Wang-Landau sampling for Quantum Monte Carlo. Stefan Wessel Institut für Theoretische Physik III Universität Stuttgart

Transcription

1 Wang-Landau sampling for Quantum Monte Carlo Stefan Wessel Institut für Theoretische Physik III Universität Stuttgart

2 Overview Classical Monte Carlo First order phase transitions Classical Wang-Landau algorithm Extension to quantum systems Applications Free energy Configurations

3 Classical Monte Carlo simulations We want to calculate a thermal average A =! c A e c "# E c! "# Ec / Z with Z = e c Exponentially large number of configurations draw a representative statistical sample by importance sampling Pick M configurations c i with probability p ci = e "#E c i /Z Calculate statistical average Within a statistical error A " A = 1 M "A = Var A M M # i=1 A c i Problem: we cannot calculate p ci = e "#E ci /Z since we do not know Z

4 Markov chains and Metropolis algorithm Metropolis algorithm builds a Markov chain Transition probabilities W x,y for transition x y need to fulfill Ergodicity: any configuration reachable from any other Detailed balance: c 1 " c 2 "..." c i " c i+1 "... "x,y #n : ( W n ) x,y 0 W x,y = p y W y,x p x Simplest algorithm due to Metropolis (1953): Wx,y = min[1, py px ] Needs only relative probabilities (energy differences) p y p x = e "# (E y "E x )

5 First order phase transitions Tunneling out of meta-stable state is suppressed exponentially gas liquid liquid gas P # " e! cl d!1 / T How can we tunnel out of metastable state?? liquid gas critical point T Critical slowing down At critical point was solved by cluster updates

6 The Wang-Landau method Directly calculates density of states r (E) instead of canonical averages Acceptance rate proportional to inverse density of states 1! Ei / kbt pi = instead of pi = e " ( Ei ) All energy levels visited equally well 1 P( E) = " E c, E P(c)! #( E) = 1 c #( E) Random walk in energy space Flat histogram in energy space => no tunneling problem Free energy accessible F = "k B T ln #(E)e "E / k B T One simulation gives results for all T E F. Wang & D.P. Landau (2001)

7 The Wang-Landau algorithm Initially, r (E) is unknown Start with r (E)=1 and adjust iteratively Only a few modifications to usual sampling needed Start with modification factor f=e do { do { Metropolis updates with transition probability W[i ] = min[1,r (E i ) / r (E j )] Adjust r (E) at each step: r (E) r (E) x f Increase histogram H(E)++ } until histogram H(E) is flat (e.g. 20%) decrease f (e.g.f f) reset H(E) } until f Comments Initially: multiplicative changes with large f allow rapid crude convergence Finally: small f means fine structure of r (E) can be mapped out

8 Example application of Wang-Landau sampling 10 states Potts model in 2D First order phase transition T c =0.701 P( E, T c ) = "( E) e! E / k B T c

9 Applications and Extensions First and second order phase transitions and disorder F. Wang and D.P. Landau, (2001) Scaling analysis for Ising spin glasses P. Dayal, et al., (2003). Improved sampling and optimized ensembles S. Trebst, D. Huse, and M. Troyer, (2003) C. Zhou and R.N. Bhatt, (2003) B.J. Schulz et al., (2002). C. Yamaguchi and N. Kawashima, (2002). Continuum simulations Q. Yan et al., (2002). M.S. Shell et al., (2002). Polymer films T.S. Jan et al., (2002). Protein folding N. Rathore et al., (2002). Try out the Ψ-Mag Wang-Landau tutorial!

10 Quantum version of the Wang-Landau algorithm Density of states not accessible directly in quantum case Classical Quantum Z = e "E c / k B T = #(E)e "E / k BT c E Z = Tre "#H Quantum version based on SSE Z = ( % n % n= 0 % " n % % n! # '(&H bi ) # # ( ) b 1,...,b We have a similar expression as a sum over orders n instead of energies i=1 ( & n)!" % % n # '(&H bi ) #! # ( ) b 1,...,b = " n g(n) n= 0 i=1

11 Quantum version of the Wang-Landau algorithm We want flat histogram in order n Use the Wang-Landau algorithm to get Z = # n= 0 " n g(n) from Z = Small change in acceptance rates for diagonal updates " & n= 0 (" # n)! & & n % '(#H bi ) % "! % ( ) b 1,...,b " " i=1 d P[1" H (i, j ) ' ] = min 1, #N d bonds H (i, j ) ) ( % & n *, + ' -- - min 1, N d bonds H (i, j ) ) ( % & n Wang&Landau - " g(n) *, g(n +1) + d P[H (i, j ) ' "1] = min) 1, ( # n +1 d %N bonds & H (i, j ) & * ' Wang-Landau, --- -" min) 1, + ( # n +1 d N bonds & H (i, j ) & g(n) * g(n 1), + Nonlocal updates (directed loops, ) do not change n and thus remain unchanged! Cutoff L limits temperatures to b < L / E 0

12 A first test L=10 site Heisenberg chain with L = 250 Free energy and entropy available One simulation for all temperatures Clear limit of reliability T>0.05J due to cutoff H v v 10 = J! S i " Si+ 1 i= 1

13 Thermal phase transitions 3D Heisenberg antiferromagnet Second order phase transition at T c =0.947J Parallelization via splitting of the n-range among several CPUs H v v = J" S i! S i, j j

14 Quantum version perturbation expansion Instead of temperature a coupling constant can be varied H H +! V = 0 Based on finite temperature perturbation expansion Z = Tr(e "#H ("#) n ) = % % % & '("H bi ) & ( n ( (b 1,...,b n ) n! n= 0 & ( ) b 1,...,b n * (* " n)!# n * ) %% % & '("H bi ) & ( n ( (b 1,...,b * ) *! n= 0 * & ( ) b 1,...,b * Flat histogram in order n l of perturbation expansion i=1 = ( n ( % g(n ( ) n " (b 1,...,b n ) counts # of " terms n ( = 0 n i=1

15 Perturbation series by Wang-Landau We want flat histogram in order n l Use the Wang-Landau algorithm to get # Z = " n " g(n " ) n " = 0 from Small change in acceptance rates for diagonal updates " (" # n)! n Z = && & % '(#H bi ) % ( n ( (b 1,...,b " ) "! n= 0 % ( ) b 1,...,b " " i=1 d P[1 " H (i, j ) ' ] = min 1, #N d bonds H (i, j ) ) ( % & n *, + ' -- - min 1, #N d bonds H (i, j ) ) ( % & n Wang&Landau - " g(n. ) *, g(n. + /n. ) + d P[H (i, j ) ' "1] = min) 1, ( # n +1 d %N bonds & H (i, j ) & * Wang-Landau, - " + ' -- - min) 1, ( # n +1 d %N bonds & H (i, j ) & g(n. ) * g(n. /n. ), + Nonlocal updates (directed loops, ) do not change n λ and thus remain unchanged! Cutoff L limits value of l for which the series converges

16 Quantum phase transition Bilayer Heisenberg Antiferromagnet Single simulation gives results for all values of λ=j/j 2 v v v v H = J Si, 2 Quantum phase transition at λ =0.396 Parallelization as in thermal case!! Si, p " S j, p + J '! Si,1 " p= 1 < i, j> i

17 Speedup at first order phase transitions Greatly reduced tunneling times at free energy barriers 2D hard-core bosons ( ) H = "t# a i a j + h.c. + V 2 # n i n j i, j i, j 8! 8 V Stripe order / t 3 V / t 3 2 = 8! 8 2 = G. Schmidt

18 Summary Extension of Wang-Landau sampling to quantum systems Stochastically evaluate series expansion coefficients High-temperature series Perturbation series Features # Z = g(n) n! n= 0 # Z = " n " g(n " ) n " = 0 " n Flat histogram in the expansion order Allows calculation of free energy Like classical systems, allows tunneling through free energy barriers

19 Qantum Wang-Landau in ALPS Application qmc/qwl Allows high temperature expansions for isotropic spin-1/2 systems You can reproduce the data presented here in the qwl tutorial.