qmc.psi - comp-phys.org

1ALPS Tutorial – PSI 08/09/2006

A Quick introduction to A Quick introduction to Quantum Monte Carlo Quantum Monte Carlo methodsmethods

Fabien Alet LPT, Univ. Paul Sabatier Toulouse

Contact : [email protected]

2F. Alet (Toulouse) – Introduction to QMC – ALPS Tutorial PSI

Quantum Monte CarloQuantum Monte CarloWhat is Quantum Monte Carlo (QMC) ?

Most general definition: “A stochastic method to solve the Schrödinger equation”

QMC comes in many different flavoursZero Temperature or

Statistics of particles : Fermions, Bosons or “Boltzmannons”

Continuum or discrete space

Finite number of particles or Thermodynamic limit

…

0≠T 2×

2×2×

3×

TodaySpin models on lattice : finite T, Bosons (spins), Discrete space, Thermodynamic limit

At least 24 different QMC communities !

PerformancesLarge systems ( thermodynamic limit, T=0) : up to spins !

Quantitative modeling of quantum magnets

Accurate description of phase transitions, Asymptotic regime (critical exponents)

≈ ≈ 610


QMC = a large family tree QMC = a large family tree « Variational Monte Carlo » family

« Diffusion Monte Carlo » familyProjector Monte Carlo, Green Function Monte Carlo, Diffusion Monte Carlo (+ Fixed Node, +

Stochastic Reconfiguration) …

(Partial) Ref. for lattice models : S. Sorella and L. Capriotti, Phys. Rev. B 61, 2599 (2000)

« Determinental Monte Carlo » familyAuxiliary Field Monte Carlo, Determinental Monte Carlo, Hirsch-Fye …

(Partial) Ref. for lattice models : F. F. Assaad, Lecture Notes, NIC series 10, 99 (2003)

« Path integral Monte Carlo » familyLattice systems (cluster algorithms)

Basic Idea : Mapping Quantum system in dimension d → Classical system in dimension d+1

Then do classical Monte Carlo on the equivalent problem


A Quick introduction to A Quick introduction to Quantum Monte Carlo Quantum Monte Carlo methodsmethodsfor for latticelattice quantum spin quantum spin modelsmodels

Derivation of configuration space : Path integral approach

Monte Carlo moves : quantum cluster algorithms

The sign problem

Practical considerations


Quantum Monte Carlo simulationsQuantum Monte Carlo simulationsNot as « easy » as classical Monte Carlo

Calculating the energy eigenvalue Ec = solving the problem

Need to find a mapping of the quantum partition function to a

classical problem

Different approachesPath integrals (time-dependent perturbation theory in imaginary time)

Stochastic Series Expansion (high temperature expansion)

Sign problem if some pc < 0 (thus try to avoid this)

Then need efficient updates for the equivalent classical problem

∑ −− ==c

EH ceeZ ββTr

∑≡= −

cc

H peZ βTr


Hamiltonian of spin ½ modelsHamiltonian of spin ½ modelsExample : XXZ model in a field

Anisotropic exchange interactions : JXY , JZ

Magnetic field h

Heisenberg model : JXY = JZ = J

Hamiltonian matrix in 2-site basis

∑∑

∑∑

−++=

−++=

+−−+

i

zi

ji

zj

zizjiji

XY

i

zi

ji

zj

ziz

yj

yi

xj

xiXYXXZ

ShSSJSSSSJ

ShSSJSSSSJH

,

,

)(2

)(

∑∑ −=i

zi

jiji ShSSJH

,

rr

{ } , , , ↓↓↓↑↑↓↑↑⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

−

−

−

+

=

hJ

JJ

JJ

hJ

H

z

zxy

xyz

z

ij

4000

042

0

024

0

0004


Trotter decompositionTrotter decompositionBasis of most QMC algorithms

Here : Generic mapping of a quantum spin system onto a classical Ising model

Not limited to special cases

Split Hamiltonian into two easily diagonalizable pieces

Obtain a decomposition of the partition function

Insert 2M sets of complete basis states

=

+

H

H1

H2

])Tr[( Tr Tr )()( 2121 MHHHHH eeeZ +Δ−+−− === τββ

)(])Tr[( 221 τττ Δ+= Δ−Δ− Oee MHH

ieiieiieiieiMii

HHM

HMM

H∑ Δ−Δ−−

Δ−Δ− ⋅⋅⋅=21

2121

,...,122312221

ττττ

)/( Mβτ =Δ

H(1) H(2) H(3) H(4)

21 HHH +=

)( 221 εεεε Oeee HHH += −−−


Example : Spin ½ Heisenberg modelExample : Spin ½ Heisenberg model

Quantum problem in d dimensions maps onto a classical problem in d+1Expand the states in the Sz eigenbasis

Effective Ising-model in d+1 dimensions with 2- and 4-sites interaction terms

Each of the matrix

elements

corresponds to a row of

shaded plaquettes and

equals the product over

those plaquettes

αi

21

2121

,...,122312221 ieiieiieiieiZ

Mii

HHM

HMM

H∑ Δ−Δ−−

Δ−Δ− ⋅⋅⋅= ττττ

jH

j iei 2,11

τΔ−+

Conservation of

magnetization :

Continuous worldlines


Weights for the spin ½ Heisenberg modelWeights for the spin ½ Heisenberg modelThe partition function becomes a sum of products of plaquette weights

The only allowed plaquette-configurations are (here h=0)

Ferromagnet (J<0) : All weights are positive

Antiferromagnet on a bipartite lattice : perform a gauge transformation on one sublattice

Frustrated antiferromagnet : we have a sign problem

)( )(

∑ ∏∑ ==C pplaquettes

pC

CwCWZ

( ) ( )+−−+±±

+−−+ +−⎯⎯⎯⎯⎯⎯ →⎯ −→+ jijii

ii

jiji SSSSJSSSSSSJ

2 )1(

2

)2/(4/ Jche J ττ ΔΔ )2/(4/ Jshe J ττ Δ−Δ4/Je τΔ−


)( )(

∏=pplaquettes

pCwCW

)2/(4/ Jche J ττ ΔΔ )2/(4/ Jshe J ττ Δ−Δ4/Je τΔ−

WorldlineWorldline approach : Summaryapproach : SummaryEach valid configuration = continuous worldlines on checkerboard

Worldline QMC = Sampling over all (important) worldline configurationsAccording to the above weight

Try to generate a new configuration from a given one

Already 2 problems arise …


Intermezzo 1 : Stochastic Series Expansion (SSE)Intermezzo 1 : Stochastic Series Expansion (SSE)

Alternative approach : Expansion in inverse temperature

Using the bond Hamiltonians

Similar to path integral approach

(Minor) difference in the treatment of diagonal terms

( )ααβ

β

α

β

∏∑ ∑∑

∑

=

∞

=

∞

=

−

−=

−==

n

ib

bbn

n

n

n

nH

i

n

Hn

Hn

eZ

1,...,0

0

)( !

)Tr(!

)Tr(

1

∑=

=),( jib

bHH

Hd(1,2)

i

54321

configuration operator

Ho(3,4)

Hd(3,4)

Ho(3,4)

1 2 3 4

Hd(1,2)

( )zj

zi

zj

ziz

dji

jijixyo

ji

SSzhSSJH

SSSSJ

H

+−=

+= +−−+

),(

),( )(2

(Sandvik, 1992)


1st solved problem : the continuous time limit1st solved problem : the continuous time limit

Systematic error due to finite value of Δτ (« Trotter error »)Need to perform an extrapolation to Δτ → 0 from simulations with different

values of Δτ (or Trotter number M)

The limit Δτ → 0 can be taken directly in the construction of the algorithm !

(Prokof’ev et al., 1996)

Number of changes

stays finite as

Different computational approach:Discrete time : store configuration at all time steps

Continuous time : store times at which configuration changes (+ initial state)

22JJMNc

βτ→

Δ=

0→Δτ

2nd solved problem solved2nd solved problem solved

Shift a kink

Insert or remove two kinks (kink-antikink pair creation process)

Problems with local updatesRestricted to canonical ensemble

No change of magnetization, particle number, winding number

Critical slowing down

Solution for classical Monte Carlo was cluster algorithms

Generalization to quantum case is possible !Loop, Directed loops, Worm algorithms


Worm algorithm : intuitive viewWorm algorithm : intuitive view

Q : How to update non-locally this configuration ?

(SSE Representation)1 2 3 4 5


1 2 3 4 5


1) Insert a worm (head and tail)


1 2 3 4 5



2) Move the worm head – Modify locally the configuration on the fly



“jump” “turn” “bounce”

3) At every “vertex” : Choose an exit



1 2 3 4 5 “straight”



1 2 3 4 5



“straight”, “jump”, “turn”, “bounce”3) At every “vertex” : Choose an exit





1 2 3 4 5






1 2 3 4 5


4) Annihilate worms





1 2 3 4 5


4) Annihilate worms

NEW CONFIGURATION


Intermezzo 2 : How to choose the worm’s moves ?Intermezzo 2 : How to choose the worm’s moves ?

Given a “vertex” (plaquette) configuration and an entrance leg, where to exit ?

Consider exit leg e, given entrance leg i at a vertex in configuration c – weight w(c)

Assign this path a probability

Sum over all paths must equal unity

Choose exit leg e with probability leading to configuration c – weight w(c)

Consider the reverted path , leading back to c

Impose Local detailed balance

),|( cieP∑ =

e

cieP 1),|(

zhJcw Z +−=

2)(

),1|1( cP ),1|2( cP ),1|3( cP ),1|4( cP

),|( cieP

),|( ceiP

)(),|()(),|( cwceiPcwcieP =

Quantum cluster algorithmsQuantum cluster algorithms

Tiny differences in how the worm exactly moves lead to different algorithmsWorm algorithm : random Metropolis move

Directed loop algorithm : locally improved « clever » move

Loop algorithm : deterministic move

Difference in RepresentationSSE : usually slighlty faster

Path Integral (PI) : good when large diagonal terms

Codes in ALPS : Which qlgorithm to choose ?looper = Loop algorithm in PI/SSE : For models with spin inversion symmetry

e.g. : Heisenberg, XY, XXZ model wihout field

dirloop-sse = Directed loops in SSE : Models without inversion symmetry (most genera

e.g. : models with field, boson models etc

worm = Worm algorithm in PI : Models with large diagonal elements

e.g. : large Ising anisotropy

All these algorithms give accurate results for large systems …


Intermezzo 3 : The sign problemIntermezzo 3 : The sign problem

In mapping of quantum to classical system

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

HHAA

ββ

p

p

iii

iiii

i

ii

Sign

SignA

ppp

pppA

p

pAA

sgn

sgn

ii

ii

i

i ≡==∑∑∑∑

∑∑

∑=i

ip pZ ||


Intermezzo 3 : The sign problemIntermezzo 3 : The sign problem

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

pi

ii

pp

eZZpp

ZSign Δ−=== ∑ βsgn1

Ne

SignNSignN

SignSign

SignSign fV

p

p

p

pp

p

Δ

=≈−

=Δ β122

)2exp( fVN Δ= β


In practice, what can I do ?In practice, what can I do ?

Simulate large samples of lattice quantum spin modelsFor non-frustrated models (Note : Diagonal (Ising) frustration is OK)

This includes most Heisenberg-like models

XY, Ising anisotropy, magnetic field, all values of S, single-ion anisotropy, …

Frustrated models are possible : but quickly sign problem arises

In all dimensions on all lattices (one can define its own specific lattice)

For all values of T (including T→0)

Which quantities are accessible ?A lot of observables : Energy, Specific heat, Susceptibilities, Structure factor,

Spin stiffness, Green functions …

In some cases : one can define its own observable

How much time ?Typically simulation time scales as Number of spins x Inverse Temperature

For more details : see hands-on session …


ThanksThanks !!

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