Entanglement and matrix elements of observables in interacting integrable systems

Abstract

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 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 chain is the same as, or very close to, 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 chain 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 − E β ({Eα} are the eigenenergies) whose magnitude scales as 1/D, where D is the Hilbert space dimension.

Speck of chaos

“…This is referred to as the diagonal ETH and has been verified for the single-defect X X Z model in [35,36] and recently in [42]. In chaotic systems, the distribution of the off-diagonal matrix elements of local operators is Gaussian [52,53], which is called the off-diagonal ETH and has been confirmed for the single-defect X X Z model as well [44].…”

“…In Ref. [52] (see also [44,[53][54][55]), the distinction between integrable and chaotic models is based on the distribution of the off-diagonal matrix elements of local observables in each subspace. In Refs.…”

Persistent dark states in anisotropic central spin models

“…Interestingly, it was recently observed that the function can be defined and remains smooth even in generic integrable systems 72 . Then vanishes as for deformations of the Hamiltonian along integrable directions 44 , 73 – 75 .…”

Random-matrix perspective on many-body entanglement with a finite localization length

“…The universality is fully established in the mesoscopic regime, where the parts are all small compared to the size of the bipartioned subsystems, and N α is moderately large, but in practice already holds well for N α = O(1). In particular, in comparison to physical models the framework turns out to be remarkably predictive for the bipartite entanglement entropy at different system sizes and choice of bipartition [11,12,21,22]. As this universality is observed also between the two variants of the model, we conjecture that it also extends to interpolating scenarios, including multifractal cases [23].…”

“…for the ensemble-averaged bipartite von Neumann entanglement entropy, assuming 1 M A M B [8]. This prediction serves as an important benchmark to detect deviations from ergodic many-body behavior, including signatures of manybody localization and topological states [9][10][11][12][13].…”