IPAM/UCLA, Sat 24 th Jan Numerical Approaches to Quantum Many-Body Systems. QS2009 tutorials. lecture: Tensor Networks.

Transcription

1 IPAM/UCLA, Sat 24 th Jan 2009 umerical Approaches to Quantum Many-Body Systems QS2009 tutorials lecture: Tensor etworks Guifre Vidal

2 Outline Tensor etworks Computation of expected values Optimization of a tensor network (energy minimization, time evolution)

3 Tensor etworks Computation of expected values Optimization of a tensor network (energy minimization, time evolution)

4 Graphical representation of matrices/tensors A a1 a = 2 M a m B b11 K b1 n = M O M bm 1 b L mn vector matrix rank 3 tensor rank p tensor a α b αβ c αβγ c αβγ K κ β α α β α γ α β γ... δ κ

5 Graphical representation of matrices/tensors product of tensors Q = RS q = r s αγ αβ βγ β α γ α γ = Q R S

6 Graphical representation of matrices/tensors other examples: x Ay c c * * αβγ αβγ αβγ? trabcd D A B C

7 Tensor etworks d L H1 H 2... H d d d GS = L ci 1i2... i i1 i2... i i = 1 i = 1 i = Monte Carlo sampling i 1 i 2 i 3 i 4... i O( β ) samples d coefficients Tensor etwork i1 i2 i3 i 4... i O( ) coefficients

8 Tensor etworks d d d GS = L ci 1i2... i i1 i2... i i = 1 i = 1 i = i 1 i 2 d i 3 i 4... i coefficients i1 i2 i3 i 4 p O( χ )... coefficients α 1 i α 2 Tensor etwork L α p = 1,2, L, χ χ = exp M H χ = 4 χ = 3 χ = 2 χ = 1 (mean field theory)

9 Tensor etworks examples of tensor network representations: Matrix Product State (MPS) Tree Tensor etwork (TT) 1D Wilson (RG) 1975 Fannes, achtergaele, Werner 1992 White 1992 (DMRG) Oestlund, Rommer 1995 time evolution in 1D D 1D/2D 1D/2D Multi-scale Entanglement Renormalization Ansatz (MERA) Tensor Product State (TPS) or Projected Entangled Pair State (PEPS) ishino 2000 Verstraete, Cirac (next lecture!)

10 What do we want to be able to do with a tensor network? efficient specification of the state = c i i... i i i... i 1 2 i i... i efficient computation of expected values σ [4] [4] σ? efficient computation of states of interest: e.g. ground state efficient simulation of time evolution GS?

11 Tensor etworks Computation of expected values Optimization of a tensor network (energy minimization, time evolution)

12 Computation of expected values Example: σ [4] GS z GS Without a tensor network: GS σ z exp( ) operations GS With a MPS: q O( χ ) operations GS σ [4] z GS σ z GS GS

13 Computation of expected values With a TT: pr [6] = σ i i i i i O χ q' ( ) operations

14 Computation of expected values with a TPS/PEPS: approximation MPS! O χ 2 ( ) O χ q'' ( ) TPS / PEPS operations exact approximate TPS / PEPS operations

15 Tensor etworks Computation of expected values Optimization of a tensor network (energy minimization, time evolution)

16 Optimization of a tensor network Ground state GS obtained from min H GS H GS H H = ij h ij K

17 Optimization of a tensor network Ground state h h12 23 x r GS H = hij min ij H K x r = x r M x r x r x r min r x r r x M x r r x x sweep over tensor network r M x r = λ x generalized eigenvalue problem

18 Optimization of a tensor network time evolution ' = U U U U = exp iht H = ij h ij break into steps of the form ' = u u

19 Optimization of a tensor network time evolution 2 ' = u u ' min u ' ψ ' = + 2 u ' u u ' u u ' ' ' c y r x r x r M x r y r x r min r r r r r r r c x y y x + x M x x r M x = r y sweep over tensor network

20 Optimization of a tensor network Therefore the ground state GS can be obtained in two ways: minimization of energy min H simulation of evolution in Euclidian or imaginary time Hτ e 0 GS = lim τ e Hτ 0 0 GS

21 Summary Tensor etworks Computation of expected values Optimization of a tensor network (energy minimization, time evolution)