We investigate the average eigenstate entanglement in a bipartite many-body system which exhibits a breaking of two local conservation laws into a global conserved quantity. Such a setting is realized by the particle number conservation in Bose-Hubbard systems. We devise a corresponding random matrix model which captures the universal features of this symmetry breaking and allows for applying powerful random matrix methods. By combining the concept of symmetry resolved entanglement with perturbation theory for quantum chaotic systems we obtain a universal entanglement transition depending on a single fundamental parameter. Furthermore, the symmetry resolved entanglement allows for separating the genuine entanglement from the part which originates from the conserved quantity. This latter contribution is quantified by the number entropy. For this it is shown that the symmetry breaking generates a localization of the eigenstates due to a banded structure of the time-evolution operator. By extrapolating the results beyond the perturbative regime we obtain an analytic description of the full transition.
We investigate the average eigenstate entanglement in a bipartite many-body system which exhibits a breaking of two local conservation laws into a global conserved quantity. Such a setting is realized by the particle number conservation in Bose-Hubbard systems. We devise a corresponding random matrix model which captures the universal features of this symmetry breaking and allows for applying powerful random matrix methods. By combining the concept of symmetry resolved entanglement with perturbation theory for quantum chaotic systems we obtain a universal entanglement transition depending on a single fundamental parameter. Furthermore, the symmetry resolved entanglement allows for separating the genuine entanglement from the part which originates from the conserved quantity. This latter contribution is quantified by the number entropy. For this it is shown that the symmetry breaking generates a localization of the eigenstates due to a banded structure of the time-evolution operator. By extrapolating the results beyond the perturbative regime we obtain an analytic description of the full transition.
While the eigenstate entanglement entropy has been extensively studied for fermionic systems, much less is known about bosonic systems. Here, we study the entanglement entropy of mid-spectrum eigenstates of Bose-Hubbard models, focusing on weakly disordered models with and without particle-number conservation, and contrasting them with the translationally-invariant model. We analyze the volume-law and O(1) contributions to the entanglement entropy via the averages over mid-spectrum eigenstates and the corresponding distributions. We derive the volume-law coefficient of the entanglement entropy by generalizing the mean-field approach from [Phys. Rev. Lett. 119, 220603 (2017)] to many-body systems with a tunable local bosonic cutoff, which agrees with previous analytical and numerical results from [Phys. Rev. B 110, 235154 (2024)]. We show that the volume-law contribution to the entanglement entropy does not change upon breaking translational invariance via on-site disorder. We then numerically study the role of the subleading O(1) contribution to the entanglement entropy. We find that, in the particle-number conserving case, it exhibits a nontrivial dependence on the particle-number density and the local bosonic cutoff, while without particle-number conservation, results suggest the emergence of a universal O(1) contribution beyond the random pure state predictions.
The entanglement of eigenstates in two coupled, classically chaotic kicked tops is studied in dependence of their interaction strength. The transition from the noninteracting and unentangled system toward full random matrix behavior is governed by a universal scaling parameter. Using suitable random matrix transition ensembles we express this transition parameter as a function of the subsystem sizes and the coupling strength for both unitary and orthogonal symmetry classes. The universality is confirmed for the level spacing statistics of the coupled kicked tops and a perturbative description is in good agreement with numerical results. The statistics of Schmidt eigenvalues and entanglement entropies of eigenstates is found to follow a universal scaling as well. Remarkably, this is not only the case for large subsystems of equal size but also if one of them is much smaller. For the entanglement entropies a perturbative description is obtained, which can be extended to large couplings and provides very good agreement with numerical results. Furthermore, the transition of the statistics of the entanglement spectrum toward the random matrix limit is demonstrated for different ratios of the subsystem sizes.
We derive universal entanglement entropy and Schmidt eigenvalue behaviors for the eigenstates of two quantum chaotic systems coupled with a weak interaction. The progression from a lack of entanglement in the noninteracting limit to the entanglement expected of fully randomized states in the opposite limit is governed by the single scaling transition parameter $\mathrm{\ensuremath{\Lambda}}$. The behaviors apply equally well to few- and many-body systems, e.g., interacting particles in quantum dots, spin chains, coupled quantum maps, and Floquet systems, as long as their subsystems are quantum chaotic and not localized in some manner. To calculate the generalized moments of the Schmidt eigenvalues in the perturbative regime, a regularized theory is applied, whose leading-order behaviors depend on $\sqrt{\mathrm{\ensuremath{\Lambda}}}$. The marginal case of the 1/2 moment, which is related to the distance to the closest maximally entangled state, is an exception having a $\sqrt{\mathrm{\ensuremath{\Lambda}}}ln\mathrm{\ensuremath{\Lambda}}$ leading order and a logarithmic dependence on subsystem size. A recursive embedding of the regularized perturbation theory gives a simple exponential behavior for the von Neumann entropy and the Havrda-Charv\'at-Tsallis entropies for increasing interaction strength, demonstrating a universal transition to nearly maximal entanglement. Moreover, the full probability densities of the Schmidt eigenvalues, i.e., the entanglement spectrum, show a transition from power laws and L\'evy distribution in the weakly interacting regime to random matrix results for the strongly interacting regime. The predicted behaviors are tested on a pair of weakly interacting kicked rotors, which follow the universal behaviors extremely well.
In a recent Letter [Phys. Rev. Lett. 125, 180604 (2020)], we introduced a closed-form analytic expression for the average bipartite von Neumann entanglement entropy of many-body eigenstates of random quadratic Hamiltonians, namely, of Hamiltonians whose single-particle eigenstates have random coefficients in the position basis. A paradigmatic Hamiltonian for which the expression is valid is the quadratic Sachdev-Ye-Kitaev (SYK2) model in its Dirac fermion formulation. Here we show that the applicability of our result is much broader. Most prominently, it is also relevant for local Hamiltonians such as the three-dimensional (3D) Anderson model at weak disorder. Moreover, it describes the average entanglement entropy in Hamiltonians without particle-number conservation, such as the SYK2 model in the Majorana fermion formulation and the 3D Anderson model with additional terms that break particle-number conservation. We extend our analysis to the average bipartite second R\'enyi entanglement entropy of eigenstates of the same quadratic Hamiltonians, which is derived analytically and tested numerically. We conjecture that our results for the entanglement entropies of many-body eigenstates apply to quadratic Hamiltonians whose single-particle eigenstates exhibit quantum chaos, which we refer to as quantum-chaotic quadratic Hamiltonians.
Recent work has shown that the entanglement of finite-temperature eigenstates in chaotic quantum many-body local Hamiltonians can be accurately described by an ensemble of random states with an internal U(1) symmetry. We build upon this result to investigate the universal symmetry-breaking properties of such eigenstates. As a probe of symmetry breaking, we employ the entanglement asymmetry, a quantum information observable that quantifies the extent to which symmetry is broken in a subsystem. This measure enables us to explore the finer structure of finite-temperature eigenstates in terms of the U(1)-symmetric random state ensemble; in particular, the relation between the Hamiltonian and the effective conserved charge in the ensemble. Our analysis is supported by analytical calculations for the symmetric random states, as well as exact numerical results for the Mixed-Field Ising spin-1/2 chain, a paradigmatic model of quantum chaoticity.
It is now well-established that classical statistical mechanics emerges from the entanglement structure of quantum chaotic systems, as quantified by the (subsystem) eigenstate thermalization hypothesis (ETH). While this statement rests on the well-studied bipartite entanglement of the eigenstates of such systems, recent years have shown that many-body states are often characterized by multipartite entanglement, raising the question: what are the universal multipartite entanglement features of chaotic eigenstates? We answer this question in part by studying a tripartite generalization of the ergodic bipartition ansatz through the lens of two tripartite entanglement probes, the Rényi tripartite multi-entropy and the Rényi reflected entropy. We derive leading-order analytical expressions for the ensemble averages of both quantities using a saddle-point analysis. Beyond these measures, we obtain the average of the Rényi-2 reflected density matrix, ρAA∗(2),which controls the Rényi-2 reflected entropy. We show that ρAA∗(2) in a chaotic eigenstate is well-described by this ensemble average which, unlike an ordinary reduced density matrix, takes three qualitatively distinct forms across subsystem-size regimes, which may be viewed as a tripartite generalization of subsystem ETH. We benchmark our analytical predictions against exact diagonalization of the mixed-field Ising model.