Numerical Simulation ofStrongly Correlated Systems

Part II: QMC for spins and bosons

Matthias Troyer (ETH)

Phases and phase transitions

Classical and quantum phase transitionsclassical phase transition

driven by thermal fluctuations

quantum phase transition

driven by quantum fluctuationsand uncertainty relatio"

increasing fluctuations

Numerical simulations“You let the computer solve the problem for you”

It’s not that easy:

Exponentially diverging number of states1 site: q states

N sites: qN states

Critical slowing down of the dynamics at phase transitions

negative sign problem for fermions (NP-hard)

✔

✔

✗

What is the probability to win in Solitaire?answer play it 100 times, count the number of wins and you have a pretty good estimate

Ulam: the Monte Carlo Method

The Monte Carlo MethodEstimate an average by a statistical sample

Need a representative sample with the correct distribution

How can this be achieved?fundamental problem of statistical mechanics

〈A〉 =1Z

N∑i=1

Aipi

1

〈A〉 ≈ A =1M

M∑i=1

Aci

∆A =

√〈A2〉 − 〈A〉2

M

M % N

1

limL→∞

〈A〉 = limL→∞

1Z

N∑i=1

Aipi

N ∝ exp(cL)

P [ci] =pci

Z

1

observable

probability

In putting together this issue of Computing in Science & Engineering, we knew three things:it would be difficult to list just 10 algorithms;it would be fun to assemble the authors andread their papers; and, whatever we came upwith in the end, it would be controversial. We

tried to assemble the 10 algorithms with the greatestinfluence on the development and practice of scienceand engineering in the 20th century. Following is ourlist (here, the list is in chronological order; however,the articles appear in no particular order):

• Metropolis Algorithm for Monte Carlo• Simplex Method for Linear Programming• Krylov Subspace Iteration Methods• The Decompositional Approach to Matrix

Computations• The Fortran Optimizing Compiler• QR Algorithm for Computing Eigenvalues• Quicksort Algorithm for Sorting• Fast Fourier Transform• Integer Relation Detection• Fast Multipole Method

With each of these algorithms or approaches, thereis a person or group receiving credit for inventing ordiscovering the method. Of course, the reality is thatthere is generally a culmination of ideas that leads to amethod. In some cases, we chose authors who had a

hand in developing the algorithm, and in other cases,the author is a leading authority.

In this issue

Monte Carlo methods are powerful tools for evalu-ating the properties of complex, many-body systems,as well as nondeterministic processes. Isabel Beichl andFrancis Sullivan describe the Metropolis Algorithm.We are often confronted with problems that have anenormous number of dimensions or a process that in-volves a path with many possible branch points, eachof which is governed by some fundamental probabilityof occurence. The solutions are not exact in a rigorousway, because we randomly sample the problem. How-ever, it is possible to achieve nearly exact results using arelatively small number of samples compared to theproblem’s dimensions. Indeed, Monte Carlo methodsare the only practical choice for evaluating problems ofhigh dimensions.

John Nash describes the Simplex method for solv-ing linear programming problems. (The use of theword programming here really refers to scheduling orplanning—and not in the way that we tell a computerwhat must be done.) The Simplex method relies onnoticing that the objective function’s maximum mustoccur on a corner of the space bounded by the con-straints of the “feasible region.”

Large-scale problems in engineering and science of-ten require solution of sparse linear algebra problems,such as systems of equations. The importance of iter-ative algorithms in linear algebra stems from the sim-ple fact that a direct approach will require O(N3) work.The Krylov subspace iteration methods have led to amajor change in how users deal with large, sparse, non-symmetric matrix problems. In this article, Henk vander Vorst describes the state of the art in terms of

22 COMPUTING IN SCIENCE & ENGINEERING

JACK DONGARRA

University of Tennessee and Oak Ridge National LaboratoryFRANCIS SULLIVAN

IDA Center for Computing Sciences

1521-9615/00/$10.00 © 2000 IEEE

G U E S T E D I T O R S ’

I N T R O D U C T I O N

the TopIn putting together this issue of Computing in Science & Engineering, we knew three things:it would be difficult to list just 10 algorithms;it would be fun to assemble the authors andread their papers; and, whatever we came upwith in the end, it would be controversial. We

tried to assemble the 10 algorithms with the greatestinfluence on the development and practice of scienceand engineering in the 20th century. Following is ourlist (here, the list is in chronological order; however,the articles appear in no particular order):

• Metropolis Algorithm for Monte Carlo• Simplex Method for Linear Programming• Krylov Subspace Iteration Methods• The Decompositional Approach to Matrix

Computations• The Fortran Optimizing Compiler• QR Algorithm for Computing Eigenvalues• Quicksort Algorithm for Sorting• Fast Fourier Transform• Integer Relation Detection• Fast Multipole Method

With each of these algorithms or approaches, thereis a person or group receiving credit for inventing ordiscovering the method. Of course, the reality is thatthere is generally a culmination of ideas that leads to amethod. In some cases, we chose authors who had a

hand in developing the algorithm, and in other cases,the author is a leading authority.

In this issue

Monte Carlo methods are powerful tools for evalu-ating the properties of complex, many-body systems,as well as nondeterministic processes. Isabel Beichl andFrancis Sullivan describe the Metropolis Algorithm.We are often confronted with problems that have anenormous number of dimensions or a process that in-volves a path with many possible branch points, eachof which is governed by some fundamental probabilityof occurence. The solutions are not exact in a rigorousway, because we randomly sample the problem. How-ever, it is possible to achieve nearly exact results using arelatively small number of samples compared to theproblem’s dimensions. Indeed, Monte Carlo methodsare the only practical choice for evaluating problems ofhigh dimensions.

John Nash describes the Simplex method for solv-ing linear programming problems. (The use of theword programming here really refers to scheduling orplanning—and not in the way that we tell a computerwhat must be done.) The Simplex method relies onnoticing that the objective function’s maximum mustoccur on a corner of the space bounded by the con-straints of the “feasible region.”

Large-scale problems in engineering and science of-ten require solution of sparse linear algebra problems,such as systems of equations. The importance of iter-ative algorithms in linear algebra stems from the sim-ple fact that a direct approach will require O(N3) work.The Krylov subspace iteration methods have led to amajor change in how users deal with large, sparse, non-symmetric matrix problems. In this article, Henk vander Vorst describes the state of the art in terms of

22 COMPUTING IN SCIENCE & ENGINEERING

JACK DONGARRA

University of Tennessee and Oak Ridge National LaboratoryFRANCIS SULLIVAN

IDA Center for Computing Sciences

1521-9615/00/$10.00 © 2000 IEEE

G U E S T E D I T O R S ’

I N T R O D U C T I O N

the Top

The Metropolis algorithm (1953)

The Metropolis algorithmus (1953)

creates a representative sample for any systemstart with a configuration i

propose a small change to a configuration j

pj

pi

P = min(

1,pj

pi

)

1

calculate the ratio of probabilities

pj

pi

P = min(

1,pj

pi

)

1

accept the new configuration with probability

Feynman (1953) lays foundation for quantum Monte CarloMap quantum system to classical world lines

Quantum Monte Carlo

“imag

inar

y tim

e”

quantum mechanical

%orld linesspac&

classical

particles

An expansion of the partition function

gives a mapping to a (d+1)-dimensional classical model

partition function of quantum system is sum over classical world lines

Discrete time path integrals

�

Z = Tre−βH = Tre−MΔτH = Tr e−ΔτH( )M = Tr 1−ΔτH( )M + O(βΔτ )

= i1 1−ΔτH i2 i2 1−ΔτH i3 iM 1−ΔτH i1{( i1 ...iM )}∑

space direction

imag

inar

y tim

e

|i1〉

|i2〉

|i3〉

|i4〉

|i5〉

|i6〉

|i7〉

|i8〉

|i1〉

place particles (spins)

for Hamiltonians conserving particle number (magnetization) we get world lines

just move the world lines locallyprobabilities given by matrix element of Hamiltonianexample: tight binding model

Monte Carlo updates

introduce or remove two kinks: shift a kink:

�

H = −t ci†ci+1 + ci+1

† ci( )i, j∑

�

P = Δτt( )2

�

P =1

�

P = Δτt

�

P→ =min 1,(Δτt)2[ ]P← =min 1,1/(Δτt)2[ ]

�

P→ = P← =1

the limit Δτ→0 can be taken in the construction of the algorithm [Prokof ’ev et al., Pis’ma v Zh.Eks. Teor. Fiz. 64, 853 (1996)]

discrete time: store configuration at all time stepscontinuous time: store times at which configuration changes

The continuous time limit

space direction

imag

inar

y tim

e

|i1〉

|i2〉

|i3〉

|i4〉

|i5〉

|i6〉

|i7〉

|i8〉

|i1〉

space directionim

agin

ary

time

τ1

τ2

τ3

τ4τ5

τ6

Path Integral Formulationinteraction representation

each term is represented by a world line configuration

H = H0 + V, H0 = JijzSi

zS jz − hSi

z ,i∑

<i , j>∑ V = Jij

xy(SixSj

x

< i, j>∑ + Si

ySjy)

Z = Tr(e− βH) = Tr(e− βH0Te− dτV (τ )0

β

∫ )

Z = Tr(e− βH0 (1 − dτV (τ )0

β

∫ + dτ1 dτ2V (τ1)τ 1

β

∫0

β

∫ V (τ 2 ) + ...))

τ τ2

τ1

based on high temperature expansion, developed by A. Sandvik

also has a graphical representation in terms of world lines

Advantage: easier algorithms since no times associated with operatorsDisadvantage: perturbation in all terms of the Hamiltonian

Stochastic Series Expansion (SSE)

Hb1 Hb1

Hb2

Z = Tr(e−βH ) = β n

n!n=0

∞

∑ Tr (−H )n⎡⎣ ⎤⎦

=β n

n!n=0

∞

∑ α1 − H α2 α2 − H α3 ⋅ ⋅ ⋅ α n − H α1α1 ,...,αn

∑

In mapping of quantum to classical system

there is a “sign problem” if some of the pi < 0Appears e.g. in simulation of electrons (Pauli principle)

Exponentially growing cancellation in the sign

The negative sign problem

�

A = Tr Aexp(−βH )[ ] Tr exp(−βH )[ ] = Ai pii∑ pi

i∑

�

�

A = Aipii∑ pi

i∑ =

Ai sgn pi pii∑ pi

i∑

sgn pi pii∑ pi

i∑

≡A ⋅ sign p

sign p

�

A ⋅ sign p ≈ sign p ≈ exp(−cβN)⇒ΔA ≈ exp(+cβN)

Autocorrelation effects

The Metropolis algorithm creates a Markov chain of configurations

successive configurations are correlated, leading to an increased statistical error

Critical slowing down at second order phase transition

Exponential tunneling problem at first order phase transition

�

c1 → c2 → ...→ ci → ci+1 → ...

�

ΔA = A − A( )2 = Var AM

(1+ 2τA )

�

τ ∝L2

�

τ ∝exp(Ld−1)

Solving slowing down at phase transitions

Changing the dynamics solves critical slowing down at second order phase transitions: make large, global, changes instead of local ones

Cluster updates for classical spins (Swendsen and Wang, 1987)

Loop algorithm for quantum spins (Evertz et al, 1993)

Worm algorithm (Prokof ’ev et al, 1998)

Changing the ensemble solves tunneling problem at first order phase transitions: remove the energy barriers

Multicanonical sampling (Berg & Neuhaus, 1991)

Wang-Landau sampling (F. Wang & D.P. Landau, 2001)

Flat histogram methods for quantum systems (Troyer et al, PRL 2003)

Updates form closed loops since world lines may not be broken

Loop-cluster updates

Cluster algorithms for quantum systems

Which system sizes can be studied?

Modern algorithms allow to accurately study quantum phase transitions

temperature Metropolis modern algorithms

1 16’000 spins 16’000’000 spins

0.1 200 spins 1’000’000 spins

0.1 32 bosons 10’000 bosons

0.005 ––– 50’000 spins

ALPS project: http://alps.comp-phys.org/

Open source software for quantum lattice models

Hands-on tutorialsClassical Monte Carlo

Spin DynamicsQuantum Monte CarloExact diagonalization

DMRG

H.G. Evertz (TU Graz)A. Läuchli (Toulouse)R. Noack (Marburg)

U. Schollwöck (RWTH)

Speakers (preliminary) T. Schulthess (ORNL)

S.Todo (Tokyo)Shan-Ho Tsai (UGA)

M. Troyer (ETH)

Workshop and tutorials Lugano, Switzerland, September 27 - October 1, 2004

Hard-core bosons

Seems trivial, but actually strongly interacting system (infinite repulsion)

equivalent to XY model in a perpendicular magnetic field

Approximately linear Tc at low doping

Agrees well with experiments on 4He filmsG. Agnolet, et al. PRB ‘89P.S. Ebey, et al., J. Low Temp. Phys. ‘98

�

Tc =π

2mρs(Tc ) with ρs (Tc) ≈ 0.74ρ

Two-dimensional hard-core boson model

�

H = −t ai†a j + aj

†ai( ) −µ nii∑

< i, j>∑

�

H = Jxy SixS j

x + SjySi

y( )< i, j>∑ − h Si

z

i∑

ρ = 1 Mott insulatorρ = 0 Mott insulator 0 < ρ < 1 superfluid

For bosons at low doping [D. Fisher and P. Hohenberg, PRB ‘88]

Fisher-Hohenberg formula only valid when

With a=10-15 m we need less than 10-18 bosons in visible universe�

ρ < a−2ee6≈10−90a−2

Fisher - Hohenberg theory

�

Tc =π2m

4ρ− lnln a2ρ

But we also know

�

ρs(Tc) ≤ ρ ρs(Tc) ≈ 0.74ρ

�

Tc =π2m

ρs

�

Tc = 0.75 π2m

ρIrrelevant for experiments, better use our result

Simple limits with longer range repulsion

t→∞: superfluid phaseU=∞ at half filling: solid phases at large V1 or V2

H = −t∑

〈i,j〉

(b†i bj + b†jbi) + U∑

i

ni(ni − 1)/2 + V1

∑

〈i,j〉

ninj + V2

∑

〈〈i,j〉〉

ninj

V1→∞checkerboard solid

V2→∞striped solid

Super solids

A super solid shows simultaneously solid order (crystalline) and superfluidity

solid (crystal) doped solid:interstitials

supersolid:superfluid interstitials

2D Bosons with nearest neighbor repulsionprevious simulations (32 particles) found supersolidnew simulations (5000 particles) instead show phase separation at first order phase transition

Do supersolids exist?

0.650.60.550.5Density !

solid

superfluid

Supersolids versus phase separation

solid doped solid supersolid

∆ε = −ρ2t2

V

doped particles gain energy by forming a domain wall

∆ε = −ρt < −ρ2t2

V

Two routes to supersolids

striped solid doped solid striped supersolid

solid supersoliddopants on same sublattice!

V ! U/4 > t

Trapped bosonic atoms

BEC in cold bosonic atomsUltra-cold trapped 87Rb atoms form BECfirst observed 1995

Standing waves from laser superimpose an optical lattice (2002)

Experiments on trapped atomsTrapped atoms in an optical latticeQuantum phase transition as lattice depth is increased

Greiner et al, Nature (2002)detected by measuring the momentum distribution function

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describes bosonic atoms in optical lattice well understood without the trap: Fisher et al, PRB 1989

Boson-Hubbard model

Phase diagram (V=0)

Ut /

1=n

2=n

0=n

U/µIncompressible Mott-insulator

Integer filling

superfluid

H = −t∑〈i,j〉

(b†i bj + b†jbi

)+ U

∑i

ni(ni − 1)/2 − µ∑

i

ni

trap introduces a site-dependent chemical potentialresults in inhomogeneous system

Boson-Hubbard model in a trap

H = −t∑〈i,j〉

(b†i bj + b†jbi

)+ U

∑i

ni(ni − 1)/2 − µ∑

i

ni + V∑

i

r2

i ni

2i

effi Vr−= µµ

ir

H = −t∑〈i,j〉

(b†i bj + b†jbi

)+ U

∑i

ni(ni − 1)/2−∑

i

µeffi ni

increasing repulsion U / t forms Mott insulator in center of trap, surrounded by superfluid shell

local density

Detecting the quantum phase transition (1)

Original proposal by Prokof ’ev and Svistunovlook for fine structure in momentum distribution

0 0.2 0.4 0.6 0.8 1k

x a / !

0

0.01

0.02

0.03

0.04

0.05

n(k

x ,0

,0)

0 2 4 6 8 10r / a

0

0.2

0.4

0.6

0.8

1

1.2

n

3D trap

µ/U=0.3125V/U=0.0122U/t=80

secondary peak in Mott insulator

0 2 4 6 8 10r / a

0

0.2

0.4

0.6

n

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8 1k

x a / !

0

0.1

0.2

0.3

0.4

n(k

x ,0

,0)

0 0.2 0.4 0.6 0.8 10

3D trap

µ/U=-0.2V/U=0.008U/t=24

no such peak in superfluid

0 0.2 0.4 0.6 0.8 10

0.004

0.008

a closer look reveals that this is wrong

Detecting the quantum phase transition (2)

Measurements at ETH by Stöferle et al [PRL, 2004]

U / J = 2.3

1 10 1000%

25%

50%

75%

Cohere

nt F

ract

ion f

c

U / J

0 2 4 6 8 10 1220

30

40

50

Inte

rfere

nce

Peak

FW

HM

[µ

m]

U / J

1D

1D-3D crossover

3D

1D hold 1ms

(a) (b)

(c)

peak height(coherence fraction)

U / J = 2.3

1 10 1000%

25%

50%

75%

Cohere

nt F

ract

ion f

c

U / J

0 2 4 6 8 10 1220

30

40

50

Inte

rfere

nce

Peak

FW

HM

[µ

m]

U / J

1D

1D-3D crossover

3D

1D hold 1ms

(a) (b)

(c)

peak width(FWHM)

can this be used to detect quantum phase transition?

Detecting the quantum phase transition (3)

uniform system (flat trap)easy to detect QPT

10 15 20 25U / t

0.0

0.1

0.2

0.3

0.4

FW

HM

/ !

0.0

0.1

0.2

0.3

0.4

0.5

0.6

cohere

nce f

racti

on, n

(0,0

)

OBC

µ/U=0.37

OBC

PBC

Mott-transitionuniform case

2D 24 " 24

Numerical simulations to check whether the QPT can be detected from peak width and height

0 10 20 30 40U / t

0.05

0.1

0.15

FW

HM

/!

0

0.1

0.2

0.3

0.4

0.5

coh

eren

ce f

ract

ion

, n

(0,0

)

Mott-plateauformation

2D trap

µ/U=0.37V/U=0.002

parabolic trapcomparison to simulation needed

a flat trap would be preferred!

Other dimensions1D results by C. Kollath et al [PRA, 2004]Our 3D results are very similar to 2D

0.0 0.2 0.4 0.6k

x a / !

0

0.1

0.2

0.3

0.4

0.5

n(k

x,0

,0)

0 0.2 0.4 0.6k

x a / !

0

0.02

0.04

n(k

x,0

,0)

U/t=20.0

53.3

16040.0

10026.7

106.7

U/t=80.03D trap

µ/U=0.25V/U=0.0125

0 20 40 60 80 100 120U / t

0

0.1

0.2

0.3

0.4

0.5

0.6

coh

eren

ce f

ract

ion

, n

(0,0

,0)

0.09

0.10

0.11

FW

HM

/ !

3D trap

µ/U=0.25V/U=0.0125

Mott-plateauformation

Quantum criticality in trapped bosonic atoms

Describing the phasesHow shall we quantitatively understand the phases?

A Mott-insulating core forms in the center of the trapsurrounded by superfluid (1D or 2D?)

quantum criticality at interface?

Measure local compressibility

Superfluid remains compressibleMott-insulator incompressible

Increased fluctuations near the edge of the superfluid

Identifying the local phases

0 5 10 15 20r / a

0.0

0.2

0.4

0.6

0.8

1.0

1.2

n(r

)

0.0

0.1

0.2

0.3

!lo

ca

l (r)

µ/U=0.37V/U=0.002U/t=25.0

2D trap

κlocali =

∂n

∂µeffi

=∂ni

∂µ= β (〈nin〉 − 〈ni〉〈n〉)

density

κlocali

Equal-time Green’s function along the superfluid ring

better described by 2D behavior

than by 1D behavior

Coherence within the superfluid ring

ji bb+

0 10 20 30 40distance along ring, d / a

0.1

0.2

0.3

0.4

corr

elat

ion

fu

nct

ion

, g

(d)

0 10 20 30r / a

0

0.05

0.1

0.15

g(!

r)

exponential fit

"/a=21.6(4)

c=0.033(3)

c + b cosh((!r-d)/")

2D trapµ/U=0.37V/U=0.002U/t=25.0

Inside superfluid ringalong r/a=12.3

g(d) = 〈b†(0)b(d)〉 = c + a cosh

(πr − d

ξ

)

g(d) = 〈b†(0)b(d)〉 ∝[d−κ + (2πr − d)−κ

]

Local potential approximation Data collapses onto a single curve

This curve varies from trap to trap

Not reducible to homogenous case

Absence of cusps and singularitiesA singularity emerges in uniform 2D

Even qualitatively different in trap

No signature of quantum criticalitySingle domain formation

No critical slowing down

Absence of quantum criticality

1ϕ1ϕ2ϕ -0.4 -0.2 0 0.2 0.4

µ / U

0.0

0.1

0.2

0.3

!lo

ca

l (µ)

0

0.2

0.4

0.6

0.8

1

1.2

n(µ

)

U/t=25.0

2D uniform

-0.4 -0.2 0 0.2 0.4

µeff

/ U

0.0

0.1

0.2

0.3

!lo

cal (µ

eff )

0

0.2

0.4

0.6

0.8

1

1.2

n(µ

eff )

µ/U=0.37V/U=0.002U/t=25.0

2D trap

Effective Ladder modelWhat causes the absence of quantum criticality?finite size effect?effect of potential gradient?structural quenching?

Structural disorder removed by considering a ladder model

SF

MI

Describe the critical region by an inhomogeneous ladder

in a linearized potential

Effective Ladder model

µµµ Δ+= iileg 0) (

-0.4 -0.2 0 0.2 0.4

µeff

/ U

0.0

0.1

0.2

0.3

!lo

cal (µ

eff )

0

0.2

0.4

0.6

0.8

1

1.2

n(µ

eff )

µ/U=0.37V/U=0.002U/t=25.0

2D trap and ladder

trap

ladder

ladder

trap

Quantitative agreementwith 2D trap

H = −t∑i,j

(b†i,jbi+1,j + b†i,jbi,j+1 + H.c.

)

+U

2

∑i,j

ni,j(ni,j − 1) −W∑

j=1

µ(j)L∑

i=1

ni,j

Usual thermodynamic limit in the trap

Changes curvatureRecover critical behavior of d-dimensional systembut not what describes experiments

What happens for a finite gradient?Is there a critical chain in the ladder?Increasing length of the ladder

No divergenceNo sign of quantum criticality

Absence of quantum criticality due toInhomogenityCoupling to the rest of the system

Thermodynamic limits

./ , consttVNN bosonsbosons =∞→

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3eff

/ U

0.0

0.2

0.4

0.6

0.8

1.0

1.2

n(

eff )

1D uniformV/U=0.0004V/U=0.00016

-0.4 -0.35 -0.30

0.1

0.2

0.15 0.2 0.250.9

0.95

1

1D traps compared to the uniform case

/U=0.37U/t=5.71

0

0.2

0.4

0.6

0.8

1

1.2

n(

eff )

-0.2 -0.1 0 0.1 0.2 0.3eff

/ U

0.0

0.1

0.2

0.3

0.4

loca

l (ef

f ) t

32 rungs

64 rungs

-0.2 0 0.2 / U

0

0.5

1

t

Effective ladder modeluniform 1D64 sitesU/t=25