Entanglement spectrum in quantum many-body systems V. Alba 1 ,M. Haque 1 , A. L¨ auchli 2 1 Max Planck Institute for the Physics of Complex Systems, Dresden 2 University of Innsbruck, Innsbruck April 29, 2012 Vincenzo Alba Entanglement spectrum
Outline Introduction Entanglement spectrum Entanglement spectra in gapped systems (1D) [arXiv:1107.1726] ES in the gapped 1 D XXZ spin chain The XXZ in the large ∆ limit Perturbative structure of the ES Boundary-locality Effective microscopic description (domain walls) Comparison with the gapped 1 D Bose-Hubbard ES of the 1 D XXZ in the limit ∆ → − 1 + ( In progress ) ES of 2 D systems: the Bose Hubbard model ( In progress ) Vincenzo Alba Entanglement spectrum
Standard tools: entropies Consider a quantum system in d dimensions in the ground state | Ψ � . ρ ≡ | Ψ �� Ψ | If the system is bipartite: 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 = Tr ρ A log ρ A = − � i λ i log λ i A = − log � Renyi’s entropies R ( n ) = − log Tr ρ n i λ n A i Area law: away from criticality the entanglement is proportional to the surface area of subsystem A (1 D S → const ) S A ≈ L d − 1 For critical systems corrections to the area law [Calabrese, Cardy,2004] 1D → S A = c 3 log L Vincenzo Alba Entanglement spectrum
Entanglement spectrum Given the reduced density matrix ρ A : ρ A = e −H A { λ } = σ ( ρ 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, 2008] ξ [L¨ auchli et al,2010] 10 [Qi et al,2010] 8 [Thomale et al,2010] 6 10 [Regnault et al,2010] 8 4 6 [Pollmann et al,2010] 4 2 ( a ) P [0 | 0] [Prodan et al,2010] 2 56 58 60 62 64 0 L A 40 45 50 55 60 65 z More general: all the relevant information to describe the physics of a system is encoded in the entanglement spectrum. Vincenzo Alba Entanglement spectrum
ES in gapped systems In gapped sytems area law holds: [Hastings,2007] Only the degrees of freedom near the boundary entangle the two subsystems. What is the consequence for the entanglement spectrum? the ES is a boundary local quantity. Vincenzo Alba Entanglement spectrum
The gapped XXZ spin chain � � H = � S + i S − i +1 + S − i S + + ∆ S z i S z i +1 i +1 i � �� � � �� � H 0 H p ( i ) In the limit (∆ → ∞ ) the ground state is a Neel state: | N 1 � ≡ | · · · ↑↓↑↓↑↓↑↓↑↓↑↓↑ · · · � | N 2 � ≡ | · · · ↓↑↓↑↓↑↓↑↓↑↓ · · · � Large ∆, series expansion | Ψ � = � ∆ − i | Ψ i � : i ≥ 0 How to calculate the entanglement spectrum: (i) Let us bipartite the system A B | Ψ � = � M i , j | v ( i ) A � ⊗ | v ( j ) ⇒ B � · · · ↓↑↓↑↓↑↓ ↑↓↑↓↑↓↑ · · · i , j (ii) Perturbative expansion for the matrix M i , j . ⇒ Boundary locality · · · ↓↑↓↑↓↑↓ ↑↓↑↓↑↓↑ · · · (iii) SVD of M i , j (Schmidt decomposition) gives the ES. Vincenzo Alba Entanglement spectrum
The perturbative ES: single boundary case A B · · · ↓↑↓↑↓↑↓ ↑↓↑↓↑↓↑ · · · Levels are organized in powers of ∆ − 1 : higher levels in the ES ⇒ higher orders. The ES is not symmetric (we selected one Neel state.) In each δ S z A sector at order n , non trivial degeneracy given by the integer partitions p ( n / 2) . The same ‘degeneracy tower’ for all the δ S z A sectors. Total degeneracy at order n is given by the restricted integer partitions q ( n ). Vincenzo Alba Entanglement spectrum
Boundary locality and domain walls Rephrase perturbation theory in terms of domain walls . H p → · · · ↓↑↓ ↑↓ ↑↓↑↓ ⇒ · · · ↓↑ ↓↓ ↑↑ ↑↓ Boundary locality : levels higher in the ES are given by excitations farther from the boundary between A and B . Boundary locality gives the rules of the game: (i) Domain walls are created at the boundary. ↓↓ ↑↓↑↓↑ · · · ≈ ∆ − 1 · · · ↓↑↓↑↓↑ ↓ ↑ ↓↑↓↑↓↑ · · · ⇒ · · · ↓↑↓↑↓ ↑↑ ↓ ↑↑ ↓↑↓↑ · · · ≈ ∆ − 2 · · · ↓↑↓↑↓ ↑↓ ↑↓ ↑↓↑↓↑ · · · ⇒ · · · ↓↑↓↑ ↓↓ ↑ (ii) Domain walls can be moved symmetrically in the bulk. ↓↑↓ ↑↑ ↓↑ · · · ≈ ∆ − 2 · · · ↓↑↓↑ ↓↓ ↑ ↓ ↑↑ ↓↑↓↑ · · · ⇒ · · · ↓↑ ↓↓ ↑↓↑ Vincenzo Alba Entanglement spectrum
The domain walls picture at work Perturbative hyerarchical structure and degeneracy counting are correctly reproduced. More than an effective microscopic picture : the domain walls configurations give the leading contribution of the reduced density matrix eigenfunctions. Vincenzo Alba Entanglement spectrum
Exact diagonalization A B · · · ↓↑↓↑↓↑↓ ↑↓↑↓↑↓↑ · · · Finite size entanglement spectrum with arbitrary precision data (10 − 90 ). Good agreement with the perturbative picture: (i) Same relation ES levels vs perturbative order. (ii) Same degeneracy structure. Finite size corrections for levels very high in the spectrum (evident in the perturbative picture). The interlevels separation is constant and is given by 2 arccosh ∆. Vincenzo Alba Entanglement spectrum
Two boundaries: ES factorization Consider the case with two boundaries: B A B · · · ↑↓↑ ↓↑↓↑↓↑↓↑ · · · ↓↑↓↑↓↑↓↑ ↑↓↑ · · · Another consequence of the boundary locality : in the limit of large A the two boundaries are decoupled. ρ ( B A B ) ≈ ρ ( B A ) ⊗ ρ ( A B ) ρ ( B A )= ρ ( A B ) | S z A →− S z A Vincenzo Alba Entanglement spectrum
The two ES combination { ξ BAB } = { ξ BA } + { ξ AB } New degeneracies (1,3,7,16,...) Parabolic envelope Vincenzo Alba Entanglement spectrum
The 1 D Bose Hubbard H = J � ( b † � + U i b i +1 + h . c . ) n i ( n i − 1) 2 i � �� � i H p In the Mott insulating phase (large U ). Expansion in J / U starting from unit filling | Ψ 0 � = | · · · 111111 · · · � perturbative order Levels are organized in powers of J / U . Levels higher in the spectrum ⇒ excitations farther from the boundary. Vincenzo Alba Entanglement spectrum
The 1 D Bose Hubbard H = J � ( b † � + U i b i +1 + h . c . ) n i ( n i − 1) 2 i � �� � i H p In the Mott insulating phase (large U ). Expansion in J / U starting from unit filling | Ψ 0 � = | · · · 111111 · · · � perturbative order Levels are organized in powers of J / U . Levels higher in the spectrum ⇒ excitations farther from the boundary. No simple rules or exact degeneracy ( integrability breaking ). Vincenzo Alba Entanglement spectrum
Summary The entanglement spectrum for gapped systems is dominated by the physics at the boundaries ( boundary-locality ). Boundary-locality provides a perturbative scheme for calculating entanglement spectra in gapped systems. The study of entanglement spectrum in the XXZ allowed to unveil a beautiful combinatorial structure ( integrability ). The entanglement spectrum is a useful tool to analyse the structure (correlations, symmetries) of the wave functions ( “state tomography” ). Vincenzo Alba Entanglement spectrum
XXZ: the ES in the limit ∆ → − 1 + GAPLESS (CFT, c = 1) GAPPED GAPPED − 1 1 ∆ Vincenzo Alba Entanglement spectrum
XXZ: the ES in the limit ∆ → − 1 + DONE! GAPLESS (CFT, c = 1) GAPPED GAPPED − 1 1 ∆ Vincenzo Alba Entanglement spectrum
XXZ: the ES in the limit ∆ → − 1 + DONE! GAPLESS (CFT, c = 1) GAPPED GAPPED − 1 1 ∆ Vincenzo Alba Entanglement spectrum
The XXZ in the limit ∆ → − 1 + L � � � S + i S − i +1 + S − i S + i +1 − S z i S z H = i +1 i ∆ = − 1 critical point ( non conformal ) Unitarily equivalent to the Heisenberg ferromagnet. � ˜ � H = − � S + i ˜ S − i +1 + ˜ S − i ˜ S + i +1 + ˜ i ˜ S z S z i +1 i The ground state is in the symmetric sector: z =3 S z =2 S z =1 S z =0 S S − | · · · ↓ · · · � tot -1.499 E | Ψ 0 � = | · · · ↑↑↑↑↑↑↑↑ · · · � ⇒ tot ) 2 ( S − | · · · ↓ · · · ↓ · · · � . . . -1.5 -1.0000 -0.9995 -0.9990 The ground state is highly degenerate ( L + 1). ∆ “ mean field ” like structure: no notion of distance the down spins are delocalized. Vincenzo Alba Entanglement spectrum
The entanglement spectrum at ∆ = − 1 The entanglement spectrum at ∆ = − 1 has a simple combinatorial structure: � � LA L − LA [Popkov,2005] � ( A )( A ) � L − LA LA 2 − Sz − Sz + Sz { ξ i } = − 2 log 2 [Doyon,2011] L ( 2 − Sz ) L One entanglement level for each sector S z A . Interpretation: ↓↓ · · · ↓ ↓↓ · · · ↓ → ��� · · · � ��� · · · � � �� � � �� � � �� � � �� � rest LB boxes LA boxes ( LA 2 − Sz A ) particles S z = 0 (i.e. the ground state at ∆ > − 1), unusual scaling of von Neumann entropy: � � 1 π L A + 1 S A = 2 log 2 + O (1 / L A ) 2 Log scaling without central charge. Vincenzo Alba Entanglement spectrum
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