TY - JOUR

T1 - Entanglement and matrix elements of observables in interacting integrable systems

AU - Leblond, Tyler

AU - Mallayya, Krishnanand

AU - Vidmar, Lev

AU - Rigol, Marcos

N1 - Funding Information:
We acknowledge discussions with S. Gopalakrishnan and M. Mierzejewski. This work was supported by the National Science Foundation under Grant No. PHY-1707482 (T.L., K.M., and M.R.), and the Slovenian Research Agency (ARRS), Research core fundings Grants No. P1-0044 and No. J1-1696 (L.V.).
Publisher Copyright:
© 2019 American Physical Society.

PY - 2019/12/26

Y1 - 2019/12/26

N2 - We study the bipartite von Neumann entanglement entropy and matrix elements of local operators in the eigenstates of an interacting integrable Hamiltonian (the paradigmatic spin-1/2 XXZ chain), and we contrast their behavior with that of quantum chaotic systems. We find that the leading term of the average (over all eigenstates in the zero magnetization sector) eigenstate entanglement entropy has a volume-law coefficient that is smaller than the universal (maximal entanglement) one in quantum chaotic systems. This establishes the entanglement entropy as a powerful measure to distinguish integrable models from generic ones. Remarkably, our numerical results suggest that the volume-law coefficient of the average entanglement entropy of eigenstates of the spin-1/2 XXZ Hamiltonian is very close to, or the same as, the one for translationally invariant quadratic fermionic models. We also study matrix elements of local operators in the eigenstates of the spin-1/2 XXZ Hamiltonian at the center of the spectrum. For the diagonal matrix elements, we show evidence that the support does not vanish with increasing system size, while the average eigenstate-to-eigenstate fluctuations vanish in a power-law fashion. For the off-diagonal matrix elements, we show that they follow a distribution that is close to (but not quite) log-normal, and that their variance is a well-defined function of ω=Eα-Eβ ({Eα} are the eigenenergies) proportional to 1/D, where D is the Hilbert space dimension.

AB - We study the bipartite von Neumann entanglement entropy and matrix elements of local operators in the eigenstates of an interacting integrable Hamiltonian (the paradigmatic spin-1/2 XXZ chain), and we contrast their behavior with that of quantum chaotic systems. We find that the leading term of the average (over all eigenstates in the zero magnetization sector) eigenstate entanglement entropy has a volume-law coefficient that is smaller than the universal (maximal entanglement) one in quantum chaotic systems. This establishes the entanglement entropy as a powerful measure to distinguish integrable models from generic ones. Remarkably, our numerical results suggest that the volume-law coefficient of the average entanglement entropy of eigenstates of the spin-1/2 XXZ Hamiltonian is very close to, or the same as, the one for translationally invariant quadratic fermionic models. We also study matrix elements of local operators in the eigenstates of the spin-1/2 XXZ Hamiltonian at the center of the spectrum. For the diagonal matrix elements, we show evidence that the support does not vanish with increasing system size, while the average eigenstate-to-eigenstate fluctuations vanish in a power-law fashion. For the off-diagonal matrix elements, we show that they follow a distribution that is close to (but not quite) log-normal, and that their variance is a well-defined function of ω=Eα-Eβ ({Eα} are the eigenenergies) proportional to 1/D, where D is the Hilbert space dimension.

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U2 - 10.1103/PhysRevE.100.062134

DO - 10.1103/PhysRevE.100.062134

M3 - Article

C2 - 31962410

AN - SCOPUS:85077237942

VL - 100

JO - Physical Review E

JF - Physical Review E

SN - 2470-0045

IS - 6

M1 - 062134

ER -