Optimized statistical ensembles for slowly equilibrating classical and quantum systems

Transcription

1 Optimized statistical ensembles for slowly equilibrating classical and quantum systems IPAM, January 2009 Simon Trebst Microsoft Station Q University of California, Santa Barbara Collaborators: David Huse, Matthias Troyer, Emanuel Gull, Helmut Katzgraber, Stefan Wessel, Ulrich Hansmann

2 Motivation Many interesting phenomena in complex many-body systems arise only in the presence of multiple energy scales complex energy landscapes slow equilibration critical behavior folding of proteins frustrated magnets quantum systems dense liquids

3 Complex energy landscapes Complex energy landscapes are characterized by many local minima. free energy phase space? Slow equilibration due to suppressed tunneling. How can we efficiently simulate such systems?

4 Simulation of Markov chains Sample configurations in phase space c 1 c 2... c i c i+1... Metropolis algorithm (1953) propose a (small) change to a configuration c i c j accept/reject the update with probability ( p acc = min 1, w(c ) j) w(c i ) How do we choose these weights?

5 Statistical ensembles Sample configurations in phase space c 1 c 2... c i c i+1... Project onto random walk in energy space E 1 E 2... E i E i+1... We define a statistical ensemble high dimensional E i = H(c i ) w(c i ) = w(e i ) = exp( βe i ) p acc (E 1 E 2 ) = min ( 1, w(e ) 2) w(e 1 ) one dimensional = min (1, exp( β E))

6 Statistical ensembles Sample configurations in phase space c 1 c 2... c i c i+1... Project onto random walk in energy space E 1 E 2... E i E i+1... Phase space: The simulated system does a biased and Markovian random walk. Energy space: The projected random walk is non-markovian, as memory is stored in the system s configuration. high dimensional E i = H(c i ) one dimensional

7 Random walks in energy Random walk in temperature space increases equilibration. fast equilibration energy / temperature energy / temperature slow equilibration

8 Extended ensemble simulations Broaden the sampled energy space, e.g. by sampling a flat histogram. histogram energy w(e) = exp( βe) w(e) = 1/g(E) histogram density of states n w (E) = w(e) g(e) weight / ensemble Wang-Landau algorithm ( 01)

9 How well does this work? z = 0.9 fully frustrated Ising model τ / N ferromagnetic Ising model z = E N The energy range scales like N. τ N 2 The round-trip time should scale like N L = N 1 / 2 Flat-histogram sampling τ N 2+z Critical slowing down.

10 The problem: local diffusivity D(E, t D ) = [E(t) E(t + t D )] 2 /t D 5 local diffusivity critical energy E / 2N The local diffusivity is NOT independent of the energy.

11 Optimizing the ensemble Measure the current in the energy interval ferromagnetic Ising model local diffusivity fraction f(e) label down current j = D(E) n w (E) df histogram de 0.2 label up E / 2N derivative of fraction Determine the local diffusivity. Maximize current by varying histogram/ensemble. Phys. Rev. E 70, (2004).

12 Optimizing the ensemble (cont d) Optimal histogram turns out to be n (opt) w (E) 1 D(E) Ensemble optimization algorithm Feedback the local diffusivity optimized ensemble w (E) w(e) df de 1 n w (E) original ensemble and iterate feedback until convergence. Phys. Rev. E 70, (2004).

13 Optimized histogram 3 2 histogram phase transition E / 2N Feedback reallocates resources towards the critical energy.

14 Performance of optimized ensemble τ / N flat-histogram ensemble speedup ~ optimized ensemble 5,000 cpu hours L = N 1 / 2 The round-trip times scale like O ( [N log N] 2).

15 Example Order by disorder transitions & spiral spin liquids D. Bergman, J. Alicea, E. Gull, ST, L. Balents Nature Physics 3, 487 (2007).

16 Frustrated magnets long-range order spin liquid high temperature paramagnet 0 T c Θ CW T frustration parameter f = Θ CW T c highly frustrated f > 5 10 spin liquid system fluctuates amongst low-energy configurations, but no long-range order Curie-Weiss law 1 χ T Θ CW

17 Diamond lattice antiferromagnets: Materials V. Fritsch et al., PRL 92, (2004); N. Tristan et al., PRB 72, (2005); T. Suzuki et al. (2007) Many materials take on the normal spinel structure AB2X4. Focus: Spinels with magnetic A-sites (only). S=5/2 S=2 CoRh2O4 Co3O4 MnSc2S4 FeSc2S4 orbital degeneracy f = Θ CW /T c MnAl2O4 CoAl2O4 S=3/2

18 Frustration in the diamond lattice Naive Hamiltonian H = J 1 ij antiferromagnetic S i S j classical spins S=3/2, S=5/2 diamond lattice two FCC lattices coupled via J 1 bipartite lattice no frustration

19 Frustration in the diamond lattice 2nd neighbor exchange H = J 1 ij S i S j + J 2 ij S i S j J 1 J 2 similar exchange path W. L. Roth, J. Phys. 25, 507 (1964) J 2 generates strong frustration

20 Phase diagram ordering temperature T c / J T c rapidly diminishes in Néel phase Néel Ordering transition with sharply reduced T c N = 512 N = 1728 N = 4096 coplanar spiral coupling J 2 / J 1

21 ordering temperature Tc / J1 Spiral surfaces Néel coplanar spiral coupling J2 / J1

22 First-order transitions ordering temperature T c / J J 2 /J 1 = J 2 /J 1 = Néel coplanar spiral coupling J 2 / J 1

23 Parallel tempering K. Hukushima and Y. Nemoto, J. Phys. Soc. Jpn. 65, 1604 (1996) Simulate multiple replicas of the system at various temperatures. T 1 T 2 T N swap temperature replicas p(e i, T i E i+1, T i+1 ) = min(1, exp( β E)) Single replica performs random walk in temperature space. How do we choose the temperature points?

24 Ensemble optimization Feedback algorithm Measure local diffusivity D(T ) of current in temperature space. Optimal choice of temperatures η opt 1 (T ) D(T ) density of T-points Iterate feedback of diffusivity. diffusivity D(T) feedback temperature T long-range order J 2 /J 1 = 0.2 spiral spin liquid temperature T

25 Example Folding of a (small) protein ST, M. Troyer, U.H.E. Hansmann J. Chem. Phys. 124, (2006).

26 A small protein: HP-36 The chicken villin headpiece project folding time: 4.3 microseconds

27 Random walk in temperature helix-coil transition local diffusivity D(T) competing ground states specific heat C V (T) temperature T [K] 0 Multiple temperature scales are revealed by the local diffusivity.

28 Optimized temperature sets competing ground states helix-coil transition feedback iteration acceptance probability A (T) temperature T [K] local diffusivity D(T) temperature T [K] Feedback reallocates resources towards the relevant temperature scales.

29 Example Quantum systems S. Wessel, N. Stoop, E. Gull, ST, M. Troyer J. Stat. Mech. P12005 (2007).

30 Quantum systems Reconsider the high-temperature series expansion Z = Tr e βh = n=0 β n n! Tr ( H)n = n=0 g(n) β n coefficients density of states We can define a broad-histogram ensemble in the expansion order. M. Troyer, S. Wessel & F. Alet, PRL 90, (2003). Stochastic series expansion (SSE) samples these coefficients n 0 high temperatures? n βn n low temperatures

31 Examples Thermal first-order transition Spin-flop transition hard-core bosons with next-nearest neighbor repulsion H = t i,j 0.12 (a i a j + a j a i)+v 2 i,k n i n k µ i n i local histogram h(n δh ) L=6 L=8 L=10 Λ δh = 500 local histogram h(n) L L = 6 L = 12 L = 10 L = 8 Λ = 20 L expansion order n / Λ 0 H = J expansion order n δh / L 2 spin-1/2 XXZ model in a magnetic field <i,j> [ S x i S x j + S y i Sy j + Sz i S z j ] h i S z i

32 Summary Metropolis cycle configuration suggest an update accept/reject an update non-local update schemes unconventional loops, worms,... statistical ensembles improve sampling efficiency! & overcome entropic barriers