Introduction aux méthodes de Monte Carlo Quantique en matière ...

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

Acknowledgments : Matthias Troyer (ETH Zürich), Stefan Wessel (Univ. Stuttgart)

2F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005

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 completeness of the sign problem ?

Implementation of algorithms : towards community codes


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 completeness of the sign problem ?

Implementation of algorithms : towards community codes


Introduction : Phase transitions in quantum systemsIntroduction : Phase transitions in quantum systems

Quantum effects induce new phases and phase transitions Metal, insulator

Superconductivity, Superfluidity

Quantum (spin) liquids

Fractionalized phases

Questions What phases are possible ?

Exotic phases with unusual properties ?

What drives the phase transitions ?

Universality classes ?

Quantitative predictions on properties of

quantum systems ?

Numerical simulations are needed Challenges similar to classical simulations

high-temperaturesuperconductors

doping

T

Fermi liquid

Non-Fermi liquid

Née

l ord

er

superconduct ivity

strange metal

pseudo gap


What can modern QMC algorithms do ?What can modern QMC algorithms do ? Quantum Monte Carlo algorithms allow

Accurate simulation of phase transitions

Reach asymptotic regime : accurate estimates of critical exponents, check of scaling

predictions

Quantitative modeling of quantum magnets and bosonic systems

Local updates (before 1994) 200 spins, T=0.1 J

Cluster algorithms (after 1995) 2D quantum phase transitions : 20.000 spins at T=0.005 J

2D square lattice : 1.000.000 spins at T=0.2 J

3D antiferromagnet : 16.000.000 spins at T = J

Extended ensembles methods (2003) Quantum Wang-Landau, parallel tempering

Allows efficient simulations of 1st order quantum phase transitions

Determination of the free energy of a quantum system

!=ji

ji SSJH,

.

Ex. : Quantum spins ininteraction


Example 1 : Quantum phase transitionsExample 1 : Quantum phase transitions

Bilayer antiferromagnet

J >> J⊥ : long range orderJ << J⊥: spin gap, no long range order

Quantum phase transition at J⊥ / J ≈ 2.524(2)Spin gap vanishes

Magnetic order vanishesUniversal properties

!!! "+"= #

= >< i

ii

p ji

pjpi SSJSSJH2,1,

2

1 ,

,, vvvv


Example 1 : Critical exponentsExample 1 : Critical exponents

2D quantum phase transitions in a quantum Heisenberg antiferromagnet

Simulations of 20.000 spins at low temperatures (Troyer et al., 1997)

Consistent with classical 3D Heisenberg model exponents

Can do quantum simulations with the same accuracy as classical Sometimes even better !!

Model ! " # z

QMC results no assumption

0.345 ±

0.025 0.685 ±

0.035 0.015 ±

0.020 1.018 ±

0.02

3D classical Heisenberg

0.3639 ±

0.0035 0.7048 ±

0.0030 0.034 ±

0.005

Mean field 1/2 1 0

0.01

0.1

0.01 0.1

Magnetization

Spin stiffnes

! = 0.345 ± 0.021

z" = 0.695 ± 0.032

# =(J0/J

1)-((J

0/J

1)c


Example 2 : Trapped atoms on optical latticesExample 2 : Trapped atoms on optical lattices

Bose-Hubbard model Both softcore and harcore

Harmonic trapping potential

Realistic modelling of 1D, 2D, 3D traps

Density profiles, Momentum distribution

Wessel et al. (2004)

U t

Lattice depth

7.6/ =tU 0.25/ =tU


Numerical methods for strongly correlated systemsNumerical methods for strongly correlated systems Exact diagonalisation

Pure gold : Allows to calculate everything for all models !

Limited to very small systems : at most 40 sites

Series Expansion High temperature, or Zero temperature in the thermodynamic limit

Can treat most models

Breaks down at phase transitions, Convergence issues

Density Matrix Renormalization Group (DMRG) Almost all models

Limited to 1d ! Difficulties with disorder

Very exciting proposals in 2004 : Time-dependent DMRG (Dynamics), DMRG in 2d ??

Quantum Monte Carlo Very large systems, any dimension, Finite temperature (T=0 essentially possible)

Sign problem !! : Efficient simulations only for spins (non frustrated) and bosons

Fermions : (mainly) only in 1d !


““Quantum Monte CarloQuantum Monte Carlo”” = zoo of methods = zoo of methods « Variational Monte Carlo » family

« Diffusion Monte Carlo » family Projector 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 » family Auxiliary Field Monte Carlo, Determinental Monte Carlo, Hirsch-Fye …

(Partial) Ref. for lattice models : F. F. Assaad, Lecture Notes (see me !)

« Path integral Monte Carlo » family Continuum (Ceperley …) or lattice 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

Q : What do you / we know about classical Monte Carlo ?

Part I :Part I :Classical Monte CarloClassical Monte Carlo


Classical Monte Carlo IClassical Monte Carlo I

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) These are easily calculated for small changes

!

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 )


Metropolis algorithmMetropolis algorithm


How does it work for the How does it work for the IsingIsing model ? model ?

Single spin flip Metropolis algorithm : Start with a random configuration c

Repeat the following many times : Randomly pick a spin

Propose to flip that single spin, leading to a new configuration c’

Calculate the energy difference ΔE=E[c']-E[c]

If ΔE<0, the next configuration is c’

If ΔE<0, accept c’ with a probability exp(-βΔE), otherwise keep c

Measure all quantities of interest

This algorithm is ergodic, it fulfills detailed balance (Metropolis update)

Before taking measurements, need to equilibrate (thermalization)

Q : How long should be the Markov chain ? (= What about errors ?)

E=3J-J=2J E’=J-3J=-2JΔE=-4J


Monte Carlo error analysis (I)Monte Carlo error analysis (I)

The simple formulas

are valid only for independent samples

The Metropolis algorithm gives us correlated samples ! The number of independent samples is reduced

where the (integrated) autocorrelation time is

( )22

1

1Var ,Var

1

AAM

MAMAA

AM

AA

M

i

i

!!

=="

=# $=

( ) MAAAVar21 !+="

22

1

2

AA

AAA

t

iti

A

!

!

=

"#

=

+

$


Digression : Autocorrelation and binning analysisDigression : Autocorrelation and binning analysis Take averages of consecutive measurements

Averages become less correlated Simple error estimates converge to real error

In practice, take enough bins

Other statistical analysis methods : bootstrap, jackknife …

!

Ai

(l ) =1

2A2i"1

( l"1) + A2i

l( )A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16

A1(1) A2

(1) A3(1) A4

(1) A5(1) A6

(1) A7(1) A8

(1)

A1(2) A2

(2) A3(2) A4

(2)

A1(3) A2

(3)

!!"

#$$%

&'=

+=())*)=(

+*

+*

1Var

Var2

2

1lim

Var)21(Var

)0(

)(

)()()(

A

A

MAAMAA

ll

lA

A

llll

,

,

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0 2 4 6 8 10

L = 4L = 48

!(l)

binning level l

100)final(!M


Why / how does it fail for the Why / how does it fail for the IsingIsing model ? model ?

Autocorrelation times diverge at criticality : CRITICAL SLOWING DOWN

Local spin flips : Dynamical exponent z ≈ 2 ⇒ effort increased by L2

Insufficient sampling leads to large statistical errors Cannot study large systems / critical properties

Advantage of Monte Carlo simulations

Can change the dynamics !

Dynamical exponent is non-universal

Need improved dynamics that change the system on length scale ξ

→ Cluster algorithms [ Swendsen-Wang (1987), Wolff (1989) ]

z

AL)],[min(!" #

A

dL !.Time CPU "


Cluster algorithm : Intuitive viewCluster algorithm : Intuitive view

Ask for each spin : “do we want to flip it against its neighbour ?” Antiparallel : yes (gain energy)

Parallel : costs energy Accept with (introduce a domain wall)

Otherwise : also flip neighbour ! Add neighbour to the cluster with

Repeat for all flipped spins → cluster updates

)2exp( JP !"=

)2exp(1 JP !""=

?

??

√?

√?√

???

√

√

?

?

√

√

√

√ √

√√

√

√ ?

√

√

√

√ √

√√

√

√√

√

√

√

√

√ √

√√

√

√

?

?

√

√

√

√

√

√ √

√√

√

√

√

√

√

√

√

√

√

√ √

√√

√

√

√

√


Cluster algorithm : Cooking recipeCluster algorithm : Cooking recipe Multiple cluster (Swendsen-Wang)

For all bonds in the system, assign “connected” or “disconnected” labels Bonds between antiparallel spins are always “disconnected”

Bonds between parallel spins are “connected” with a probability 1-exp(-2βJ)

Identify connected clusters of spins

Flip each cluster independently with a probability ½

Drastic decrease of autocorrelations : No critical slowing down !

Single cluster (Wolff)

Start from a random site

Construct the cluster from there

Flip all spins in this cluster

Wolff algorithm simpler / more efficient Picks large clusters with larger probability

Dynamical critical exponent z≈0


Cluster Algorithm : formal view (I)Cluster Algorithm : formal view (I)

Extension of phase space From configurations C to configurations + graphs (C+G)

For each configuration C, assign a graph G with probability

With G fixed, go to a new configuration C’ with probability

!! !! ===GC C G

GCWCWGCWCWZ ),()( : rule Sum with ),()(

)(

),(),(

CW

GCWGCp =

]),'[],([ GCGCp !

11 ),( ),( ++ !!!!iiiiCGCGGCC

1. Pick a graph G

2. Discard configuration

3. Pick a new configuration

4. Discard graph

NB : Same framework for the quantum case


Cluster Algorithm : formal view (II)Cluster Algorithm : formal view (II) Good choices

Weight W(C,G) of allowed graph G independent of configuration C

From a graph, pick any allowed new configuration

!

W (C,G) = "(C,G)V (G) where "(C,G) =1 graph G allowed for C

0 otherwise

# $ %

CNGCGCp

1]),'[],([ =!

Detailed balance

)],[(

)],'[(

)(),(

)(),'(1

N/1

N/1

)],(),'[(

)],'(),[(

C

C

GCP

GCP

GVGC

GVGC

GCGCP

GCGCP=

!

!===

"

"

Weights are usually product of local (bond) weights

!=b

bcwCW )()( !! "==

b

bbb

b

bb gvgcgcwGCW )(),(),(),(


Cluster Algorithm : formal view (III)Cluster Algorithm : formal view (III) Graphs for the Ising model

Local bond configurations

Graphs are simply “connected” or “disconnected” bonds

{ }!""!!!""# ,,,bc

{ } , !bg

Graphs weights for the Ising model

bc !! !" !!!" )( bgv

) ,(bc!

) ,(bc!

)(bcw

Je! J

e!J

e!" J

e!"

if=0 otherwise

And for : Configuration must be allowed ⇒ connected spins must be parallel

⇒ connected spins flipped as one cluster!

G " (Ci+1

,G) " Ci+1

)(

),(),(

CW

GCWGCp =


Cluster algorithms : variantsCluster algorithms : variants

Improved estimators : Same average but smaller variance Idea : Measure in all equally probable 2Nc configurations before flip

E.g. : Spin – Spin correlation function

Imagine <A> small (large distance …)

Unimproved estimator

Improved estimator

Clusters are physical

!"#

==otherwise 0,

cluster same theone are and if ,1 jiA jiimp $$

Generalization to other models possible Potts, O(N), field, …

11222!"=" AAA

122

2

. <<!"=" impimpimpimp AAAAAimp

Part II :Part II :Quantum Monte Carlo IQuantum Monte Carlo I


Quantum Monte Carlo simulationsQuantum Monte Carlo simulations Not 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 approaches Path 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) [LAST WEEK]

Then need efficient updates for the equivalent classical problem

[NEXT WEEK]

! ""==

c

EH ceeZ##

Tr

!"=#

c

c

HpeZ

$Tr



1. PATH INTEGRAL APPROACH


Hamiltonian of spin ½ modelsHamiltonian of spin ½ models Example : 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

z

i

ji

z

j

z

izjijiXY

i

z

i

ji

z

j

z

iz

y

j

y

i

x

j

x

iXYXXZ

ShSSJSSSSJ

ShSSJSSSSJH

,

,

)(2

)(

!! "=i

z

i

ji

ji ShSSJH,

rr

{ } , , , !!!""!""!!!!!!!!!

"

#

$$$$$$$$$

%

&

'

'

'

+

=

hJ

JJ

JJ

hJ

H

z

zxy

xyz

z

ij

4000

042

0

024

0

0004


The The worldlineworldline approach approach

Representation based on mapping of quantum spin-1/2 system onto a

classical Ising model

Traditionally employing local updates using Metropolis sampling [TODAY]

Continuous time version can be constructed (Prokof’ev et al., 1996) [TODAY]

Cluster updates using the loop algorithm (Evertz et al., 1993) [NEXT WEEK]


Trotter-Suzuki decompositionTrotter-Suzuki decomposition Basis of most (path integral) 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+!"+""

===#$$

)(])Tr[( 221 !!!

"+="#"#

OeeMHH

ieiieiieiiei

Mii

HH

M

H

MM

H! "#"#

#

"#"# $$$=21

2121

,...,

122312221

%%%%

)/( M!" =#

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

21HHH +=

)( 221 !!!!

OeeeHHH+=

"""


Example : Spin ½ Heisenberg modelExample : Spin ½ Heisenberg model

Quantum problem in d dimensions maps onto a classical problem in d+1 Expand 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

!i

21

2121

,...,

122312221 ieiieiieiieiZ

Mii

HH

M

H

MM

H! "#"#

#

"#"# $$$=%%%%

j

H

j iei 2,1

1

!"#

+

Conservation of

magnetization :

Continuous worldlines


Weights for the spin ½ Heisenberg modelWeights for the spin ½ Heisenberg model The 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

p

C

CwCWZ

( ) ( )+!!+±±

+!!+ +!"""""" #"!#

+ jijii

i

ijiji SSSS

JSSSSSS

J

2

)1(

2


WorldlineWorldline approach : Summary approach : Summary Each valid configuration = continuous worldlines on checkerboard

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

Try to generate a new configuration from a given one

)( )(

!=pplaquettes

pCwCW


Local updates (I)Local updates (I)

Move the world lines locally using Metropolis Probabilities given by the resulting plaquette weights

Consider the limit of small Δτ

Insert or remove two kinks

Shift a kink1=w 2)2/ ( Jw !"=

])2/ /(1,1min[

])2/ (,1min[

2

2

JP

JP

!

!

"=

"=

#

$

2/ Jw !"=

1== !" PP


Local updates (II)Local updates (II)

Problems with local updates Restricted 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 algorithm, Worm algorithms [NEXT WEEK]

Improvement : “Directed loops” [NEXT WEEK]


The continuous time limit : Intuitive viewThe continuous time limit : Intuitive view Systematic error due to finite value of Δτ (« Trotter error »)

Need to perform an extrapolation to Δτ → 0 from simulations with differentvalues 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)

22

JJMN

c

!"#

$=

0!"#


Path integral representation (I)Path integral representation (I) Based on the perturbation expansion of the path integral

Continuous time representation

Discrete local objects (kinks, changes in worldline configuration)

Local update of kinks using Metropolis

Improved update scheme using « Worm update » (Prokof’ev et al., 1997)[NEXT WEEK]


Path integral representation (II)Path integral representation (II) Perturbation expansion in interaction representation :

Each term represented by a worldline configuration

...)))()()(1(Tr(

)Tr()Tr(

)(,,

1

0 0

212

0

)(

,,

00

2

0

00

++!=

"==

+=!=+=

" ""

## #

!

!!!

><><

$$$$$$% $%

%

$$%%

%

VVddVdeZ

eeeZ

SSSSJVhSSSJHVHH

H

VdHH

y

j

y

i

ji

x

j

x

ixy

ji i

z

i

z

j

z

iz

T

! !2

!1


Local update in continuous timeLocal update in continuous time

Shift a kink

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

Vanishing acceptance rate :

Solution : Integrate over all possible insertions in a finite time window

1=P 0)2/ ( 2!"= JP #

0])2/ (,1min[ 2!"=! JP #

0 8

)2/(22

12

2

0 1

!"

#= $ $" "

#

JddJP %%

%



2. SERIES EXPANSION APPROACH


Stochastic Series Expansion (SSE) approachStochastic Series Expansion (SSE) approach

Based on a high temperature series expansion of the partition function

Original formulation based on local updates using Metropolis ≈ local

updates in discrete time path integral representation [NEXT WEEK]

Cluster updates (Sandvik, 1999), Directed loop updates (Syljuasen and

Sandvik, 2002) [NEXT WEEK]


Decomposing the Hamiltonian for SSEDecomposing the Hamiltonian for SSE

Break up the (general) Hamiltonian in off-diagonal and diagonal bond terms

Example : Heisenberg antiferromagnet

( )

!!

!!!

!!!

+=

+"++=

"++=

+""+

+""+

ji

d

ji

ji

o

ji

ji

z

j

z

i

ji

z

j

z

iZ

ji

jijiXY

i

z

ji

z

j

z

iZ

ji

jijiXY

XXZ

HH

SSz

hSSJSSSS

J

ShSSJSSSSJ

H

,

),(

,

),(

,,,

1

,,

)(2

)(2

convert site terms into bond terms

split into diagonal andoffdiagonal bond terms

( )zjz

i

z

j

z

iz

d

ji

jiji

xyo

ji

SSz

hSSJH

SSSSJ

H

+!=

+= +!!+

),(

),( )(2

with


High temperature series expansionHigh temperature series expansion

Expansion in inverse temperature

Using the bond Hamiltonians

( )

!!"

"

!

"

#$ $$

$

=

%

=

%

=

&

&=

&==

n

i

b

bbn

n

n

n

n

H

i

n

Hn

Hn

eZ

1,...,0

0

)( !

)Tr(!

)Tr(

1

{ }Uji

o

ji

d

jib HHHi

,

),(),( ,!

!=b

bHH

Hd(1,2)

i

54321

configuration operator

Ho (3,4)

Hd (3,4)

Ho (3,4)

1 2 3 4

Hd(1,2)


Ensuring positive diagonal bond weightsEnsuring positive diagonal bond weights

SSE Expansion

Sign problem ? Need to make all matrix elements non-positive for a correct sampling

Diagonal matrix elements : subtract an energy shift

Does not change the physics

( )

( )

( )( )n

n bb

n

n

i

b

bb

n

bbW

Hn

Z

n

i

n

,...,,

)(!

1

0 ,...,

0 1,...,

1

1

!

!!"

!

!

## #

# $# #

%

=

%

= =

=

&=

!

C "Jz

4+ h

( ) CSSz

hSSJH z

j

z

i

z

j

z

iz

d

ji !+!=),(


PositivityPositivity of off-diagonal bond weights of off-diagonal bond weights

Energy shift will not help for off-diagonal matrix elements, e.g.

Ferromagnet (Jxy <0) : no problem

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

Frustrated antiferromagnet : Sign problem similar to the worldline approach ! [LAST WEEK]

)(2

),(

+!!++= jiji

xyo

ji SSSSJ

H

( ) ( )+!!+±±

+!!+ +!"""""" #"!#

+ jijiXYi

i

ijiji

XY SSSSJSS

SSSSJ

2

)1(

2


Fixed length operator stringsFixed length operator strings

SSE sampling requires variable length n operator strings

Extend operator string to fixed length Λ by adding extra Identity operators

n: number of non-unit operators

Ensure Λ is large enough during thermalization

Such that e.g.

!

Z =" n

n!n= 0

#

$ %b1 ,...,bn( )

$%

$ (&Hbi

)i=1

n

' %

!

Hbi" H

(i, j )

d,H

(i, j )

o{ }i, j

U

!<4

3

maxn Vnn !">

max

!

Hid

= "1

!

Z =(" # n)!$ n

"!%

b1 ,...,b"( )

&%

& (#Hbi

)i=1

"

' %n= 0

"

&

{ } { }Uji

o

ji

d

jiidb HHHHi

,

),(),( ,!"


SSE approach : SummarySSE approach : Summary

Each SSE configuration is given by an initial state and a fixed length

operator string

Example for a four site system

SSE QMC = Sample over all possible configurations According to the above weights

Need a way to generate new configurations from a given one [NEXT WEEK]

!!"

!#$$%

=

&%

&%=

% 1

)(!

)!(

i

b

S

n

i

Hn

Z

Hd(1,2)

index

54321


Ho (3,4)1

Ho (3,4)

1 2 3 4

Hd(1,2)),, 1 ,,(

4

5

)2,1()4,3()4,3()2,1(

doodHHHHS

n

=

=

=

=!

!

"


Measurements in SSEMeasurements in SSE

Some observables are very simple : Energy :

Specific Heat :

Uniform Susceptibility :

Some more involved … : Equal time diagonal correlations :

Time dependent diagonal correlations :

nHE!

1"==

nnnCV

!!=22

2

!!"# $=i

z

iS

Hd(1,2)

index

54321


Ho (3,4)1

Ho (3,4)

1 2 3 4

Hd(1,2)( )0!

( )1!

( ) ( )50 !! =

( )4!

( )2!

( )3!

)()(][ where,][][1

1

0

1221 pDppdpdpdn

DD ii

n

p

!!=+

= "=

][][1

1)( ,)(1)0()( 2

0

112

0

1221 pdppdn

pCpCp

nDD

n

p

n

p

pnp

!++

=!!""#

$%%&

'(""

#

$%%&

'""#

$%%&

'

!= ))

==!

!(!

*

+

*

++


Comparing path integrals and SSE (I)Comparing path integrals and SSE (I)

Perturbation expansion :

Decomposition into bond terms :

Compare to SSE

VHH +=0

...)))()()(1(Tr( 1

0 0

212

0

2

0 ++!= " ""! ######

$ #$$

VVddVdeZH

!=b

o

bHV

( )}){},{,( ...

2

1 0

1

0

1

0 ,..., 0

!"!!!!!

"

#

o

n

n bb

nbWdddZ

n

noo

$$%% % $ &

'

=

=

!!"!"# $$

==

%%%

&&'

())*

+%= %

n

p

o

b

n

p

EEEno

p

ppp HeebW11

)( 10)1(}){},{,(

( )

( )( )m

m bb

bbWZ

n

,...,, 1

0 ,...,1

!!

"" "#

=

=


Comparing path integrals and SSE (II)Comparing path integrals and SSE (II)

Relation between the weights :

Given :

Consequences : Operator string longer in SSE, includes diagonal terms

In perturbation expansion, time integration complicated → need to also

sample over continuous times (Sandvik et al., 1997)

),...,(}{ 1 mooobbb =

}){},{,( ...}){,( 2

1 0

1

0}{|}{

),...,(}{

!"!!"!#

o

n

nm

bb

bbb

bWddbW

o

o

m

$$% % =&

=

=

=


Comparing path integrals and SSE (III)Comparing path integrals and SSE (III)

Worldlines in path integrals

Advantage Diagonal terms treated exactly

Drawback Continuous imaginary time

Best when large diagonal terms

Worldlines in SSE

Drawback Perturbation also in diagonal terms

Advantage Integer index instead of time

Other cases

space direction

imag

inar

y t

ime

0

!

space direction

inte

ger

in

dex

1

!


Partial conclusionPartial conclusion

What we saw today : Classical Monte Carlo (including framework for cluster algorithms)

QMC : Path integral approach

QMC : Series expansion approach

Local moves, continuous time limit

Next week : QMC : non-local moves, cluster algorithm approach

Deep into the most recent algorithms : loop, SSE, worm algorithms …


ReferencesReferences Classical Monte Carlo

Landau and Binder, A Guide to Monte Carlo Simulations in Statistical Physics,

Cambridge University Press (2000)

Newman and Barkema, Monte Carlo methods in statistical physics, Oxford University

Press (1999)

Cluster algorithms : Swendsen and Wang, Phys. Rev. Lett. 58, 86 (1987); Wolff, Phys

. Rev. Lett. 62, 361 (1989)

Statistical analysis : Book by Efron

Quantum Monte Carlo Brief introduction to lattice path integral methods : Troyer et al., physics/0306128

Continuous time limit : Prokof’ev et al., cond-mat/9612091; JETP 87, 310 (1998)

Basics of Stochastic Series Expansion (SSE) :

A. Sandvik and J. Kurkijarvi, Phys. Rev. B 43, 5950 (1991)

A. W. Sandvik, J.Phys. A 25, 3667 (1992)

Relation Path Integral – SSE : Sandvik et al., Phys. Rev. B 56, 14510 (1997)

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