optimized statistical ensembleshelper.ipam.ucla.edu/publications/qs2009/qs2009_8075.pdf · 2009. 2....
TRANSCRIPT
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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
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• 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
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• 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
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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).
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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.
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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).
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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
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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))
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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
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ExampleFolding of a (small) protein
ST, M. Troyer, U.H.E. HansmannJ. Chem. Phys. 124, 174903 (2006).
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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
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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
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!
ExampleQuantum systems
S. Wessel, N. Stoop, E. Gull, ST, M. TroyerJ. Stat. Mech. P12005 (2007).
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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
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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
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Summary
Metropolis cycle
configuration suggest an update
accept/rejectan update
non-local update schemesloops, worms, ...
unconventionalstatistical ensembles!improve sampling efficiency
& overcome entropic barriers