Time Evolving Block Decimation Algorithm

Transcription

1 Time Evolving Block Decimation Algorithm Application to bosons on a lattice Jakub Zakrzewski Marian Smoluchowski Institute of Physics and Mark Kac Complex Systems Research Center, Jagiellonian University, Kraków, Poland November 2006

2 What will it be about Intro Motivation Example: Bose-Hubbard Time-Evolving Block Decimation Entanglement Vidal s construction Time evolution algorithm Finding the ground state Costs and Improvements Bose-Hubbard example Extension to an infinite chain Bose-Hubbard example Further reading

3 Motivation Many interacting particles systems (bosons, fermions, spins - typical problems of statistical physics, condensed matter, e.g. Ising model) Static properties, ground state, low excitations, quantum phase transitions Dynamics, time evolution,... i ψ = H ψ t Typowe metody Typical approaches: Finite difference method Basis sets expansions (or in overcomplete coherent states..) Large scale diagonalizations but...

4 Example: Bose-Hubbard Hamiltonian Ĥ = J d i=0 ( ) â i+1âi + h.a. + d ɛ iˆn i + U 2 i=0 d ˆn i (ˆn i 1), i=1 â i (â i ) are creation (annihilation) operators for a boson at site i ˆn i = â i âi occupation number operator J - hopping rate, U - interaction energy, d maximal occupation ( dim(h) = d + 1 on each site) Very sparse matrix of dimension: D = (N + M 1)! N!(M 1)!, the number of particles is N and number of lattice sites M. D = 6435 for N = M = 8 D = for N = M = 10 D = for N = M = 12

5 Bose-Hubbard Hamiltonian possibilities: Basis truncation Lanczos scheme s...

6 Bose-Hubbard Hamiltonian possibilities: Basis truncation Lanczos scheme s... yield a partial gain only.. Alternatives for a static (ground state) problem Quantum Monte Carlo (QMC) Density Matrix Renormalization Group (DMRG) approximations (mean field etc.)... as well as novel ideas coming from Quantum Entanglement machinery

7 Quantum Entanglement Primer Consider a system composed of two parts iff Ψ = Φ [A] Φ [B] the state is separable otherwise Schmidt s decomposition possible χ A Ψ = λ α Φ [A] α Φ [B] α, Φ [A] α ( Φ [B] ρ [A] = Tr B ( Ψ Ψ α=1 α ) is an eigenvector with eigenvalue λ α 2 > 0 of (ρ [B] ) also Φ [A] α Ψ = λ α Φ [B] α. The entanglement of Ψ is e.g. χ max A χ A. We assume χ is small.

8 Entanglement Primer 2 d Ψ = C ij i A j B Then Singular Value Decomposition does the job i,j C = ÛˆDV ; ˆD = diagλ i Now we shall concentrate on states with hopefully small entanglement. Also the type of time-evolution that will slowly increase χ only.

9 Vidal s construction Consider now a general state for M-site chain: d d Ψ = c i1 i n i 1 i n, i 1 =0 i n=0

10 Vidal s construction Consider now a general state for M-site chain: d d Ψ = c i1 i n i 1 i n, i 1 =0 i n=0 The key ingredient of Vidal s simulation protocol: Ψ Γ [1] λ [1] Γ [2] λ [2] Γ [l] λ [n 1] Γ [n].

11 Vidal s construction Consider now a general state for M-site chain: d d Ψ = c i1 i n i 1 i n, i 1 =0 i n=0 The key ingredient of Vidal s simulation protocol: Ψ Γ [1] λ [1] Γ [2] λ [2] Γ [l] λ [n 1] Γ [n]. Γ [l]i αα - tensor associated with site l λ [l] α - the Schmidt splitting [1 l]:[(l +1) n]. explicite :-) c i1 i 2 i n = Γ [1]i 1 α 1 α 1,,α n 1 λ [l] α 1 Γ [2]i 2 α 1 α 2 λ [2] α 2 Γ [n]in α n 1.

12 Vidal s construction 2 How to get Γ s (i) expand Φ [2 n] α Ψ = α 1 λ [1] α 1 Φ [1] α 1 Φ [2 n] α 1 = i 1,α 1 Γ [1]i 1 α 1 λ [1] α 1 i 1 Φ [2 n] α 1, Φ [2 n] α 1 = i 2 i 2 τ [3 n] α 1 i 2 ; (ii) express via Schmidt decomposition of 2 : [3..n] (iii) make all substitutions τ [3 n] α 1 i 2 = α 2 Γ [2]i 2 α 1 α 2 λ [2] α 2 Φ [3 n] α 2 ; Ψ = Γ α [1]i1 1 λ α [1] 1 Γ [2]i 2 α 1 α 2 λ [2] α 2 i 1 i 2 Φ [3 n] α 1. i 1,α 1,i 2,α 2

13 (iv) move to site 4,5,...

14 Remarks: close links to finitely correlated states (Fannes et al.) close links to matrix product states (Öslund and Rommer) thus very similar to DMRG scheme

15 Remarks: close links to finitely correlated states (Fannes et al.) close links to matrix product states (Öslund and Rommer) thus very similar to DMRG scheme So what is new?

16 Remarks: close links to finitely correlated states (Fannes et al.) close links to matrix product states (Öslund and Rommer) thus very similar to DMRG scheme So what is new? simple study of time dependence

17 Remarks: close links to finitely correlated states (Fannes et al.) close links to matrix product states (Öslund and Rommer) thus very similar to DMRG scheme So what is new? simple study of time dependence similar approach for ground state solution

18 Remarks: close links to finitely correlated states (Fannes et al.) close links to matrix product states (Öslund and Rommer) thus very similar to DMRG scheme So what is new? simple study of time dependence similar approach for ground state solution natural extension to truly infinite systems

19 Time evolution Consider a Hamiltonian H = M i=1 H [i] 1 + M i=1 H [i,i+1] 2 ;

20 Time evolution Consider a Hamiltonian H = M i=1 H [i] 1 + M i=1 H [i,i+1] 2 ; Time evolution: Ψ(T ) = exp( iht ) Ψ(0) done, as usual, in small steps Ψ(t + τ) = exp( ihτ) Ψ(t)

21 Time evolution Consider a Hamiltonian H = M i=1 H [i] 1 + M i=1 H [i,i+1] 2 ; Time evolution: Ψ(T ) = exp( iht ) Ψ(0) done, as usual, in small steps Ψ(t + τ) = exp( ihτ) Ψ(t) Decompose H as H = F + G, F F [l] even l even l G G [l] odd l odd l (H [l] 1 + H[l,l+1] 2 ), (H [l] 1 + H[l,l+1] 2 ),

22 Time evolution Consider a Hamiltonian H = M i=1 H [i] 1 + M i=1 H [i,i+1] 2 ; Time evolution: Ψ(T ) = exp( iht ) Ψ(0) done, as usual, in small steps Ψ(t + τ) = exp( ihτ) Ψ(t) Decompose H as H = F + G, F F [l] even l even l G G [l] odd l odd l (H [l] 1 + H[l,l+1] 2 ), (H [l] 1 + H[l,l+1] 2 ), Observe: [F [l], F [l ] ] = 0 ([G [l], G [l ] ] = 0) but [F, G] 0.

23 Time evolution 2 Apply Trotter expansion of order p for exp( ihτ) exp( i(f + G)τ) f p [exp( if τ), exp( igτ)], f 1 (x, y) = xy, f 2 (x, y) = x 1/2 yx 1/2,

24 Time evolution 2 Apply Trotter expansion of order p for exp( ihτ) exp( i(f + G)τ) f p [exp( if τ), exp( igτ)], f 1 (x, y) = xy, f 2 (x, y) = x 1/2 yx 1/2, And we get a chain of two-site gates: exp( if τ) = Π l exp( if [l] τ) = Π l exp( i(h [l] 1 +H[l,l+1] 2 )τ) = Π l U [l] Vidal s lemma: Updating Ψ expressed as Ψ Γ [1] λ [1] Γ [2] λ [2] Γ [l] λ [n 1] Γ [n]. is a LOCAL operation affecting Γ [l], λ [l] and Γ [l+1].

25 Time evolution 3

26 Time evolution 3 Recall that there are at most χ links of our l and l + 1 sites χ d Ψ = αijγ, α,β,γ=1 i,j=0 Γ [l]i αβ λ[l] β Γ[l+1]j βγ where Ψ = U (l) Ψ = Θ ij αγ = β kn χ α,γ=1 i,j=0 d Θ ij αγ αijγ, V ij kn Γ[l]k αβ λ[l] β Γ[l+1]n βγ.

27 Time evolution 4 Now singular value decomposition of Θ grouping indexes as [αi] : [jγ]: Θ = β X (l) [αi]β λ β Y β[jγ]

28 Time evolution 4 Now singular value decomposition of Θ grouping indexes as [αi] : [jγ]: Θ = β X (l) [αi]β λ β Y β[jγ] And Γ [l]i αβ = X [αi]β/λ (l 1) α ; Γ [l+1]i αβ = Y [αi]β /λ (l+1) β

29 Time evolution 4 Now singular value decomposition of Θ grouping indexes as [αi] : [jγ]: Θ = β X (l) [αi]β λ β Y β[jγ] And Γ [l]i αβ = X [αi]β/λ (l 1) α ; Γ [l+1]i αβ = Y [αi]β /λ (l+1) β Note: α [1, χ], i.e. [αi] leads to χ(d + 1) as dimension of Θ.

30 Time evolution 4 Now singular value decomposition of Θ grouping indexes as [αi] : [jγ]: Θ = β X (l) [αi]β λ β Y β[jγ] And Γ [l]i αβ = X [αi]β/λ (l 1) α ; Γ [l+1]i αβ = Y [αi]β /λ (l+1) β Note: α [1, χ], i.e. [αi] leads to χ(d + 1) as dimension of Θ. SVD yields λ [l] of that length, normalized so as ( λ[l] α ) 2 = 1.

31 Time evolution 4 Now singular value decomposition of Θ grouping indexes as [αi] : [jγ]: Θ = β X (l) [αi]β λ β Y β[jγ] And Γ [l]i αβ = X [αi]β/λ (l 1) α ; Γ [l+1]i αβ = Y [αi]β /λ (l+1) β Note: α [1, χ], i.e. [αi] leads to χ(d + 1) as dimension of Θ. SVD yields λ [l] of that length, normalized so as ( λ[l] α ) 2 = 1. Numerically we want to limit the length of λ to some χ e so we drop smallest and renormalize: α = [ λ [l] χ ɛ α=1 And that s the end of the story!! λ [l] α 2 ] 1 2 λ [l] α

32 Finding the ground state (one of the methods) Start with Φ φ [1] φ [n], Φ Ψ gr 0, (expansion into λ s and Γ s) trivial.

33 Finding the ground state (one of the methods) Start with Φ φ [1] φ [n], Φ Ψ gr 0, (expansion into λ s and Γ s) trivial. Use the scheme to simulate an evolution in imaginary time t according to H, Ψ gr = lim t exp( Ht) Φ exp( Ht) Φ ;

34 Finding the ground state (one of the methods) Start with Φ φ [1] φ [n], Φ Ψ gr 0, (expansion into λ s and Γ s) trivial. Use the scheme to simulate an evolution in imaginary time t according to H, Ψ gr = lim t exp( Ht) Φ exp( Ht) Φ ; The procedure converges almost always :-)

35 Costs and Improvements Generally execution time scales as ((d + 1)χ e ) 3 Further simple improvements:

36 Costs and Improvements Generally execution time scales as ((d + 1)χ e ) 3 Further simple improvements: Global selection rules ((d + 1)χ e ) 3 (d eff χ e ) 3

37 Costs and Improvements Generally execution time scales as ((d + 1)χ e ) 3 Further simple improvements: Global selection rules ((d + 1)χ e ) 3 (d eff χ e ) 3 The procedure is easy to parallelize :-)

38 Costs and Improvements Generally execution time scales as ((d + 1)χ e ) 3 Further simple improvements: Global selection rules ((d + 1)χ e ) 3 (d eff χ e ) 3 The procedure is easy to parallelize :-) But yet to be done :-( Further difficult (under construction) improvements Long range interactions Renormalizable entanglement Tree structures multidimensions - - projected entanged pair states, MERA

39 Bose-Hubbard example Comparison with high order perturbation theory (1) Ĥ = J d i=0 ( ) â i+1âi + h.a. + d ɛ iˆn i + U 2 i=0 d ˆn i (ˆn i 1), i=1 1 B. Damski and J. Zakrzewski, Phys. Rev. A74, (2006)

40 Bose-Hubbard example Comparison with high order perturbation theory (1) Ĥ = J d i=0 ( ) â i+1âi + h.a. + d ɛ iˆn i + U 2 i=0 d ˆn i (ˆn i 1), i=1 1 B. Damski and J. Zakrzewski, Phys. Rev. A74, (2006)

41 Bose-Hubbard example Comparison with high order perturbation theory (1) Ĥ = J d i=0 ( ) â i+1âi + h.a. + d ɛ iˆn i + U 2 i=0 d ˆn i (ˆn i 1), i=1 1 B. Damski and J. Zakrzewski, Phys. Rev. A74, (2006)

42 Bose-Hubbard example Comparison with high order perturbation theory (1) Ĥ = J d i=0 ( ) â i+1âi + h.a. + d ɛ iˆn i + U 2 i=0 d ˆn i (ˆn i 1), i=1 1 B. Damski and J. Zakrzewski, Phys. Rev. A74, (2006)

43 Treatment of infinite systems Let H = r H[r,r+1] be invariant under r r + 1 transformation. If Ψ 0 is also translationally invariant, evolution preserves that. Now we can treat truly infinite systems χ Ψ = λ (r) where Φ ( r+1) α = Ψ = χ Φ ( r) α = D α,β,γ=1 i,j=1 α=1 χ β=1 i=1 χ α Φ ( r) α D D β=1 i=1 λ (r 1) α Γ (r) iαβ λ(r) Φ ( r+1) α ; λ (r) β Γ(r+1) iβα Φ( r) Γ (r+1) iαβ β Γ(r+1) jβγ λ (r+1) γ β i (r+1), λ(r+1) β i (r) Φ ( r+1) β. i (r) j (r+1) Φ α ( r) Φ γ ( r+2).

44 Infinite systems 2 Ψ shift invariant (r) is not needed, but.. We need to act with the two-side gate: U [r,r+1] exp( ih [r,r+1] δt), δt 1 that breaks for a moment the symmetry. Thus Γ [2r] = Γ A, λ [2r] = λ A, Γ [2r+1] = Γ B, λ [2r+1] = λ B, r Z.

45 Infinite systems 2 Ψ shift invariant (r) is not needed, but.. We need to act with the two-side gate: U [r,r+1] exp( ih [r,r+1] δt), δt 1 that breaks for a moment the symmetry. Thus Γ [2r] = Γ A, λ [2r] = λ A, Γ [2r+1] = Γ B, λ [2r+1] = λ B, r Z.

46 Infinite systems 3

47 Bose-Hubbard example Ĥ = J d i=0 ( ) â i+1âi + h.a. + d ɛ iˆn i + U 2 i=0 d ˆn i (ˆn i 1) µ i=1 d ˆn i, Note the presence of the chemical potential µ. Challenging problem: The tip of the quantum phase transitions i=1

48 Bose-Hubbard example 2 Various methods used (mean-field, Bethe-ansatz, diagonalization N = M = 12, QMC, DMRG...) The most stable approach - the slope of the correlation function C(s) = Ψ gr â r+sâr Ψ gr. exponential decay for large s in Mott Insulator Phase. power law decay for large s in Superfluid Phase C(s) s K /2 with K = 1/2 at transition.

49 Bose-Hubbard example 3 T. Kühner, S.R. White, H. Monien, Phys. Rev. B61, (2000) give J c = ± 0.01 (DMRG). Our simulation (2) yields J c = ± Table: The location of the critical point J c for different ranges of s value in the correlation function C(s) s KWM2000 This work 4 s ± ± s ± ± s ± ± s ± ± s ± ± s ± s ± J. Zakrzewski and D. Delande, in preparation (2006)

50 Bose-Hubbard example 4 And that is converged (thanks to ICM...)

51 Further reading the basic algorithm: G. Vidal Phys. Rev. Lett. 91, (2003). the infinite case: G. Vidal cond-mat/ relation to DMRG A.J. Daley et al. cond-mat/ ; DMRG review U. Schollwoeck, Rev. Mod. Phys. 77, 259 (2005) and cond-mat/ attempts for multidimensions F. Verstraete and J.I. Cirac cond-mat/ (PEPS); G. Vidal cond-mat/ Entanglement renormalization: G. Vidal, cond-mat/