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arXiv:2311.18045 · Physical review. B./Physical review. B

Page curve entanglement dynamics in an analytically solvable model

Stefan K. Kehrein

Published Jun 17, 2024• 11 citations
arXivHTMLPDF

Abstract

The entanglement entropy of black holes is expected to follow the Page curve. After an initial linear increase with time the entanglement entropy should reach a maximum at the Page time and then decrease. This paper introduces an exactly solvable model of free fermions that explicitly shows such a Page curve: The entanglement entropy vanishes asymptotically for late times instead of saturating at a volume law. The bending down of the Page curve is accompanied by a breakdown of the semiclassical connection between particle current and entanglement generation, a quantum phase transition in the entanglement Hamiltonian and nonanalytic behavior of the $q\ensuremath{\rightarrow}\ensuremath{\infty}$ R\'enyi entropy. These observations are expected to hold for a larger class of systems beyond the exactly solvable model analyzed here.

DOI
10.1103/physrevb.109.224308

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The entanglement entropy of black holes is expected to follow the Page curve. After an initial linear increase with time the entanglement entropy should reach a maximum at the Page time and then decrease. This paper introduces an exactly solvable model of free fermions that explicitly shows such a Page curve: The entanglement entropy vanishes asymptotically for late times instead of saturating at a volume law. The bending down of the Page curve is accompanied by a breakdown of the semiclassical connection between particle current and entanglement generation, a quantum phase transition in the entanglement Hamiltonian and nonanalytic behavior of the q\ensuremath→\ensuremath∞q\ensuremath{\rightarrow}\ensuremath{\infty}q\ensuremath→\ensuremath∞ R'enyi entropy. These observations are expected to hold for a larger class of systems beyond the exactly solvable model analyzed here.
The entanglement entropy of a black hole and that of its Hawking radiation are expected to follow the so-called Page curve: After an increase in line with Hawking’s calculation, it is expected to decrease back to zero once the black hole has fully evaporated, as demanded by unitarity. Recently, a simple system-plus-bath model has been proposed which shows a similar behavior. Here, we make a general argument as to why such a Page-curve-like entanglement dynamics should be expected to hold generally for system-plus-bath models at small coupling and low temperatures, when the system is initialized in a pure state far from equilibrium. The interaction with the bath will then generate entanglement entropy, but it eventually has to decrease to the value prescribed by the corresponding mean-force Gibbs state. Under those conditions, it is close to the system ground state. We illustrate this on two paradigmatic open-quantum-system models, the exactly solvable harmonic quantum Brownian motion and the spin-boson model, which we study numerically. In the first example we find that the intermediate entropy of an initially localized impurity is higher for more localized initial states. In the second example, for an impurity initialized in the excited state, the Page time—when the entropy reaches its maximum—occurs when the excitation has half decayed. Published by the American Physical Society 2024
Generic nonequilibrium many-body systems display a linear growth of bipartite entanglement entropy in time, followed by a volume law saturation. In stark contrast, the Page curve dynamics of black-hole physics shows that the entropy peaks at the Page time t Page and then decreases to zero. Here, we investigate such Page-like behavior of the von Neumann entropy in a model of strongly correlated spinless fermions in a typical system-environment setup and characterize the properties of the Page curve dynamics in the presence of interactions using numerically exact matrix product states methods. The two phases of growth, namely, linear growth and bending down, are shown to be separated by a nonanalyticity in the min-entropy before t Page , which separates two different quantum phases, realized as the respective ground states of the corresponding entanglement (or equivalently, modular) Hamiltonian. We confirm and generalize, by introducing interactions, the findings of for a free spinless fermionic chain where the corresponding entanglement Hamiltonian undergoes a quantum phase transition at the point of nonanalyticity. However, in the presence of interactions, a scaling analysis gives a nonzero critical time for the nonanalyticity in the thermodynamic limit only for weak to intermediate interaction strengths, while the dynamics leading to the nonanalyticity becomes for interactions large enough. We present a physical picture explaining these findings.
We consider an analytically tractable model that exhibits the main features of the Page curve characterizing the evolution of entanglement entropy during evaporation of a black hole. Our model is a gas of noninteracting fermions on a lattice that is released from a box into the vacuum. More precisely, our Hamiltonian is a tight-binding model with a defect at the junction between the filled box and the vacuum. In addition to the entanglement entropy we consider several other observables, such as the spatial density profile and current, and show that the semiclassical approach of generalized hydrodynamics provides a remarkably accurate description of the quantum dynamics including that of the entanglement entropy at all times t. Our hydrodynamic results agree closely with those obtained via exact microscopic numerics. We find that the growth of entanglement with time is linear and universal, i.e, independent of the details of the defect. The decay shows 1/t scaling for conformal defect while for nonconformal defects, it is slower. Our analytical results from hydrodynamics enables us to make predictions for long time behaviors of observables that are otherwise highly inaccessible computationally by exact microscopic numerics. Our study shows the power of the semiclassical approach and could be relevant for discussions on the resolution of the black hole information paradox.
We study the dynamics of entanglement in a one-dimensional XXZXXZXXZ spin-1/21/21/2 chain with and without integrability-breaking interactions and connected to a bath. We start from a state where the system and bath are completely unentangled, and the bath is polarized spin-down. We consider two different initial states for the system: (i) a polarized spin-up state, and (ii) an infinite-temperature state. In the particle representation of the spin chain, the polarized spin-up state corresponds to a filled state, while the polarized spin-down state corresponds to an empty state. Starting from these inhomogeneous quenches, in all the above-mentioned cases, we obtain the Page curve like behavior in the entanglement. We report different power-law behavior in the growth of entanglement for different initial states and different kinds of baths (interacting and noninteracting). In an attempt to explore plausible deep connections between entanglement and Boltzmann entropy, we investigate the latter in both the filled and the infinite-temperature case, for the system and the bath. For the filled case, the Boltzmann entropy of the system has the form of a Page curve but quantitatively deviates from the entanglement. On the other hand, the entropy of the bath keeps increasing. Remarkably, for the infinite-temperature case, we find that the system and bath Boltzmann entropies agree with the entanglement entropy, after and before the Page time, respectively. Our findings are expected to hold for generic interacting quantum systems and could be of relevance to black-hole physics.
Abstract We study entanglement dynamics in toy models of black hole information built out of chaotic many-body quantum systems, by utilising a coarse-grained description of entanglement dynamics in such systems known as the ‘entanglement membrane’. We show that in these models the Page curve associated to the entropy of Hawking radiation arises from a transition in the entanglement membrane around the Page time, in an analogous manner to the change in quantum extremal surfaces that leads to the Page curve in semi-classical gravity. We also use the entanglement membrane prescription to study the Hayden-Preskill protocol, and demonstrate how information initially encoded in the black hole is rapidly transferred to the radiation around the Page time. Our results relate recent developments in black hole information to generic features of entanglement dynamics in chaotic many-body quantum systems.
Analyses suggest that the quantum RT construction can be effectively extended beyond the asymptotically AdS spacetime for the calculation of the entanglement entropy. We study the entanglement entropy of a one-parameter family of exactly solvable gravities in the 2-dimensional asymptotically-flat space, and investigate the evaporating and the bath-removed black hole solutions including both the RST and the BPP black holes as two special cases. We generalize the results on the Page curves of evaporating (dynamical) black holes to this family of gravities and compute the corresponding entanglement entropy explicitly. The Page curves and the configuration of islands directly follow from the calculation. In the case of evaporating black holes formed by collapsing matter shells, the Page time is around a third of the black hole evaporating time and the island lies inside the horizon. In the bath-removed black hole case, the position of the islands is discontinuous across two different spacetime geometries and two transition points are witnessed in this model. We analyze the island configurations and the Page curves in the large-N limit in both cases and restore the unitary Page curves.