introduction aux méthodes de monte carlo quantique en ... · de pablo et al. (2004), swendsen et...

1Cours de Physique Théorique du SPhT/Saclay

Introduction aux méthodes de Monte CarloIntroduction aux méthodes de Monte CarloQuantique en matière condenséeQuantique en matière condensée

Fabien Alet

SPhT, CEA-Saclay

2F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 3 – 18/02/2005

LecturesLectures’’ roadmaproadmap

Quantum Monte Carlo : why and when ?

Lecture I : Introduction Classical Monte Carlo techniques in statistical physics : a brief review

Quantum Monte Carlo I : Path integral vs. series expansion approach,

sign problem, (local update, continuous time)

Lecture II : QMC : How-to’s Quantum Monte Carlo II : (local update, continuous time)

Algorithms : loop algorithm, Stochastic Series Expansion, worm

algorithm

Lecture III : Recent developments Multicanonical sampling, Quantum Wang-Landau algorithm

NP hardness of the sign problem ?

Implementation of algorithms : towards community codes


Quantum Monte Carlo simulationsQuantum Monte Carlo simulations Not as « easy » as classical Monte Carlo

Calculating the energy eigenvalue Ec = solving the problem

Find a mapping of the quantum partition function to a classical problem

Different approaches Path integrals (time-dependent perturbation theory in imaginary time)

Stochastic Series Expansion (high temperature expansion)

At second order phase transitions : critical slowing down Solved by

efficient non-local cluster updates [PREVIOUS WEEK]

First order phase transitions and long tunneling times [THIS WEEK]

Sign problem if some pc < 0 [THIS WEEK]

! ""==

c

EH ceeZ##

Tr

!"=#

c

c

HpeZ

$Tr


TodayToday’’s plans plan

Multicanonical sampling of quantum systems Classical Monte Carlo and first order phase transitions

Classical Wang-Landau algorithm

Extension to quantum systems

Applications

The sign problem Origin

Complexity classes

NP hardness of the sign problem ?

(Meron cluster Approach)

Implementation of QMC algorithms Towards community codes : The ALPS project

Free energy

Configurations

0<cp

Part I :Part I :MulticanonicalMulticanonical sampling of sampling of

quantum systemsquantum systems


Classical Monte Carlo simulationsClassical Monte Carlo simulations

We want to calculate a thermal average

Exponentially large number of configurations

→ draw a representative statistical sample by important sampling

Pick M configurations ci with probability

Calculate statistical average

Within a statistical error

Problem : we cannot calculate since we do not know Z

!! ""=

c

=Z with / ccE

c

E

ceZeAA

##

!

pci = e"#Eci /Z

!=

="M

i

ci

AM

AA

1

1

M

AA

Var!"

!

pci = e"#Eci /Z


Markov chains and Metropolis algorithmMarkov chains and Metropolis algorithm

Idea : build a Markov chain

It’s sufficient for transitions probabilities Wx,y for transition x→y to fulfill

Ergodicity : any configuration reachable from any other

Detailed balance :

Simplest algorithm due to Metropolis et al. (1953) :

Needs only relative probabilities (energy differences)

!

c1" c

2" ..." c

i" c

i+1" ...

!

"x,y #n : Wn( )

x,y$ 0

!

Wx,y

Wy,x

=py

px

!

Wx,y =min[1, py px ]

!

py px = e"# (Ey "Ex )


First order phase transitionsFirst order phase transitions

Tunneling out of meta-stable states is suppressed exponentially

T

P

gas

liquidcriticalpoint

Critical slowing down

At critical point wassolved by cluster updates

liquid

gas liquid gas

How can we tunnelout of metastable state? ??

TcLd

e/1!

"#


MulticanonicalMulticanonical sampling sampling Idea : Directly work with density of states g(E)

Acceptance rate proportionnal to inverse density of states

All energy levels visited equally well

Random walk in energy space Flat histogram in energy space no tunneling problem

With g(E), we have access to everything Free energy available

One simulation gives results for all T

Problem : we don’t know g(E) !!

)(

1

i

iEg

p = iE

i ep!"

#instead of

1)(

1)( )( i, =!="

EgEgpEP

i

EEi#

! ""=E

TkE

BBeEgTkF

/)(ln


Estimation of density of statesEstimation of density of states

Sketch of « traditional » multicanonical method 1. Start with some estimate of g(E)

2. Do a full MC simulation with the related weight, record Energy histogram H(E)

3. Calculate new estimate : g(E) H(E).g(E)

4. Reiterate 2 and 3 until “convergence”

Wang-Landau algorithm Step 2 too slow !! Modify g(E) on the fly at every single MC step

If , then energy histogram H(E) flat

Turn the argument around : For a given level of precision, we have a correct estimation

of g(E) if we force the histogram to be flat !!

Condition enforced by putting a penalty to energy level each time this level is visited

g(E) is consequently modified

When histogram flat, increase precision

Berg, Neuhaus, 1992

Wang, Landau (2001)

)(

1

i

iEg

p =


The Wang Landau algorithmThe Wang Landau algorithm Initially, g(E) is unknown

Start with g(E)=1 and adjust iteratively

Only a few modifications to usual sampling neededStart with modification factor f=e

do {

do { Metropolis updates with transition probability W[i→j]=min[1,g(Ei)/g(Ej)]

Adjust g(E) at each step : g(E)←g(E).f

Increase histogram H(E)++

} until histogram H(E) is “flat” (e.g. 20%)

decrease f (e.g. f ← f1/2)

reset H(E)

} until f-1≈ 10-8

Comments : Initially : multiplicative changes with large f allow rapid crude convergence

Finally : small f means fine structure of g(E) can be mapped out


Example of application of Wang-Landau algorithmExample of application of Wang-Landau algorithm

10 states Potts model in 2d First order phase transition

Tc=0.701

cBTkE

c eEgTEP/

)(),(!

=


Applications and extensionsApplications and extensions

First and second order phase transitions and disorder (spin systems) Wang and Landau (2001)

Systems with rough energy landscape Molecular systems (Calvo, 2002)

Polymer films (Jan et al., 2002)

Protein folding (Rathore et al., 2002)

Scaling analyses for spin glasses (Dayal et al., 2004) Continuum simulations

Yan et al. (2002)

Shell et al. (2002)

And many others …

Improved sampling and optimized ensembles Zhou and Bhatt (2003)

Schulz et al. (2002)

Yamaguchi and Kawashima (2002)

…


Limitations of Wang-Landau algorithmLimitations of Wang-Landau algorithm

Convergence issues at very small f de Pablo et al. (2004), Swendsen et al. (2004)

Not a perfect energy random walk … Dayal et al. (2004), Trebst et al. (2004)

Today’s consensus Convergence problems when asking for extremely precise density of states

However, perfect method to obtain a “very good” density of states, to be fed

to “traditional” multicanonical sampling


Quantum version of Wang-Landau algorithm (I)Quantum version of Wang-Landau algorithm (I) Density of states not accessible directly in quantum case

Classical

Quantum

Quantum version based on SSE

Idea : Make a flat histogram in the expansion order n

Acceptance rate proportional to 1/g(n)

! ! ""==

c E

TkETkE BBc eEgeZ//

)(

!

Z = Tre"#H

!

Z =" n

n!#

b1 ,...,b$( )

%#

% (&Hbi)

i=1

$

' #n

%

(($ & n)!" n

$!#

b1 ,...,b$( )

%#

% (&Hbi)

i=1

$

' #n= 0

$

%

= " ng(n)

n= 0

$

%


Quantum version of Wang-Landau algorithm (II)Quantum version of Wang-Landau algorithm (II) “Diagonal Update” : Walk through operator string

Propose to insert diagonal operators instead of Identity operators

Propose to remove diagonal operators

Changes the expansion order

Small change in acceptance rates for diagonal updates

Loop/Worm/Directed loop update does not change n and is thus unchanged !

Cutoff Λ limits temperatures to β < Λ / E0

!

P[1" H(i, j )

d] =min 1,

#Nbonds $ H(i, j )

d $

% & n

'

( )

*

+ ,

!

P[H(i, j )

d "1] =min 1,# $ n +1

%Nbonds & H(i, j )

d &

'

( ) )

*

+ , ,

Hd(1,2)

index

54321

configuration operator

Ho (3,4)1

Ho (3,4)

1 2 3 4Hd

(1,2)1

1

!!

"

#

$$

%

&

+'(=)

)1(

)(,1min]1[

),(

),(ng

ng

n

HNHP

d

jibondsd

ji

**

!!

"

#

$$

%

&

'

+'(=)

)1(

)(1,1min]1[

),(

),(ng

ng

HN

nHP

d

jibonds

d

ji**


A first testA first test L=10 sites Heisenberg chain with Λ=250

Free energy and entropy available

One simulations for all temperatures

Clear limit of reliability T>0.05J due to cutoff

1

10

1

+

=

! "=i

i

iSSJHvv


Thermal phase transitionsThermal phase transitions

3D Heisenberg antiferromagnet Second order phase transition at Tc=0.947 J

Parallelization via splitting of the n-range among several CPUs

j

ji

i SSJHvv

,

!= "


Speedup at first order phase transitionsSpeedup at first order phase transitions

Greatly reduced tunneling times at free energy barriers 2D Hardcore bosons

!

H = "t ai†a j + h.c.( )

i, j

# +V2

nin j

i, j

#

3/V

88

2=

!

t


Perturbation expansion and Wang-Landau algorithmPerturbation expansion and Wang-Landau algorithm

Quantum Phase Transition

Expansion in the coupling constant

At fixed temperature

Same procedure as thermal Wang-Landau

VHH !+=0

!!"

=

"

=

=##=00

0 )()(!

$

$$$%

$n

n

n

nn

ngVHTrn

Z

!

P[1" H(i, j )

d ] =min 1,#Nbonds $ H(i, j )

d $

% & n

'

( )

*

+ , Wang&Landau- " - - - - min 1,

#Nbonds $ H(i, j )

d $

% & n

g(n.)

g(n. + /n.)

'

( )

*

+ ,

!

P[H(i, j )

d "1] =min 1,# $ n +1

%Nbonds & H(i, j )

d &

'

( ) )

*

+ , ,

Wang-Landau- " - - - - min 1,# $ n +1

%Nbonds & H(i, j )

d &

g(n.)

g(n. $ /n.)

'

( ) )

*

+ , ,


Quantum phase transitionQuantum phase transition

Bilayer Heisenberg antiferromagnet Single simulation gives results for all values of λ=J/J’

Quantum phase transition at λ=0.396

Parallelization as in thermal case!!! "+"=

= >< i

ii

p ji

pjpi SSJSSJH2,1,

2

1 ,

,, ' vvvv


Partial SummaryPartial Summary

Extension of multicanonical sampling / Wang-Landau algorithm to quantum

systems

Stochastically evaluate series expansion coefficients High temperatures series

Perturbation series

Features Flat histogram in the expansion order

Allows calculation of free energy

Like classical systems, allows tunneling through free energy barriers

Like classical systems, Wang-Landau algorithm has some limitations ...

!"

=

=0

)(n

nngZ #

)(0

!

!

!! ngZn

n"#

=

=


The sign problem and the Wang-Landau algorithmThe sign problem and the Wang-Landau algorithm

Idea : Perform simulations on unfrustrated systems, then insert factors of (-1)n

into the expansion series

Unfortunately … Does not work, even without sign problem !

Precision and cancellation issues …

Sign problem : related to cancellation of large positive and negative numbers …

n

n

ngZ !"= )(

n

n

nngZ !" #= )()1(

10 sites chain AF→F

Part II :Part II :The sign problemThe sign problem


The The negative signnegative sign problem problem

In mapping of quantum to classical system

« Sign problem » if some of the pi<0 Cannot interpret pi as probabilities

Appears in simulation of fermions and frustrated magnets

“Way out” : Perform simulations using |pi| and measure the sign :

Sampling

[ ][ ] !

!=

"

"=

i

i

)exp(Tr

)exp(Tr

i

ii

p

pA

H

HAA

#

#

p

p

iii

iiii

i

ii

Sign

SignA

ppp

pppA

p

pA

A

sgn

sgn

ii

ii

i

i !==""

""

"

"

!=i

ip pZ ||


The negative sign The negative sign problemproblem

The average sign becomes very small

Both in system size and inverse temperature

This is the origin of the sign problem !

The error of the sign:

Need of the order measurements for sufficient accuracy

Similar problem occurs for the observables

Exponential growth ! Impossible to treat large systems or low temperatures

fV

p

i

i

i

p

pe

Z

Zpp

ZSign !"

=== # $sgn

1

N

e

SignNSignN

SignSign

Sign

Sign fV

p

p

p

pp

p

!

="

#

=! $1

22

)2exp( fVN != "


Is the sign problem exponentially hard ?Is the sign problem exponentially hard ?

The sign problem is basis-dependent Diagonalize the Hamiltonian matrix

All weights are positive : no sign problem in the eigenbasis

But this is an exponentially hard problem since dim(H)=cN ! Good news : the sign problem is basis-dependent ! (see also Nakamura, 1998)

But: the sign problem is still not solved Despite decades of attempts

Reminiscent of the NP-hard problems No proof that they are exponentially hard

No polynomial solution either

!

H i = "ii

!

A = Tr Aexp("#H)[ ] Tr exp("#H)[ ] = i Aii exp("#$

i)

i

% exp("#$i)

i

%


Complexity of decision problemsComplexity of decision problems

Some problems are harder that others

Complexity class P

• Can be solved in polynomial time on a Turing (deterministic) machine

• Eulerian circuit problem

• Minimum spanning tree (decision version)

Complexity class NP

• Polynomial complexity using non-deterministic algorithms

• Hamiltonian circle problem

• Traveling salesman problem (decision version)

See introduction by Mertens, 2002


The complexity class PThe complexity class P

The Eulerian circuit problem Seven bridges in Konigsberg crossed the river Pregel

Can we do a roundtrip by crossing each bridge exactly once ?

Is there a closed walk on the graph going through each edge exactly once ?

Looks like an extensive task by testing all possible paths

Euler : Desired path exits only if the coordination of each vertex is even

This is of order O(N2)

Concerning Konigsberg : NO !


The complexity class NPThe complexity class NP The Hamiltonian cycle problem

Hamilton’s Icosian game

Is there a closed walk on the graph going through each vertex exactly once ?

Looks like an extensive task by testing all possible paths

No polynomial algorithm is known

Nor a proof that it cannot be constructed

Concerning the Icosian game : YES !


Non-deterministic algorithmsNon-deterministic algorithms

Only a theoretical concept

Like ordinary algorithms, but can use additional instruction goto both label 1, label 2

Parallel branching of the computation

Performing an exponential number of

computations is in polynomial time

Definition of NP : A decision problem is in the class NP, iif

it can be solved in polynomial time by a

non-deterministic algorithm

Obviously Just don’t use the goto both instruction

NPP !


Complexity of decision problemsComplexity of decision problems

Some problems are harder that others

Complexity class P

• Can be solved in polynomial time on a Turing machine

• Eulerian circuit problem

• Minimum spanning tree (decision version)

Complexity class NP

• Polynomial complexity using non-deterministic algorithms

• Hamiltonian circle problem

• Traveling salesman problem (decision version)

Are there relations between these different problems ?

Often can use a solution for one problem to solve another


NP-hardness and NP-completenessNP-hardness and NP-completeness

Polynomial reduction Two problems P and Q

: there is a polynomial algorithm for Q, provided there is one for P

Typical proof : Use the algorithm for P as a subroutine in an algorithm for Q

Many problems have been reduced to other problems

NP-hardness A problem P is NP-hard if

NP-completeness A problem P is NP-complete

If P is NP-hard and

Lots of problems in NP were shown to

be NP-complete

PQ !

PQNPQ !"# :

NPP!


The P versus NP problemThe P versus NP problem

Hundreds of importante NP-complete problems in computer science Despite decades of research no polynomial time algorithm has been found

Exponential complexity has not been proven either

The P versus NP problem Is P=NP or is P≠NP ?

1 million US$ for proving

either P=NP or P≠NP

(millenium challenges of the

Clay Math foundation)

The situation is similar to the sign problem Despite decades of research no general polynomial

time algorithm has been found

Exponential complexity has not been proven either

?


The sign problem is NP-hardThe sign problem is NP-hard according to M. Troyer and U.-J. Wiese, cond-mat/0408370

Use the following theorem A problem P is NP-hard if

Q is the 3d classical spin glass problem

PQNPQQ !"#$ complete :

Getting the ground state of the 3d Ising spin glass is NP-hard

The NP-complete question is : « Is there a configuration with Energy ≤ E0 ? »

!

H = " Jij# j# j

i, j

$ with Jij = 0,±1Barahona, 1982


Frustration and Monte CarloFrustration and Monte Carlo

Frustration and Quantum Monte Carlo Negative probabilities for a worldline configuration

Due to exchange term

-Jdt

-Jdt-Jdt

Negative weight (-Jdt)3

Frustration and classical Monte Carlo Frustration leads to NP-hardness of Monte Carlo

In Monte Carlo : exponentially long tunneling and autocorrelation times

!

c1" c

2" ..." c

i" c

i+1" ...

!

"A = A # A( )2

=Var A

M(1+ 2$

A)

??


Solutions of the sign problemSolutions of the sign problem

Condition 2 : The sampling with respect to | pi | scales polynomially

i.e. The “bosonic” problem must be easy !

Condition 1 : Quantum system with a sign problem (some pi < 0)

[ ] [ ] !!=""=ii

)exp(Tr)exp(Tr iii ppAHHAA ##

mnN !" 2

TimeCPU#$

A solution to the sign problem is defined as an algorithm that can calculate

the average (of say, Energy) with respect to pi also in polynomial time


Solving a NP-hard problem by QMCSolving a NP-hard problem by QMC

View it as a quantum problem in basis where H is not diagonal

The randomness ends up in the sign of off diagonal matrix elements

Ignoring the sign gives the ferromagnet and we can use loop algorithm which scales

polynomially

A solution to the sign problem would answer the spin glass NP-complete question (just

calculate <E> at low enough temperatures)

The sign problem causes NP-hardness

Take 3D Ising spin glass

A solution to this sign problem (and thus a general solution of the sign

problem) would solve the NP-complete problems (and prove NP=P)

!

H = Jij" j" j

i, j

# with Jij = 0,±1

!

H(SG )

= Jij"xj" x

j

i, j

# with Jij = 0,±1

!

H(FM )

= " # xj# x

j

i, j

$


How can we deal with the sign problem ?How can we deal with the sign problem ?

The sign problem is NP-hard (worst-case complexity) A general solution is almost certainly impossible

What can we do ? Simulate models without a sign problem

• Non-frustrated quantum magnets

• Bosonic models (atomic BEC condensates)

• (Some) fermionic models in 1d

Brute force approach• Live with the exponential scaling of the sign problem and stay on small lattices

Other exact algorithms• DMRG, exact diagonalization, series expansion … might be better

Special solutions for certain models• Meron-Cluster approach

• Special choice of basis

The The meronmeron approach approach Works for specific models and parameter regimes

Uses the loop algorithm to eliminate the sign problem

Flipping a single loop can change the sign of a configuration Such loops are called “merons”

Need specific structure of the loops Must be able to flip them independently from each other

The magnitude of the weight is not affected by the meron flip All contributions to the partition function with merons then cancel

The subspace of zero merons is positive definite Can be sampled without a sign problem

Sign = -1, |W(C)| Sign = +1, |W(C’)|=|W(C)|

MeronChandrasekharan and

Wiese, 1999


MeronMeron solution for frustrated magnets (I) solution for frustrated magnets (I)

Works only for the « semifrustrated » Heisenberg model

Square lattice with next nearest neighbour bonds

Frustrated because of exchange terms

Sign in SSE :

There exists a deterministic update (loop algorithm) : always “jump”

Zero-Meron sector is positive definite

Sign problem disappears in another representation (y-Basis)

Employ improved estimators Sum over all possible loop configurations

0 ,)(,

>!+=" ij

z

j

z

i

ji

ij

y

j

y

i

x

j

x

iij JSSJSSSSJH

p

p

Sign

ASignA = !

=

=

LN

L

l

lNCSign

2

1

)sgn(2

1

loffdiagonanCSign )1()( !=


MeronMeron solution for frustrated magnets (II) solution for frustrated magnets (II)

Improved estimators

Sign :

Energy :

Uniform susceptibility :

Need to sample both the 0- and 2-meron sector 2-meron sector dominates, thus decreases <Sign>

Reweighting the 0-meron sector helps• Reduce acceptance of leaving the 0-meron sector

0,MnSign !=

0,

0,1

M

M

n

nn

E

!

!

"#=

0,

22,210,

22

1

M

MM

LN

n

nMMnl lmmm

!

!!

"

+

=

# =

0-meron sector

2-merons sector


MeronMeron solution for frustrated magnets (III) solution for frustrated magnets (III)

The sign

Scaling reduced from

exponential to algebraic

By changing the quantization axis to the

y-direction, we can escape the sign

problem from the very beginning

0- and 2-meron sector

Unrestricted

The error of the susceptibility

Without reweighting

With optimal reweighting

Henelius and Sandvik, 2002

Part III :Part III :Implementation ofImplementation ofQMC algorithmsQMC algorithms


Implementation of algorithmsImplementation of algorithms

Condensed matter simulations : The great status quo No “community codes” available

• Individual code for each project

• Model specific implementations

Growing complexity of methods

• Weeks to months of software development

Inputs/Outputs in non-portable format

Specialist’s work

The ALPS project (Algorithms and Libraries for Physics Simulations) International project for open source software for quantum lattice models

Generic implementation, “advanced” technologies (C++, MPI, XML…)

ALPS applications = collection of modern algorithms such as DMRG,

Exact Diagonalization, Quantum Monte Carlo (loop/worm/SSE algorithms)

Easy to use, “black box” codes

http://alps.comp-phys.org


Simulations with ALPS (I)Simulations with ALPS (I)


Simulations with ALPS (II)Simulations with ALPS (II)

Typical output output.xml

All QMC algorithms have already been implemented ! Loop algorithm

Worm Algorithm

Directed Loops

Quantum Wang Landau

Check by yourself ! http://alps.comp-phys.org


ReferencesReferences Wang-Landau algorithm

F. Wang and D.P. Landau, Phys. Rev. Lett. 86, 2050 (2001); Phys. Rev. E 64,

056101 (2001)

Quantum version : M. Troyer, S. Wessel and F.A., Phys. Rev. Lett. 90, 120201 (2003)

Sign Problem Introduction to complexity classes : S. Mertens, CISE 4, 31 (2002) [also available as

cond-mat/0012185]

NP hardness of sign problem : M. Troyer and U.-J. Wiese, cond-mat/0408370

Meron-cluster approach : S. Chandrasekharan and U.-J. Wiese, Phys. Rev. Lett. 83,

3116 (1999); P. Henelius and A.W. Sandvik, Phys. Rev. B 62, 1102 (2000)

Implementation of QMC algorithms ALPS project : F.A. et al., cond-mat/0410407; http://alps.comp-phys.org


Summary of these lecturesSummary of these lectures

(Path integral) Quantum Monte Carlo first maps quantum systems to

classical ones Path integrals

Stochastic Series Expansion (SSE)

All classical algorithms can be used for quantum systems Local updates

Cluster updates

Multicanonical / Wang-Landau algorithms

Additional algorithms available Worm algorithm

Directed-Loop algorithm

Can simulate quantum systems as accurately as classical ones As long as there is no sign problem

Merci !Merci !


Speedup at first order phase transitionsSpeedup at first order phase transitions

Greatly reduced tunneling times at free energy barriers 2D Hardcore bosons

!

H = "t ai†a j + h.c.( )

i, j

# +V2

nin j

i, j

#

Stripe order3/V

88

2=

!

t 3/V

88

2=

!

t

G. Schmidt

introduction aux méthodes de monte carlo quantique en ... · de pablo et al. (2004), swendsen et...

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