Optimized statistical ensembleshelper.ipam.ucla.edu/publications/qs2009/qs2009_8075.pdf · 2009. 2. 5. · Reconsider the high-temperature series expansion M. Troyer, S. Wessel &

© Simon Trebst

Optimized statistical ensemblesfor slowly equilibrating classical and quantum systems

IPAM, January 2009

Simon TrebstMicrosoft Station Q

University of California, Santa Barbara

Collaborators: David Huse, Matthias Troyer, Emanuel Gull, Helmut Katzgraber, Stefan Wessel, Ulrich Hansmann

© Simon Trebst

• multiple energy scales

Many interesting phenomena in complex many-body systems arise only in the presence of

• slow equilibration• complex energy landscapes

Motivation

!

critical behavior

folding ofproteins

frustratedmagnets

quantumsystems

denseliquids

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Complex energy landscapes are characterized by many local minima.

?phase space

free

ene

rgy

How can we efficiently simulate such systems?Slow equilibration due to suppressed tunneling.

Complex energy landscapes

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c1 ! c2 ! . . . ! ci ! ci+1 ! . . .

• Sample configurations in phase space

How do we choose these weights?

Simulation of Markov chains

ci cj

propose a (small) change to a configuration

pacc = min!

1,w(cj)w(ci)

"accept/reject the update with probability

Metropolis algorithm (1953)

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c1 ! c2 ! . . . ! ci ! ci+1 ! . . .

• Sample configurations in phase space

high dimensional

• Project onto random walk in energy space

E1 ! E2 ! . . .! Ei ! Ei+1 ! . . .

pacc(E1 ! E2) = min!

1,w(E2)w(E1)

"= min (1, exp("!!E))

• We define a statistical ensemble

w(ci) = w(Ei) = exp(!!Ei)one dimensional

Statistical ensembles

Ei = H(ci)

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c1 ! c2 ! . . . ! ci ! ci+1 ! . . .

• Sample configurations in phase space

high dimensional

• Project onto random walk in energy space

• Phase space: The simulated system does a biased and Markovian random walk.

E1 ! E2 ! . . .! Ei ! Ei+1 ! . . .

one dimensional

Ei = H(ci)

• Energy space: The projected random walk is non-Markovian, as memory is stored in the system’s configuration.

Statistical ensembles

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Random walk in temperature space increases equilibration. en

ergy

/ te

mpe

ratu

re

slow equilibration

fast equilibration

ener

gy /

tem

pera

ture

Random walks in energy

© Simon Trebst

• Broaden the sampled energy space, e.g. by sampling a flat histogram.

energyhistogram

density of states

weight /ensemble

nw(E) = w(E) g(E)

w(E) = exp(!!E)

w(E) = 1/g(E)

hist

ogra

m

Wang-Landau algorithm (’01)

Extended ensemble simulations

© Simon Trebst

8 16 32 64L = N 1 / 2

1 1

10 10

100 100

τ / N

2

ferromagnetic Ising model

fully frustrated Ising model

z = 0.4

z = 0.9

Flat-histogram sampling

! ! N2+z

Critical slowing down.

! ! N2

The round-trip timeshould scale like N

2.

The energy range scales like N.E ! N

How well does this work?

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“critical energy”

-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0E / 2N

1

2

3

4

5lo

cal d

iffus

ivity

• The local diffusivity is NOT independent of the energy.

D(E, tD) = ![E(t) " E(t + tD)]2#/tD

1

The problem: local diffusivity

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current

local diffusivity

histogram derivative offraction

Measure the current in the energy interval

j = D(E) nw(E)df

dE

Determine the local diffusivity.

Maximize current by varying histogram/ensemble.

Optimizing the ensemble

label up

label down

-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0E / 2N

0

0.2

0.4

0.6

0.8

1

fract

ion

f(E

)

ferromagnetic Ising model

Phys. Rev. E 70, 046701 (2004).

© Simon Trebst

original ensemble

optimized ensemble

w!(E) ! w(E) ·

!df

dE· 1nw(E)

1

Feedback the local diffusivity

and iterate feedback until convergence.

Optimal histogram turns out to be

n(opt)w (E) ! 1!

D(E)

Optimizing the ensemble (cont’d)

Phys. Rev. E 70, 046701 (2004).

Ensemble optimization algorithm

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phase transition

Optimized histogram

-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0E / 2N

0

1

2

3histogram

• Feedback reallocates resources towards the critical energy.

© Simon Trebst

8 16 32 64 128

L = N 1 / 2

0.1 0.1

1 1

10 10

100 100

τ / N

2

optimized ensemble

flat-histogram ensemble

speedup~ 100

5,000 cpu hours

Performance of optimized ensemble

The round-trip times scale like .O![N log N ]2

"

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ExampleOrder by disorder transitions

& spiral spin liquidsD. Bergman, J. Alicea, E. Gull, ST, L. Balents

Nature Physics 3, 487 (2007).

© Simon Trebst

spin liquid

spin liquid

system fluctuatesamongst low-energyconfigurations, butno long-range order

long-rangeorder

frustration parameter

f =!CW

Tc

“highly frustrated”

f > 5! 10

high temperatureparamagnet

Curie-Weiss law

! ! 1T "!CW

Frustrated magnets

!CWTc T0

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Diamond lattice antiferromagnets: Materials

Many materials take on the normal spinel structure AB2X4.

V. Fritsch et al., PRL 92, 116401 (2004); N. Tristan et al., PRB 72, 174404 (2005); T. Suzuki et al. (2007)

Focus: Spinels with magnetic A-sites (only).

CoRh2O4 Co3O4

MnAl2O4

MnSc2S4

CoAl2O4

FeSc2S4

S=5/2

S=3/2

S=2orbital

degeneracy

f = !CW/Tc1 5 10 20 900

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classical spinsS=3/2, S=5/2

antiferromagnetic

Frustration in the diamond lattice

Naive Hamiltonian H = J1

!

!ij"

!Si · !Sj

diamond latticetwo FCC lattices

coupled via J1

bipartite latticeno frustration

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W. L. Roth, J. Phys. 25, 507 (1964)

J1 ! J2similar exchange path

Frustration in the diamond lattice

H = J1

!

!ij"

!Si · !Sj + J2

!

!!ij""

!Si · !Sj2nd neighbor

exchange

strong frustrationgeneratesJ2

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Phase diagram

rapidly diminishes in Néel phaseTc

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1coupling J2 / J1

0 0

0.1 0.1

0.2 0.2

0.3 0.3

0.4 0.4

0.5 0.5

orde

ring

tem

pera

ture

T c /

J1

Néel coplanar spiral

Ordering transitionwith sharply reduced Tc

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

coupling J2 / J

1

0 0

0.1 0.1

0.2 0.2

0.3 0.3

0.4 0.4

0.5 0.5

ord

erin

g t

emp

erat

ure

T

c / J

1

Néel 111 111* 110 100*

N = 512N = 1728N = 4096

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Spiral surfaces

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1coupling J2 / J1

0 0

0.1 0.1

0.2 0.2

0.3 0.3

0.4 0.4

0.5 0.5

orde

ring

tem

pera

ture

T c /

J1

Néel coplanar spiral

© Simon Trebst

J2/J1 = 0.85-2 -1 0 1 2

-0.01 0 0.01

J2/J1 = 0.2

First-order transitions

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1coupling J2 / J1

0 0

0.1 0.1

0.2 0.2

0.3 0.3

0.4 0.4

0.5 0.5

orde

ring

tem

pera

ture

T c /

J1

Néel coplanar spiral

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Parallel tempering

Single replica performs random walk in temperature space.

K. Hukushima and Y. Nemoto, J. Phys. Soc. Jpn. 65, 1604 (1996)

Simulate multiple replicas of the system at various temperatures.

How do we choose the temperature points?

T1 T2 TN

temperatureswap

replicas

p(Ei, Ti ! Ei+1, Ti+1) = min(1, exp(!!!E))

© Simon Trebst

Ensemble optimization

Feedback algorithm

Measure local diffusivity of current in temperature space.

D(T )

density of T-points

Optimal choice of temperatures

!opt(T ) ! 1!D(T )

Iterate feedback of diffusivity.spiral spin liquidlong-range

order

feed

back

0.04 0.05 0.06 0.07 0.08 0.09 0.1temperature T

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

temperature T

10-7 10-7

10-6 10-6

10-5 10-5

10-4 10-4

10-3 10-3

10-2 10-2

diffu

sivity

D(T)

J2/J1 = 0.2

© Simon Trebst

ExampleFolding of a (small) protein

ST, M. Troyer, U.H.E. HansmannJ. Chem. Phys. 124, 174903 (2006).

© Simon Trebst

The chicken villin headpiece

folding time: 4.3 microseconds

fold

ing@

hom

e pr

ojec

t

A small protein: HP-36

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300 400 500 600 700temperature T [K]

103

104

105

106

107lo

cal d

iffus

ivity

D(T)

0

5

10

15

20

spec

ific

heat

C

V (T)

• Multiple temperature scales are revealed by the local diffusivity.

helix-coil transition

competingground states

Random walk in temperature

© Simon Trebst

competing ground states helix-coil transition

200 300 400 500 600 700 800 900 1000temperature T [K]

0

1

2

3

feed

back

iter

atio

n

• Feedback reallocates resources towards the relevant temperature scales.

300 400 500 600 700temperature T [K]

103

104

105

106

107

loca

l diff

usiv

ity

D(T

)

0

0.2

0.4

0.6

0.8

1

acce

ptan

ce p

roba

bilit

y A

(T)

Optimized temperature sets

© Simon Trebst

!

ExampleQuantum systems

S. Wessel, N. Stoop, E. Gull, ST, M. TroyerJ. Stat. Mech. P12005 (2007).

© Simon Trebst

coefficients“density of states”

Quantum systems

Z = Tr e!!H ="!

n=0

!n

n!Tr (!H)n =

"!

n=0

g(n) !n

Reconsider the high-temperature series expansion

!

M. Troyer, S. Wessel & F. Alet, PRL 90, 120201 (2003).We can define a broad-histogram ensemble in the expansion order.

Stochastic series expansion (SSE) samples these coefficients

high temperaturesn ! 0 n ! "

low temperatures?

!n" # !N

© Simon Trebst

Examples !

0 1 2 3 4 5expansion order n

δh / L2

0

0.005

0.01

0.015

0.02

loca

l hist

ogra

m

h(n δ

h ) L=6

L=8L=10

Λδh = 500

Spin-flop transition

H = J!

<i,j>

"Sx

i Sxj + Sy

i Syj + !Sz

i Szj

#

!h!

i

Szi

spin-1/2 XXZ model in a magnetic field

0 0.2 0.4 0.6 0.8 1expansion order n / Λ

0

0.04

0.08

0.12

loca

l hist

ogra

m h

(n) L

2

Λ = 20 L2

L = 12

L = 10L = 8

L = 6

Thermal first-order transition

H = !t!

!i,j"

(a†iaj + a†jai) + V2

!

!!i,k""

nink ! µ!

i

ni

hard-core bosons with next-nearest neighbor repulsion

© Simon Trebst

Summary

Metropolis cycle

configuration suggest an update

accept/rejectan update

non-local update schemesloops, worms, ...

unconventionalstatistical ensembles!improve sampling efficiency

& overcome entropic barriers