Many-body quantum eigenstates of generic Hamiltonians at finite-energy density typically satisfy the ``volume law'' of entanglement entropy: the von Neumann entanglement entropy and the Renyi entropies for a subregion scale in proportion to its volume. Here we provide a connection between the volume law and the sign structure of eigenstates. In particular, we ask the following question: Can a positive wave function support a volume law entanglement? Remarkably, we find that a typical random positive wave function exhibits a constant law for Renyi entanglement entropies ${S}_{n}$ for $n>1$, despite arbitrary large-amplitude fluctuations. We also provide evidence that the modulus of the finite-energy density eigenstates of generic local Hamiltonians shows similar behavior.
Many-body quantum eigenstates of generic Hamiltonians at finite-energy density typically satisfy the ``volume law'' of entanglement entropy: the von Neumann entanglement entropy and the Renyi entropies for a subregion scale in proportion to its volume. Here we provide a connection between the volume law and the sign structure of eigenstates. In particular, we ask the following question: Can a positive wave function support a volume law entanglement? Remarkably, we find that a typical random positive wave function exhibits a constant law for Renyi entanglement entropies Sn for n>1, despite arbitrary large-amplitude fluctuations. We also provide evidence that the modulus of the finite-energy density eigenstates of generic local Hamiltonians shows similar behavior.
The entanglement entropy of subsystems of typical eigenstates of quantum many-body Hamiltonians has recently been conjectured to be a diagnostic of quantum chaos and integrability. In quantum chaotic systems it has been found to behave as in typical pure states, while in integrable systems it has been found to behave as in typical pure Gaussian states. In this tutorial, we provide a pedagogical introduction to known results about the entanglement entropy of subsystems of typical pure states and of typical pure Gaussian states. They both exhibit a leading term that scales with the volume of the subsystem, when smaller than one half of the volume of the system, but the prefactor of the volume law is fundamentally different. It is constant (and maximal) for typical pure states, and it depends on the ratio between the volume of the subsystem and of the entire system for typical pure Gaussian states. Since particle-number conservation plays an important role in many physical Hamiltonians, we discuss its effect on typical pure states and on typical pure Gaussian states. We prove that, while the behavior of the leading volume-law terms does not change qualitatively, the nature of the subleading terms can change. In particular, subleading corrections can appear that depend on the square root of the volume of the subsystem. We unveil the origin of those corrections. Finally, we discuss the connection between the entanglement entropy of typical pure states and analytical results obtained in the context of random matrix theory, as well as numerical results obtained for physical Hamiltonians.
Using arguments built on ergodicity, we derive an analytical expression for the Renyi entanglement entropies which, we conjecture, applies to the finite-energy density eigenstates of chaotic many-body Hamiltonians. The expression is a universal function of the density of states and is valid even when the subsystem is a finite fraction of the total system-a regime in which the reduced density matrix is not thermal. We find that in the thermodynamic limit, only the von Neumann entropy density is independent of the subsystem to the total system ratio V_{A}/V, while the Renyi entropy densities depend nonlinearly on V_{A}/V. Surprisingly, Renyi entropies S_{n} for n>1 are convex functions of the subsystem size, with a volume law coefficient that depends on V_{A}/V, and exceeds that of a thermal mixed state at the same energy density. We provide two different arguments to support our results: the first one relies on a many-body version of Berry's formula for chaotic quantum-mechanical systems, and is closely related to the eigenstate thermalization hypothesis. The second argument relies on the assumption that for a fixed energy in a subsystem, all states in its complement allowed by the energy conservation are equally likely. We perform an exact diagonalization study on quantum spin-chain Hamiltonians to test our analytical predictions.
Entanglement comes in different forms, some more complex than others. In this paper we study the transitions of entanglement complexity in an exemplary family of states---the Rokhsar-Kivelson-sign wavefunctions---whose degree of entanglement is controlled by a single parameter. This family of states is known to feature a transition between a phase exhibiting volume-law scaling of entanglement entropy and a phase with subextensive scaling of entanglement, reminiscent of the many-body-localization transition of disordered quantum Hamiltonians. We study the singularities of the Rokhsar-Kivelson-sign wavefunctions and their entanglement complexity across the transition using several tools from quantum information theory: fidelity metric, entanglement spectrum statistics, entanglement entropy fluctuations, stabilizer R\'enyi entropy, and the performance of a disentangling algorithm. Across the whole volume-law phase the states feature universal entanglement spectrum statistics. Yet a ``superuniversal'' regime appears for small values of the control parameter in which all metrics become independent of the parameter itself, the entanglement entropy as well as the stabilizer R\'enyi entropy appear to approach their theoretical maximum, the entanglement fluctuations scale to zero as in output states of random universal circuits, and the disentangling algorithm has essentially null efficiency. All these indicators consistently reveal a complex pattern of entanglement. In the sub-volume-law phase, on the other hand, the entanglement spectrum statistics is no longer universal, entanglement fluctuations are larger and exhibiting a nonuniversal scaling, and the efficiency of the disentangling algorithm becomes finite. Our results, based on model wavefunctions, suggest that a similar combination of entanglement scaling properties and of entanglement complexity features may be found in high-energy Hamiltonian eigenstates---a very strong candidate being offered by the many-body localization transition of disordered lattice-spin models.
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=E_{α}-E_{β} ({E_{α}} are the eigenenergies) proportional to 1/D, where D is the Hilbert space dimension.
Thermal equilibrium in quantum many-body systems can be fully described by pure quantum states as far as local observables are concerned, while the conventional description of it is based on mixed-state ensembles.The thermodynamic entropy, in that case, is identified with the entanglement entropy of small subsystems.It scales with the size of the subsystem, which is called the volume law of entanglement.When the size of the subsystem increases, however, the correspondence between thermal and entanglement entropies fails and there appears a quantum correction to the simple volume-law scaling.Explicating the scaling of the entanglement entropy with the subsystem size is hence of great importance in connecting quantum physics to thermodynamics as well as analyzing recent experiments on ultracold atoms.In this thesis, we study the volume law of entanglement for pure quantum states representing thermal equilibrium.We derive an analytic formula of the volume law of entanglement for a certain class of pure states called canonical Thermal Pure Quantum (cTPQ) states which represent thermal equilibrium.We illustrate an advantage of the formula by numerically calculating the entanglement entropy of the cTPQ states and applying the formula to it.Furthermore, we argue that our formula universally applies to any sufficiently scrambled pure state, even if the state is not thermal.We consider two examples of such scrambled states, namely, stationary pure states after quantum quench and energy eigenstates of general Hamiltonians with and without integrability.The entanglement entropy of the former states, including actual experimental data in ultracold atoms, is in an excellent agreement with our formula as long as the states are scrambled.For the latter states, we find that our formula works as a good fitting function in non-integrable models to extract information in the thermodynamic limit from finite size systems.We also show that our formula can distinguish eigenstates of integrable models from non-integrable ones.4.3.1 Review of result by Lu and Grover for eigenstates of non-integrable models . . . . . . . . . . . .
A bstract We investigate various aspects of capacity of entanglement in certain setups whose entanglement entropy becomes extensive and obeys a volume law. In particular, considering geometric decomposition of the Hilbert space, we study this measure both in the vacuum state of a family of non-local scalar theories and also in the squeezed states of a local scalar theory. We also evaluate field space capacity of entanglement between interacting scalar field theories. We present both analytical and numerical evidences for the volume law scaling of this quantity in different setups and discuss how these results are consistent with the behavior of other entanglement measures including Renyi entropies. Our study reveals some generic properties of the capacity of entanglement and the corresponding reduced density matrix in the specific regimes of the parameter space. Finally, by comparing entanglement entropy and capacity of entanglement, we discuss some implications of our results on the existence of consistent holographic duals for the models in question.