Introduction to Quantum Monte Carlo

Transcription

1 Entanglement in Strongly Correlated Benasque Feb. 6-17, 2017 Introduction to Quantum Monte Carlo Naoki KAWASHIMA (ISSP)

2 Why bother? Estimating scaling dimension by TRG, TNR, etc is very elegant, and when it works it produces very accurate results. But so far not many universality classes have been examined in this way. Quantum Monte Carlo is still one of the standard methods for studying large systems to obtain reliable characterization of critical points of a given system.

3 Importance Sampling Problem: Compute the momentum of inertia of a circle. X dxdy 2 2 y 2 2 x y 1 x Strategy I (random sampling) 1) Generate a point in [-1,1]x[-1,1] uniform randomly. 2) If the point is in the circle, S += x 2 +y 2. 3) Repeat 1) and 2), N times. 4) X := 4 x S / N

4 Importance Sampling Problem: Compute the moment of inertia of a circle. X 2 2 dxdy x y 2 2 x y 1 Strategy II (importance sampling) 0) Choose any point (x,y) in the circle. 1) Shift the point by a uniform random vector (dx,dy) where dx, dy is a uniform random number in [-D,D] 2) If the point (x+dx,y+dy) is still in the circle x := x+dx, y := y+dy 3) S += x 2 +y 2. 4) Repeat 1) through 3), N times. 5) X := π x S / N

5 Importance Sampling Random Sampling Importance Sampling V 2 d d 2 d d d 2 / 2 1 e 2d d 2 0 High dimensions kill random sampling.

6 Markov Chain X t : state at the t-th step X 0 X1 X 2 X 3 T X X : transition probability P X : the probability of having X at the t-th step t t t or simply, P X T X X P X t1 t1 t1 t t t X t P t1 T P t

7 Detailed Balance Designing a Markov chain We demand, T W T W ij j ji i e.g. W i e E i W i : the target distribution TW W

8 Ergodicity "After a sufficiently long time, any state can appear with non-zero probability." t t t t i, j T ij - T must be irreducible. - Cyclic solution should be excluded.

9 Renyi (1960) Gubernatis, Werner, NK (2016) Convergence and H-Theorem p "Error" W W 1 p p p 1 : the target distribution Free energy (or Kulback-Leibler Information), p F TI KL I KL p p pi log p p t IKL pt p Both and monotocally decrease, and converge to 0 in the limit t. i i i

10 Local Update for Ising Model W i e K W j e K T ji W i W j W j e K K e e K satisfies the detailed balance condition, T ij W j T ji W i

11 Failure of Local Update The configuration changes only locally at each step. Similar to a diffusion process, or a random walk. It takes a very long time to create/annihilate large magnetic clusters. Autocorrelation time diverges near the critical point.

12 Fortuin-Kasteleyn Formula Ising Model Fortuin and Kasteleyn (1969), KS1S 2 K K K e e S 1 S e e v 2 g g S S SG, g=0 g=1, Z V G S G, 1 2 g0,1 S S i G g ij S G

13 Markov Chain in (S,G) Space Z W S, G, W S, G V G S, G SG, S G S G t t t1 t1 W S, G W S, G T G S, T S G W ( S) W ( G)

14 Correlation Functions, S S Z V G S G S S ij G 1 i j i j SG, N 2 c G Z V G ij G ij G G if i and j are connected in G otherwise i and j are correlated i and j belong to the same cluster The size of the clusters in G is O(ξ)

15 Path-Integral

16 Graph Expansion H ij a ˆ g g ij g Ex: S=1/2 Antiferromagnetic Heisenberg Model 0 J J 1 1 Hij J J S, S H S, S S, S S, S 4 2 J J H ˆ ij ij gh 4 2 i j ij i j i j i j AF

17 Loop/Cluster Algorithm Z S S 1 H S S S Sik, k, bij k, b ij gkb,, i, k 1 j, k 1 b ik jk ˆ S S 1 a g S S S S G k b ij S G W S, G i, k 1 j, k 1 g ij ik jk g G ˆ a S S g S S g i, k 1 j, k 1 ij k, b ik jk kb, The same as the Fortuin-Kasteleyn formula!

18 Loop/Cluster Algorithm S=1/2 XY Model

19 Continuous-Time Limit Placing a stone with probability lim a in a cell of height Placing a stone with density a

20 S=1/2 XY Model (Loop Algorithm) Super Fluid Density Stiffness t 2 F s 2 e F H t ij Tr e ij ij e i H t ij ij T L 1 2 Wx y b j b i h. c. ij iff Lx i interaction ij // e x i Harada-Kawashima 1998 Tc = (5)J

21 When Loop Update Fails A two-site problem H J S i S j H S z i S z j Magnetic Field competing against exchange couplings The cause of the problem... The effect of magnetic field is NOT taken into account in the graph generation.

22 High-T Expansion and Worms Ising Model W G ij Z 1 tanh K SiS j tanh K S tanh G G 0 closed graphs K 0 G How can we construct a Markov chain for effective sampling?

23 Extending Graph Space Ising Model graphs with 2 odd vertices G G tanh W G a K "the number of odd vertices in G" G

24 Worm Update for Ising Model Prokof'ev and Svistunov (2001)

25 SSE (Worm Update Version) Worm update for S=1/2 AF Heisenberg model The effect of magnetic field is taken into account in the "scattering probability" of the worm.

26 Application of Classical Monte Carlo --- AKLT state ---

27 TN representation of AKLT State Unitary transformation for bipartite lattice vac!! 1 m z j m i z i b a m z m m S vac vac j i j i j i j i b b a a a b b a Schwinger boson representation spin singlet projector P Projection to S=1 states T T T α β

28 Reduced Density Operator U V (SVD) A Tr B U 4 U "Overlap Matrix" V 2 V Studying is sufficient. Katsura, NK, Kirillov, Korepin and Tanaka (2010)

29 2D AKLT State Katsura, NK, Kirillov, Korepin and Tanaka (2010) Lou, Tanaka, Katsura, and NK (2011) 0,1 of n.n. :, 1, ij i j ij i i i i n n n n z i i

30 Monte Carlo sampling of Overlap Matrix pair creation/ annihilation Local update is sufficient horizontal flip P const, P, freq.of having, Normalization... Tr on the boundary 1 Katsura, NK, Kirillov, Korepin and Tanaka (2010)

31 Boundary Hamiltonian e H B ( is arbitrary) Cirac, Poilblanc, Schuch and Verstraete (2011) Lou, Tanaka, Katsura, and NK (2011) square honeycomb U H H B B U Is H B meaningful? 2 des Cloizeaux- Pearson mode?

32 CFT of Boundary Hamiltonian B e H B The ground state of the boundary Hamiltonian Entanglement entropy of T=0 boundary density operator S B T 0 B l Tr l1, l2,, L y 0 T 0 T 0 l Tr l log l cf: Calabrese-Cardy 2004 S B l c 3 ln B f l const, f l B 0 Lou, Tanaka, Katsura, and NK (2011) L y l sin Ly S B l H B Ly=16 Lx=5 c 1.01(7) l

33 Application of Loop Update --- SU(N) Heisenberg Model ---

34 SU(N) Heisenberg Model A natural extension of the SU(2) AF Heisenberg model S S r r H J N r, r' generators of SU(N) rotation represented by some representation the same with the conjugate S r S representation r R S, S S S,,, 1,2,, N Representation: n=1 (fundamental representation.),,,,etc n=2 n=3 n=4

35 2D Analogue of "Haldane" States Prediction from 1/N expansion Arovas & Auerbach (1988) Read & Sachdev (1989) Nc ~ 5.3 n For numerical evidences, see n=1 : Tanabe & NK: PRL (2007) n=2,3 : Okubo,Harada,Lou & NK : PRB (2015)

36 Magnetic/Non-Magnetic Transition (SU(N) JQ-Model) SU(2) symmetry broken. Space symmetry not broken. SU(2) symmetry not broken. Space symmetry broken. Text book symmetry breaking paradigm predict the 1st order transition 1 1 H JSi S j Q Si S j Sk Sl ( ij) p i, j, k, l 4 4 =Q/J

37 SU(3) and SU(4) J-Q Models J. Lou, A. Sandvik, N.K (2009) SU(3) J-Q s 0.38(3), 0.65(3) SU(4) J-Q s 0.42(5), 0.70(2)

38 Universality? For N 1, s 1. T. Senthil, et al, Science 303, 1490 (2004) M. Levin and T. Senthil, Phys. Rev. B 70, R (2004). For N 1, d N 1. CP N-1 Field Theory: M. A. Metlitski, et al, PRB 78, (2008) G. Murthy and S. Sachdev, Nucl. Phys. B 344, 557 (1990). 2D JQ-Model (Lou, Sandvik, N.K.)

39 Monopole Scaling Dimension up to O(N -1 ) D O 2 N N N N D N

40 Further Evidences for DCP Kaul, Sandvik: PRL108_ (2012) Introducing "ferromagnatic" interaction to favor the Neel state for large N. Kaul, Melko and Sandvik: arxiv: Bilayer system show 1st order transition.

41 An inconvenient truth Harada et al (2013) System-size-restricted finite-size scaling yields strongly size dependent estimates of scaling dimensions

42 Application of worm update --- Bose Hubbard Model ---

43 "Big Wedding Cake" in ultracold atoms t / U 0.05, r 0.08, L 3 0 / U 3.3, ~ 2.610, t 5.0. Fisher et al (1989) The number of bosons N (1) 64 ~ Kato and N.K., PRE (2009) 3,

44 Supersolid on Triangular Lattice Wessel and Troyer: PRL 95, (2005) BHM with hardcores and n.n. interaction T=0 "superflow path" No vacancy condensation below n=1/3, and no interstitial condensation above n=1/3.

45 Continuous Space MC Corboz, Boninsegni, Pollet and Troyer (2008) Can Helium-4 on graphite surface be super-solid? T 0.5K, A IC, A 2 C, F C, A Solid state was observed only when the first layer is fixed.

46 Remaining Puzzle... Experiment (Nakamura et al (2016)) suggests existence of an intermediate phase "C2". What is "C2" phase? Why two peaks in C-T curve in C2 phase?

47 Summary When it is not poisoned by negative signs, Monte Carlo method is suitable for many systems, and complementary to the tensor network methods.

48 END