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Entanglement Entropy of Typical Eigenstates of Translationally Invariant Quadratic Hamiltonians

Lev Vidmar, Lucas Hackl, Eugenio Bianchi A, Marcos Rigol

Published Mar 8, 2017• 0 citations
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Abstract

In a seminal paper [Phys. Rev. Lett. 71, 1291 (1993)], Page conjectured that the entanglement entropy of typical pure states is $S_{\rm typ}\simeq\ln{\cal D}_{\rm A} - (1/2) {\cal D}_{\rm A}^2/{\cal D}$, where ${\cal D}_{\rm A}$ and ${\cal D}$ are the Hilbert space dimensions of the subsystem and the system, respectively, and typicality is with respect to the Haar measure. Typical pure states are hence (nearly) maximally entangled. We prove that $S^+_1=\ln{\cal D}_{\rm A} - [1/(2\ln2)] (\ln{\cal D}_{\rm A})^2/\ln{\cal D}$ is an upper bound for the average entanglement entropy of all many-body fermionic eigenstates of a class of translationally invariant quadratic Hamiltonians in one dimension, and that the variance vanishes with increasing system size. Consequently, whenever the size of the subsystem is not a vanishing fraction of the size of the system, typical eigenstates of such Hamiltonians have a volume-law entanglement but do not become maximally entangled as ${\cal D}\rightarrow\infty$, i.e., they are qualitatively different from typical pure states. We also prove that, in the limit in which the subsystem size is a vanishing fraction of the system size, the average entanglement entropy is maximal.

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In a seminal paper [Phys. Rev. Lett. 71, 1291 (1993)], Page conjectured that the entanglement entropy of typical pure states is Styp≃ln⁡DA−(1/2)DA2/DS_{\rm typ}\simeq\ln{\cal D}_{\rm A} - (1/2) {\cal D}_{\rm A}^2/{\cal D}Styp​≃lnDA​−(1/2)DA2​/D, where DA{\cal D}_{\rm A}DA​ and D{\cal D}D are the Hilbert space dimensions of the subsystem and the system, respectively, and typicality is with respect to the Haar measure. Typical pure states are hence (nearly) maximally entangled. We prove that S1+=ln⁡DA−[1/(2ln⁡2)](ln⁡DA)2/ln⁡DS^+_1=\ln{\cal D}_{\rm A} - [1/(2\ln2)] (\ln{\cal D}_{\rm A})^2/\ln{\cal D}S1+​=lnDA​−[1/(2ln2)](lnDA​)2/lnD is an upper bound for the average entanglement entropy of all many-body fermionic eigenstates of a class of translationally invariant quadratic Hamiltonians in one dimension, and that the variance vanishes with increasing system size. Consequently, whenever the size of the subsystem is not a vanishing fraction of the size of the system, typical eigenstates of such Hamiltonians have a volume-law entanglement but do not become maximally entangled as D→∞{\cal D}\rightarrow\inftyD→∞, i.e., they are qualitatively different from typical pure states. We also prove that, in the limit in which the subsystem size is a vanishing fraction of the system size, the average entanglement entropy is maximal.
In a seminal paper [D. N. Page, Phys. Rev. Lett. 71, 1291 (1993)], Page proved that the average entanglement entropy of subsystems of random pure states is ${S}_{\text{ave}}\ensuremath{\simeq}\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}\ensuremath{-}(1/2){\mathcal{D}}_{\mathrm{A}}^{2}/\mathcal{D}$ for $1\ensuremath{\ll}{\mathcal{D}}_{\mathrm{A}}\ensuremath{\le}\sqrt{\mathcal{D}}$, where DA{\mathcal{D}}_{\mathrm{A}}DA​ and D\mathcal{D}D are the Hilbert space dimensions of the subsystem and the system, respectively. Hence, typical pure states are (nearly) maximally entangled. We develop tools to compute the average entanglement entropy ⟨S⟩⟨S⟩⟨S⟩ of all eigenstates of quadratic fermionic Hamiltonians. In particular, we derive exact bounds for the most general translationally invariant models $\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}\ensuremath{-}(\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}{)}^{2}/\mathrm{ln}\mathcal{D}\ensuremath{\le}⟨S⟩\ensuremath{\le}\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}\ensuremath{-}[1/(2\mathrm{ln}2)](\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}{)}^{2}/\mathrm{ln}\mathcal{D}$. Consequently, we prove that (i) if the subsystem size is a finite fraction of the system size, then ⟨S⟩<lnDA⟨S⟩<\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}⟨S⟩<lnDA​ in the thermodynamic limit; i.e., the average over eigenstates of the Hamiltonian departs from the result for typical pure states, and (ii) in the limit in which the subsystem size is a vanishing fraction of the system size, the average entanglement entropy is maximal; i.e., typical eigenstates of such Hamiltonians exhibit eigenstate thermalization.
In quantum statistical mechanics, it is of fundamental interest to understand how close the bipartite entanglement entropy of eigenstates of quantum chaotic Hamiltonians is to maximal. For random pure states in the Hilbert space, the average entanglement entropy is known to be nearly maximal, with a deviation that is, at most, a constant. Here we prove that, in a system that is away from half filling and divided in two equal halves, an upper bound for the average entanglement entropy of random pure states with a fixed particle number and normally distributed real coefficients exhibits a deviation from the maximal value that grows with the square root of the volume of the system. Exact numerical results for highly excited eigenstates of a particle number conserving quantum chaotic model indicate that the bound is saturated with increasing system size.
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.
In a quantum system in a pure state, a subsystem generally has a non-zero entropy because of entanglement with the rest of the system. Is the average entanglement entropy of pure states also the typical entropy of the subsystem? The exact formula for the average ⟨SA⟩\langle{S_{A}}\rangle⟨SA​⟩ was conjectured by Page in 1995 and later proved. Here we compute the exact formula for the average entropy and variance (ΔSA)2=⟨SA2⟩−⟨SA⟩2(\Delta S_A)^2 = \langle{S_A}^2\rangle - \langle{S_A}\rangle{}^2(ΔSA​)2=⟨SA​2⟩−⟨SA​⟩2 both for random pure states and for energy eigenstates of a non-interacting system. We show that for large systems the variance is small, ΔSA/⟨SA⟩≪1\Delta S_A/\langle{S_{A}}\rangle\ll 1ΔSA​/⟨SA​⟩≪1, and therefore the average entanglement entropy is typical. We compare exact and numerical results for the probability distribution P(SA)dSAP(S_A) \mathrm{d}S_AP(SA​)dSA​, and comment on the relation to previous results on concentration of measure bounds. We discuss the application to physical systems in an energy eigenstate. In particular, for a system of non-interacting spins in a magnetic field, we show how the thermal entropy arises as the typical entanglement entropy of a subsystem.
The eigenstate entanglement entropy is a powerful tool to distinguish integrable from generic quantum-chaotic models. In integrable models, the average eigenstate entanglement entropy (over all Hamiltonian eigenstates) has a volume-law coefficient that generally depends on the subsystem fraction. In contrast, it is maximal (subsystem fraction independent) in quantum-chaotic models. Using random matrix theory for quadratic Hamiltonians, we obtain a closed-form expression for the average eigenstate entanglement entropy as a function of the subsystem fraction. We test it against numerical results for the quadratic Sachdev-Ye-Kitaev model and show that it describes the results for the power-law random banded matrix model (in the delocalized regime). We show that localization in quasimomentum space produces (small) deviations from our analytic predictions.
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.