Entanglement Dynamics for the Quantum Disordered XY Chain

Transcription

1 Entanglement Dynamics for the Quantum Disordered XY Chain Houssam Abdul-Rahman Joint with: B. Nachtergaele, R. Sims, G. Stolz AMS Southeastern Sectional Meeting University of Georgia March 6, 2016 Houssam Abdul-Rahman Entanglement Dynamics for XY Chains 1 / 16

2 The Many-Body Hilbert Space Λ = [1, n] := {1, 2,..., n}. For each vertex x Λ we associate the Hilbert space H x := C 2. The Hilbert space associated with the system is H := H x = (C 2 ) n x Λ ρ B(H) is a pure state if ρ 0, Tr ρ = 1, and ρ 2 = ρ. There is a one to one correspondence between pure states and rank one projections. Houssam Abdul-Rahman Entanglement Dynamics for XY Chains 2 / 16

3 The Bipartite Entanglements Λ 0 Fix Λ 0 Λ, consider the decomposition: H = H Λ0 H Λ\Λ0, where H Λ0 = x Λ 0 H x, H Λ\Λ0 = Let ρ be a pure state in B(H), then ρ is separable: if there exist pure states ρ (1) B(H Λ0 ) and ρ (2) B(H Λ\Λ0 ), such that ρ = ρ (1) ρ (2). ρ is entangled: if it is not separable. x Λ\Λ 0 H x. (1) Houssam Abdul-Rahman Entanglement Dynamics for XY Chains 3 / 16

4 Entanglement Entropy The Entanglement Entropy of a pure state ρ with respect to the decomposition H Λ0 H Λ\Λ0 is defined as follows: E(ρ) = Tr [ρ 1 log ρ 1 ], where ρ 1 = Tr H2 ρ. For any pure state ρ B(H): E(ρ) 0. E(ρ) = 0 if and only if ρ is product state (Not Entangled). E(ρ) (log 2) Λ 0 (volume scaling). Houssam Abdul-Rahman Entanglement Dynamics for XY Chains 4 / 16

5 An XY Chain in Transversal Magnetic Field n 1 n H = µ j [(1 + γ j )σj x σj+1 x + (1 γ j )σ y j σy j+1 ] ν j σj z j=1 j=1 Hilbert space H = ( C 2) Λ. Λ = [1, n], Λ 0 a block of spins (subinterval of Λ). µ j, γ j and ν j are i.i.d. σ x = ( ) 0 1, σ y = 1 0 ( ) ( ) 0 i, σ z 1 0 =. i A j acts on the j th component of the tensor product, i.e. A j = 1 (j 1) A 1 (n j) Houssam Abdul-Rahman Entanglement Dynamics for XY Chains 5 / 16

6 Jordan-Wigner Transform n 1 n H = µ j [(1 + γ j )σj x σj+1 x + (1 γ j )σ y j σy j+1 ] ν j σj z a = j=1 ( ) 0 0, a 1 0 = ( ) j=1 σ x j = a j + a j, σy j = i(a j a j ), and σz j = 2a j a j 1. c 1 := a 1, c j := σ z 1 σz 2... σz j 1 a j, j > 1. H = C MC, where C := (c 1, c 1, c 2, c 2,..., c n, c n) t. Houssam Abdul-Rahman Entanglement Dynamics for XY Chains 6 / 16

7 Effective One-Particle Hamiltonian H = C MC, C := (c 1, c 1, c 2, c 2,..., c n, c n) t. M is the block Jacobi matrix ν 1 σ z µ 1 S(γ 1 ) M := µ 1 S(γ 1 ) t µn 1 S(γ n 1 ), µ n 1 S(γ n 1 ) t ν n σ z ( ) 1 γ where S(γ) = σ z + iγσ y =. Recall that: σ γ 1 z = ( ) Houssam Abdul-Rahman Entanglement Dynamics for XY Chains 7 / 16

8 Motivation Question For 1 l n, let H [1,l] and H [l+1,n] be the restrictions of H to the corresponding interval. Let ρ (1) and ρ (2) be any eigenstates of H [1,l] and H [l+1,n], respectively. We study the Schrödinger dynamics ρ t of ρ (1) ρ (2) with respect to the full chain: ρ t := e ith ( ρ (1) ρ (2)) e ith. Note that ρ t is an Entangled state with respect to H [1,l] H [l+1,n]. Question: What can we say about the Entanglement Entropy of ρ t? Houssam Abdul-Rahman Entanglement Dynamics for XY Chains 8 / 16

9 Motivation Question For 1 l n, let H [1,l] and H [l+1,n] be the restrictions of H to the corresponding interval. Let ρ (1) and ρ (2) be any eigenstates of H [1,l] and H [l+1,n], respectively. We study the Schrödinger dynamics ρ t of ρ (1) ρ (2) with respect to the full chain: ρ t := e ith ( ρ (1) ρ (2)) e ith. Note that ρ t is an Entangled state with respect to H [1,l] H [l+1,n]. Question: What can we say about the Entanglement Entropy of ρ t? Houssam Abdul-Rahman Entanglement Dynamics for XY Chains 8 / 16

10 Problem Setting Λ 0 Λ 1 Λ 2 Λ 3 Λ 4 In general Decompose Λ into disjoint intervals Λ 1, Λ 2,..., Λ m. H Λk is the restriction of H to Λ k. ψ k is an eigenfunction of H Λk, and ρ k = ψ k ψ k. Define ρ = m k=1 ρ k, and its dynamics ρ t = e ith ρe ith. Houssam Abdul-Rahman Entanglement Dynamics for XY Chains 9 / 16

11 Assumptions Assumptions: The XY chain H has almost sure simple spectrum. M( satisfies eigencorrelator ) localization, i.e E sup g 1 g(m) jk C 0 (1 + j k ) β, for some β > 6. Applications: µ j = µ, γ j = γ for all j N. ν j are i.i.d from an absolutely continuous, compactly supported distribution. Isotropic case (γ = 0): M Anderson Model. Anisotropic case (γ 0): Large disorder case: Elgart, Shamis, and Sodin (2012). Uniform Spectral gap for M around zero: Chapman and Stolz (2014). Houssam Abdul-Rahman Entanglement Dynamics for XY Chains 10 / 16

12 An Area Law Theorem Then there exists C < such that ( E sup E(ρ t ) t,{ψ k } k=1,2,...,m ) C for all n, m, any choice of the interval Λ 0 Λ and all decompositions Λ 1,..., Λ m of Λ = [1, n]. Hamza/Sims/Stolz (2012). Nachtergale/Sims/Stolz (2013). Sims/Warzel (2016). Houssam Abdul-Rahman Entanglement Dynamics for XY Chains 11 / 16

13 Corollaries Dynamics of Up-Down Spins If m = n number of decompositions is n. eigenfunctions are up and down spins: [ ] 1 e :=, e 0 := [ ] 0. 1 For α = (α 1, α 2,..., α n ) {, } n, define the up-down configuration associated with α: e α = e α1 e α2... e αn Result: ( ) Eigencorrelator localization of M E sup E(e ith e α e α e ith ) α < C. Houssam Abdul-Rahman Entanglement Dynamics for XY Chains 12 / 16

14 Corollaries Dynamics of Up-Down Spins If m = n number of decompositions is n. eigenfunctions are up and down spins: [ ] 1 e :=, e 0 := [ ] 0. 1 For α = (α 1, α 2,..., α n ) {, } n, define the up-down configuration associated with α: e α = e α1 e α2... e αn Result: ( ) Eigencorrelator localization of M E sup E(e ith e α e α e ith ) α < C. Houssam Abdul-Rahman Entanglement Dynamics for XY Chains 12 / 16

15 Corollaries Entanglement of Eigenstates For m = 1 (No Decomposition) Let ψ be an eigenfunction of the full XY chain H. Result: Eigencorrelator localization of M E ( sup E( ψ ψ ) ψ ) < C. Pasture/Slavin (2014). AR/Stolz (2015). Elgart/Pasture/Shcherbina (2015). Houssam Abdul-Rahman Entanglement Dynamics for XY Chains 13 / 16

16 Corollaries Entanglement of Eigenstates For m = 1 (No Decomposition) Let ψ be an eigenfunction of the full XY chain H. Result: Eigencorrelator localization of M E ( sup E( ψ ψ ) ψ ) < C. Pasture/Slavin (2014). AR/Stolz (2015). Elgart/Pasture/Shcherbina (2015). Houssam Abdul-Rahman Entanglement Dynamics for XY Chains 13 / 16

17 Particle Number Transport n 1 n H 0 = [σj x σj+1 x + σ y j σy j+1 ] ν j σj z j=1 The particle number operator N := n e e j. j=1 N e α = ke α, where k = #{j : α j = }. j=1 Let ρ = e α e α then N ρ := Tr N ρ = k. [e ith 0, N ] = 0 the number of up-spins is conserved in time. N S := j S e e j counts the number of up-spins in S Λ. Houssam Abdul-Rahman Entanglement Dynamics for XY Chains 14 / 16

18 d(λ 0,S) Λ 0 S Fix Λ 0 Λ and S Λ \ Λ 0. Initial state: ρ = φ φ, where φ = (e ) Λ 0 (e ) Λ\Λ 0 Theorem For the isotropic XY chain, there exist constants C, η < such that ( ) E sup N S ρt Ce ηd(λ 0,S) t Similar results for disordered Tonks-Girardeau Gas, Seiringer and Warzel (2016) Houssam Abdul-Rahman Entanglement Dynamics for XY Chains 15 / 16

19 Thank you. Houssam Abdul-Rahman Entanglement Dynamics for XY Chains 16 / 16