

(πβ (−t+ iεSign(t))

)sinh

(πβ (t− iεSign(t))

)]1/4K .(A.8)

α is a small cutoff, β is the inverse temperature and u is the sound velocity (the otherTLL parameter). Using the relation connecting the retarded correlations with thetime-ordered ones, and Eq. (C.61) in Ref. [7], we get that

χR+−(x, t) = −2Θ(t)Im(χT+−(x, t)

)= −2Θ(t)

(πα

βu

) 12K

sin( π

4K

) ∣∣∣∣sinh

(π

βt

)∣∣∣∣− 12K

(A.9)

where Θ(t) is the step function. Now,

χR+−(x = 0, ω) =

∫ +∞

−∞dt eiωtχR+−(x = 0, t) =

= −2

(πα

βu

) 12K

sin( π

4K

)∫ ∞0

dt eiωt∣∣∣∣sinh

(π

βt

)∣∣∣∣− 12K

(A.10)

78

A.2 Derivation of the result in Eq. (4.11)

To solve the integral we make use of Eq. (C.65) in Ref [7] to get (with some simplealgebra)

χR+−(x = 0, ω) = − 2

usin( π

4K

)( 2π

βu

) 12K−1

B

(−iβω

2π+

1

4K, 1− 1

2K

),

(A.11)where B(. . . ) is the Euler beta function. Let’s now then recall the first line ofEq. (4.11) and introduce in it the previous result:

1

T1' limω→0− 2

βωImχR+−(x=0, ω0) =

=4

u

(2π

βu

) 12K−1

sin( π

4K

)limω→0

[1

βωIm(B

(−iβω

2π+

1

4K, 1− 1

2K

))].

(A.12)

Since we are working in the low energy limit, ω is small and one can developB(. . . , . . . ) with respect to the first argument:

B

(−iβω

2π+

1

4K, 1− 1

2K

)≈(

1 + iβω

2cot( π

4K

))·B(

1

4K, 1− 1

2K

).

(A.13)This result makes the limit in Eq. (A.12) easily computable, and we get

1

T1=

2

ucos( π

4K

)( 2π

βu

) 12K−1

B

(1

4K, 1− 1

2K

), (A.14)

which is exactly Eq. (4.11) except for an overall factor of 2Ax which has to be addedif one takes into account the proper definition of the operators S+ and S−.

79

A. TECHNICAL ASPECTS OF CHAPTER 4

80

APPENDIX B

Extrapolation Method for Spin-Lattice RelaxationRate

As discussed in Section 4.2, the numerical results for correlations are only availableup to a certain time tmax. Since in principle the time integral of these correlationsshould be performed up to∞, one needs to find a way to approximate the value ofthe extended integral and to associate an error bar to it. In order to do this we studythe behavior of these integrals as a function of 1/tmax for 1/tmax → 0. We computethe integrals for the 20 largest available values of tmax, then we perform a linear fitand we extrapolate the result for 1/tmax = 0. If the value of the integral still showsa considerable trend, we associate to the extrapolated value a one-sided errorbarcorresponding to the difference between the extrapolated value and the value of theintegral for the maximum tmax available. An example is shown in Fig. B.1 for thedimerized chain in the TLL phase.

For the case where the integral oscillates around a certain value and no clear trendis visible, we choose to associate a symmetric errorbar with semi-amplitude equal tothe distance between the extrapolated value itself and the value of the integral forthe maximum tmax available. An example is shown in Fig. B.2 for the dimerizedchain in the gapped phase. In both cases, the extracted error bars should give a (mostprobably pessimistic) estimate of the uncertainty on the value of the integral.

81

B. EXTRAPOLATION METHOD FOR SPIN-LATTICE RELAXATIONRATE

0 0.005 0.01 0.015 0.02 0.025 0.03

2.1

2.15

2.2

2.25

2.3

2.35

2.4

h/(tmaxJw)

(Jw/h)·

∫+t m

ax

−t m

axdtS−+(t)

0 10 20 30 400

0.2

0.4

tJw/h

S−+(t)

Figure B.1: Integral over time from −tmax to +tmax of the onsite S−+ correlations as afunction of ~/(tmaxJw) at h ≈ 3.02Jw & hc1, at the temperature kBT ≈ 0.0814Jw forthe dimerized model. The extrapolation is shown as a solid (red) line. The extrapolatedpoint is reported with its error bar. The inset shows the correlations as a function oftJw/~. Picture taken from Ref. [29].

0 0.005 0.01 0.015 0.02

−0.04

−0.02

0

0.02

0.04

h/(tmaxJw)

(Jw/h)·

∫+t m

ax

−t m

axS−+(t)

0 20 40 60−0.5

0

0.5

tJw/h

S−+(t)

Figure B.2: Integral over time from −tmax to +tmax of the onsite S−+ correlations as afunction of ~/(tmaxJw) at h = 0, at the temperature kBT ≈ 0.081Jw for the dimerizedmodel. The extrapolation is shown as solid (red) line. The extrapolated point is reportedwith its error bar. The inset shows the correlations as a function of tJw/~. Picture takenfrom Ref. [29].

82

APPENDIX C

Consistency Test Using the XX Model

To test the accuracy of the numerical calculations, we derived some results for theXX model under a magnetic field along the z direction:

H =J

2

∑j

(S+j S−j+1 + h.c.

)− h

∑j

Szj . (C.1)

We focused on two specific cases, h = 0 (gapless phase) and h = 5J (gapped phase),and on Szz correlations. In particular, we determine for different temperatures theratio 1/T1 for Szz correlations, which we define here as:(

1

T1

)zz

=

tmax∫0

dt Re⟨Szj (t)Szj (0)

⟩T. (C.2)

We compare our numerical results obtained using our MPS code against exact an-alytically results [7]. In the limit of an infinite-size system, the exact result for theonsite correlations at a temperature T and h = 0 is given by

⟨Szj (t)Szj (0)

⟩T

=J0 (Jt/~)

2π·

+π∫−π

dk eiλkt/~ · fk(β) −

− 1

4π2·

∣∣∣∣∣∣+π∫−π

dk eiλkt/~ · fk(β)

∣∣∣∣∣∣2

, (C.3)

83

C. CONSISTENCY TEST USING THE XX MODEL

where J0(. . . ) is the 0th-order Bessel function of the first kind, i is the imaginaryunit, λk = J cos (k), where k is the dimensionless momentum, and

fk(β) =1

1 + eβλk(C.4)

is the Fermi function, where β is the inverse temperature.

0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

kBT/J

J(1/T

1) z

z/h

h=0

MPSAnalytics

0 1 2 3 4 50

1

2

3

4

5

kBT/J

J(1/T

1) z

z/h

h=5J

MPSAnalytics

Figure C.1: (1/T1)zz as defined in Eq. (C.2), multiplied by J/~, as a function ofkBT/J , for the XX model at zero magnetic field (left) and at h = 5J (right). Theanalytical results correspond to infinite system size and tmax = 20~/J . The numericalresults are obtained for L = 100 and tmax = 20~/J . The agreement is excellent at alltemperatures. Picture taken from Ref. [29].

For h 6= 0 one obtains

⟨Szj (t)Szj (0)

⟩T

=J0 (Jt/~)

2π· eiht/~ ·

+π∫−π

dk eiλ′kt/~fk(β) −

− 1

4π2·

∣∣∣∣∣∣+π∫−π

dk eiλ′kt/~fk(β)

∣∣∣∣∣∣2

+1

4+

+1

4π2·

+π∫−π

dk fk(β)− 2π

· +π∫−π

dk fk(β). (C.5)

84

Here λ′k = J cos (k)− h and

fk(β) =1

1 + eβλ′k

. (C.6)

Simulations are performed for a chain of L = 100 spins. Onsite correlations aremeasured in the center of the chain (j = 50) to minimize boundary effects. Imagi-nary time evolutions are performed using the following parameter set: minimal trun-cation εβ = 10−20, retained states maximum 400 and temperature step kBδT =0.01J . Real time evolutions are performed using: minimal truncation εt = 10−10,a retained states maximum of 800, maximal truncated weight 10−6, and time stepδt = 0.05~/J up to tmax = 20~/J . Results of the comparison theory-numerics arereported in Fig. C.1. The agreement between the analytical and the numerical resultsis extremely good at all temperatures.

85

C. CONSISTENCY TEST USING THE XX MODEL

86

APPENDIX D

Dynamics of a single excitation in the strongdimerization limit

In the strong dimerization limit, we can restrict ourselves to the single strong bondpicture. The four-dimensional Hilbert space on a single strong bond is spanned bythe four states |s〉, |t+〉,

∣∣t0⟩ and |t−〉. If we put ourselves in a gapped regime, whichmeans h = 0 or h > hc2, the ground state of the system within this picture presentsa singlet or a triplet “+”, respectively, on each strong bond. The application of aspin operator (S+, S− or Sz) on a site induces a transition on the correspondingstrong bond to an excited state. This excitation can then propagate along the chainsince the strong bonds are not totally disconnected from each other (Jw finite). Inthe following we will do simple calculations for the two cases (h = 0 or h > hc2)which make the interpretation of the correlation plots easier in those regions.

D.1 h = 0

In absence of magnetic field the ground state of the system in the strong dimerizationlimit is made of singlets on each strong bond. The application of any of the threeoperators indicated above induces a transition from a singlet to one of the triplets.Taking into account that we always act on the first site of a strong bond we get:

S+1 |s〉 −→ − 1√

2|t+〉 ,

S−1 |s〉 −→ 1√2|t−〉 ,

Sz1 |s〉 −→ 12

∣∣t0⟩ . (D.1)

87

D. DYNAMICS OF A SINGLE EXCITATION IN THE STRONGDIMERIZATION LIMIT

Let us now consider a couple of strong bonds connected by the coupling Jw, andlet’s try to see how one of these excitations can hop from one strong bond to thenext one. We start by |ϕα〉 = |tα〉 |s〉, being α = +, 0 or −, and we apply Hw =JwSj,2Sj+1,1, where the two spin operators act respectively on the second spin ofthe first bond, and on the first spin of the second bond. What one gets is

Hw |ϕ+〉 = −Jw

4 |s〉 |t+〉+ Jw

4 |t+〉∣∣t0⟩− Jw

4

∣∣t0⟩ |t+〉 ,Hw

∣∣ϕ0⟩

= −Jw

4 |s〉∣∣t0⟩+ Jw

4 |t+〉 |t−〉 − Jw

4 |t−〉 |t+〉 ,

Hw |ϕ−〉 = −Jw

4 |s〉 |t−〉+ Jw

4

∣∣t0⟩ |t−〉 − Jw4 |t

−〉∣∣t0⟩ . (D.2)

The most interesting resulting terms are the three highlighted in bold, which expressthe hopping of the excitations from one strong bond to the next one. The otherterms represent higher energy processes and therefore will be neglected. From theseresults it can be easily seen that the spectrum of the system in all the three cases canbe approximated by the one of an isolated particle (the triplet) moving with a tightbinding Hamiltonian with hopping matrix element −Jw/4.

D.2 S+− correlations for h > hc2

At very high magnetic field the ground state of the system in the strong dimerizationlimit is made of triplets “+” on each strong bond. The application of S− inducesa transition from a triplet “+” to the state |↓↑〉, which is a linear combination of∣∣t0⟩ and |s〉 on a strong bond. For these two excitations, following the same line ofreasoning adopted above, one gets that{

Hw |t+〉 |s〉 = −Jw

4 |s〉 |t+〉+ Jw

4 |t+〉∣∣t0⟩− Jw

4

∣∣t0⟩ |t+〉 ,Hw |t+〉

∣∣t0⟩ = Jw

4

∣∣t0⟩ |t+〉+ Jw4 |s〉 |t

+〉+ Jw4 |t

+〉 |s〉 .(D.3)

The most interesting resulting terms are again those highlighted in bold, expressingthe hopping of the excitations, while the other terms can be neglected since theirenergy is too high. In this case the spectrum can be approximated by the one of twoisolated particles (the triplet “0” at high energy, the singlet at low energy) movingwith a tight binding Hamiltonian with hopping matrix element respectively Jw/4and −Jw/4.

88

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