of the D Bose-Hubbard model V. Alba,M. Haque, A. Läuchli 3 Ludwig-Maximilians Universität, München Max Planck Institute for the Physics of Complex Systems, Dresden 3 University of Innsbruck, Innsbruck November 3,
Outline Introduction. (ES). in the D Bose-Hubbard. In the Mott insulator: ES is boundary local. Perturbative structure. Entanglement dispersions. Effective boundary entanglement Hamiltonian. In the superfluid phase: Tower of states structure (symmetry breaking). Entanglement gap. Entanglement entropies and the gap.
Introduction Consider a quantum system in d dimensions in the ground state Ψ. If the system is bipartite: ρ Ψ Ψ H = H A H B ρ A = Tr B ρ How to quantify the entanglement (quantum correlations) between A and B? von Neumann entropy S A = Trρ A log ρ A = i λ i log λ i Area law: away from criticality the entanglement is proportional to the surface area of subsystem A S A L d For critical systems violations of the area law [Vidal et al,3][calabrese, Cardy,4] D S A = c 3 log L (D S const)
(ES) Given the reduced density matrix ρ A : {λ} = σ(ρ A ) ρ A = e H A σ(h A ) log({λ}) Quantum Hall: the ES retains all the features of the critical edge modes (ES edge spectra correspondence) [Li,Haldane, 8] ξ 8 6 4 8 6 4 56 58 6 6 64 (a) P[ ] 4 45 5 55 6 65 L A z More general: all the relevant information to describe the physics of a system is encoded in the entanglement spectrum.
D Bose-Hubbard on a cylinder D Bose-Hubbard at unit filling on a cylinder. H = ij (b i b j + h.c.) + U i n i (n i ) L B A [Fisher,eichman,Grinstein,Fisher,989] [Jaksch,Bruder,Cirac,Gardiner,998]
Boundary locality of the ES in gapped systems Gapped systems area law (in D [Hastings,7]). The ES is a boundary local quantity. [V.A.,M. Haque,A.M. Läuchli,] A S S S B Lower part of ES is dominated by the physics at the boundary between A and B. Higher levels in the ES involve degrees of freedom deeper into the subsystems bulk.
Mott insulator: ES structure 3 DMRG U = ( )(-)+ ( )(-)+ ( 3 ) ( 3 ) A L B ξ ( ) ( ( ) ) -4-3 - - 3 4 Linear envelope (incompressibility of Mott phase). Multiplet structure (entanglement dispersions) Simple multiplicity counting only function of (boundary locality). δn A
Boundary-local perturbation theory for ES At U = the ground state is a product (Mott) state Ψ : H = H + H p H i n i(n i ) H p U i,j (b i b j + h.c.) At large U perturbative expansion Ψ α U α Ψ α. Ψ α (H p ) α Ψ A S S S B Boundary locality and perturbation theory [V.A.,M. Haque,A.M. Läuchli,]: To have the ES accurate up to order α it is sufficient to consider only the perturbative terms living in a box S (α)
Boundary-local perturbation theory at work ξ 3 ( )(-)+ ( )(-)+ ( 3 3 ) (d) (e) ( 3 ) ( ) (c) ( ( ) ) (b) (a) -4-3 - - 3 4 δn A perturbative order α
Boundary-local perturbation theory at work ξ 3 ( )(-)+ ( )(-)+ ( 3 ) (d) (e) ( ) (c) ( ) (b) ( ) ( 3 ) 3 perturbative order α (a) (d) A (b) L (e) B 3 (c) (a) -4-3 - - 3 4 δn A Boundary locality: The ES structure (multiplicities, envelope behavior,...) is understood in terms of boundary perturbative processes.
Entanglement dispersions ξ 3 ( )(-)+ ( )(-)+ ( ( 3 3 ) (d) (e) 3 ) ( ) (c) ( ( ) ) (b) (a) -4-3 - - 3 4 δn A α ξ ξ 7.5 7 8.8 8.6 8.4 4 6 8 k 4 6 k k boundary momentum The envelope ES multiplets show simple entanglement dispersions.
Boundary entanglement Hamiltonians ξ 8.8 8.6 8.4 Let us focus on the sector δn A = : 4 6 k ξ k = log(u /) + ( + )/U 4 h/u + ( 5)/U 3i cos k {z } {z } A A 68/U {z } A cos k + 55/U 3 cos 3k {z } A 3 Single particle D tight binding entanglement Hamiltonian: H E = ir A r(b i b i+r + h.c.) Effective degrees of freedom localized at the boundary (boundary locality). Gapped phase H E is short range (A r U r ).[Cirac et al.,]
ES across the Mott superfluid transition 3 ξ U=5 Mott Mott superfluid δ - δn A 3 U= - δn A 5 U=5-4 - 4 δn A δ ξ (δna =,k=) ξ ξ (δna =,k min ) ξ (δna =,k=) 4 S 3 U M 4 δ =3 =4 =5 =6 =8 3 U Dramatic change in the ES structure across the transition (U c 7). Dual behavior of the gap and the level spacing δ in the two phases.
ES structure in the superfluid phase DMRG U =, L = ξ 5 - δn A - δn A δ =5 =6 =7 =8 =9 = Drastic differences with the ES in the Mott phase. In the superfluid parabolic envelope. As the envelope curvature vanishes (i.e. δ ). Entanglement gap signature of finite condensate fraction.
Tower of states mechanism H = ij (b i b j + h.c.) + U n i (n i ) In the superfluid Bose condensate U() (continuous) symmetry breaking. Problem: the system is in a finite volume V (no symmetry breaking). b i = e iφ i n i n i = n + δn i [n i, φ j ] = iδ ij Jn U H i (δn i) + U (φ i φ j ) ij Anderson tower of states structure: i H U H tos (δn tot) V H sw + ( U k n k + ǫ k φ k ) E ~/V / ~/V ǫ k v s k ( E) sw / V while ( E) tos /V. - δn
Tower of states structure and ES envelope The lower part of the ES (ES envelope) is described by the tower of states Hamiltonian (H tos ) restricted to the subsystem. [Grover,Metlitski,()] H E H (A) tos /T E T E v s /V / For the Bose-Hubbard: H E U V (δn A) the ES envelope is parabolic and the level spacing δ U/V.
Tower of states scenario: numerical checks DMRG data at fixed aspect ratio /L = /. δ,4, U= U=4 U=6 U=8 U= Behavior of the level spacing δ : fixed U δ A/ + B/. fixed /L δ α + β U δ,4,,,,3 / =3 =4 =5 =6 =8 Theoretical prediction δ U/(δN A ) fully confirmed. 5 U
Entanglement entropies across the phase transition Consider the Renyi S () A S () A = log Trρ A,5 () S A =3 =4 =5 =6 =7 =8,5 3 4 U Sharp change of S () A at the transition.
Entanglement entropies: Mott phase,5 () S A,5 =3 =4 =5 =6 =7 =8,8 S () A /,4 3 4 U 3 4 U In the Mott phase S () c (area law) in agreement with boundary perturbation theory: S () A / = c = 8 U + 968 U 4 + O(U 6 )
Entanglement entropies: the superfluid S () A = log Trρ A,3 S () A /, =3 =4 =5 =6, =7 =8 5 5 U - -4 envelope higher levels 5 5 U (Trρ A )part. Finite gap only the envelope contributes to S () A. ES envelope gives a subleading logarithmic correction S () A = c + c log No enough ES levels above the gap to build the area law. [Metlitski,Grover,]
Summary e have studied the ES of the D Bose-Hubbard in both the Mott insulating and the superfluid phase. Mott insulator: Boundary locality of the ES. ES shows a perturbative structure. Entanglement dispersions. Boundary local D entanglement Hamiltonians. Superfluid: Signatures of the symmetry breaking. Lower part of the ES (ES envelope) related to tower of states. Finite gap. Entropies: Area law behavior in the Mott phase. No area law (entropies are dominated by the envelope levels).
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