introduction aux méthodes de monte carlo quantique en matière
TRANSCRIPT
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
3F. 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
4F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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
5F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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
6F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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
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= >< i
ii
p ji
pjpi SSJSSJH2,1,
2
1 ,
,, vvvv
7F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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
8F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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
9F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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 !
10F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
““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
12F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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
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13F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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
!
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i" c
i+1" ...
!
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x,y$ 0
!
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=py
px
!
Wx,y =min[1, py px ]
!
py px = e"# (Ey "Ex )
14F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
Metropolis algorithmMetropolis algorithm
15F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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
16F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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
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17F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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 …
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18F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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) ]
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AL)],[min(!" #
A
dL !.Time CPU "
19F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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
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20F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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
21F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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
22F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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
# $ %
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Detailed balance
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Weights are usually product of local (bond) weights
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b
bbb
b
bb gvgcgcwGCW )(),(),(),(
23F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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 =
24F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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
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cluster same theone are and if ,1 jiA jiimp $$
Generalization to other models possible Potts, O(N), field, …
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122
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Part II :Part II :Quantum Monte Carlo IQuantum Monte Carlo I
26F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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]
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Tr
!"=#
c
c
HpeZ
$Tr
27Cours de Physique Théorique du SPhT/Saclay
Part II :Part II :Quantum Monte Carlo IQuantum Monte Carlo I
1. PATH INTEGRAL APPROACH
28F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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
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29F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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]
30F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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
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31F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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
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,...,
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1
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+
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magnetization :
Continuous worldlines
32F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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
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33F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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
34F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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
35F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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]
36F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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
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c
!"#
$=
0!"#
37F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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]
38F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
Path integral representation (II)Path integral representation (II) Perturbation expansion in interaction representation :
Each term represented by a worldline configuration
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39F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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
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%
40Cours de Physique Théorique du SPhT/Saclay
Part II :Part II :Quantum Monte Carlo IQuantum Monte Carlo I
2. SERIES EXPANSION APPROACH
41F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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]
42F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
Decomposing the Hamiltonian for SSEDecomposing the Hamiltonian for SSE
Break up the (general) Hamiltonian in off-diagonal and diagonal bond terms
Example : Heisenberg antiferromagnet
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with
43F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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)
44F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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 !+!=),(
45F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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
46F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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
,
),(),( ,!"
47F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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
configuration operator
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
=
=
=
=!
!
"
48F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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
configuration operator
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
!++
=!!""#
$%%&
'(""
#
$%%&
'""#
$%%&
'
!= ))
==!
!(!
*
+
*
++
49F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
Comparing path integrals and SSE (I)Comparing path integrals and SSE (I)
Perturbation expansion :
Decomposition into bond terms :
Compare to SSE
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212
0
2
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!=b
o
bHV
( )}){},{,( ...
2
1 0
1
0
1
0 ,..., 0
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o
n
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nbWdddZ
n
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=
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bbWZ
n
,...,, 1
0 ,...,1
!!
"" "#
=
=
50F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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}{|}{
),...,(}{
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o
n
nm
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bbb
bWddbW
o
o
m
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=
=
51F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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
!
52F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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 …
53F. Alet, (SPhT, CEA Saclay) – Introduction to QMC – Lecture 1 – 04/02/2005
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)