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arXiv:2305.18329 · Journal of High Energy Physics

Perturbative Page curve induced by external impulse

Pengfei Zhang

Published Sep 11, 2023• 2 citations
arXivHTMLPDF

Abstract

Abstract In this work, we extend the recent study of entropy dynamics induced by an external impulse in open quantum systems, where the entropy response follows the Page curve. For small system-bath coupling κ, we expect that the entropy first increases exponentially κ2eϰt in the early-time regime t ≲ |log κ| due to quantum many-body chaos, and then decreases as e−λ0t with λ0 ∝ κ2 due to the energy relaxation. These results are confirmed through explicit calculations using two methods: (1) generalized Boltzmann equation for systems with quasi-particles; (2) scramblon effective theory in the early-time regime and perturbation theory in the late-time regime for 0+1-d systems. We also prove that in the second stage, the entropy of the system is equal to the coarse-grained entropy.

DOI
10.1007/jhep09(2023)056

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Abstract In this work, we extend the recent study of entropy dynamics induced by an external impulse in open quantum systems, where the entropy response follows the Page curve. For small system-bath coupling κ, we expect that the entropy first increases exponentially κ2eϰt in the early-time regime t ≲ |log κ| due to quantum many-body chaos, and then decreases as e−λ0t with λ0 ∝ κ2 due to the energy relaxation. These results are confirmed through explicit calculations using two methods: (1) generalized Boltzmann equation for systems with quasi-particles; (2) scramblon effective theory in the early-time regime and perturbation theory in the late-time regime for 0+1-d systems. We also prove that in the second stage, the entropy of the system is equal to the coarse-grained entropy.
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
A bstract In closed quantum many-body systems, initially localized information spreads throughout the system and becomes highly complex. This phenomenon, known as information scrambling, is closely related to entropy growth and quantum thermalization. Recent studies have shown that dissipation in open systems can hinder information scrambling, driving the system into a dissipative phase when the system-bath coupling is strong. However, the signature of this scrambling transition in entropy dynamics remains unexplored. In this work, we unveil a novel phenomenon in open quantum systems, termed entropy accumulation, which occurs exclusively within the scrambling phase. By generalizing the study of perturbative Page curve in closed systems, we consider a setup in which a probe is weakly coupled to a system that is already interacting with a bath. We calculate the increase in the second Rényi entropy induced by an external impulse on the system, after tracing out the probe. Despite the system-probe coupling being weak, the entropy continues to increase and eventually saturates at a finite value due to operator growth. In contrast, the entropy increase is limited by the coupling strength in the dissipative phase. The theoretical prediction is derived from both general arguments and explicit examples using generalized Boltzmann equations or SYK solvable models. Our results offer new insights into the intriguing relationship between entropy dynamics and information scrambling in open quantum systems.
Abstract In this paper, we use the exactly solvable Sachdev-Ye-Kitaev model to address the issue of entropy dynamics when an interacting quantum system is coupled to a non-Markovian environment. We find that at the initial stage, the entropy always increases linearly matching the Markovian result. When the system thermalizes with the environment at a sufficiently long time, if the environment temperature is low and the coupling between system and environment is weak, then the total thermal entropy is low and the entanglement between system and environment is also weak, which yields a small system entropy in the long-time steady state. This manifestation of non-Markovian effects of the environment forces the entropy to decrease in the later stage, which yields the Page curve for the entropy dynamics. We argue that this physical scenario revealed by the exact solution of the Sachdev-Ye-Kitaev model is universally applicable for general chaotic quantum many-body systems and can be verified experimentally in near future.
Abstract This paper is an attempt to extend the recent understanding of the Page curve for evaporating black holes to more general systems coupled to a heat bath. Although calculating the von Neumann entropy by the replica trick is usually a challenge, we have identified two solvable cases. For the initial section of the Page curve, we sum up the perturbation series in the system-bath coupling κ; the most interesting contribution is of order 2s, where s is the number of replicas. For the saturated regime, we consider the effect of an external impulse on the entropy at a later time and relate it to OTOCs. A significant simplification occurs in the maximal chaos case such that the effect may be interpreted in terms of an intermediate object, analogous to the branching surface of a replica wormhole.
Abstract We developed a perturbative calculation for entropy dynamics, which considers a sudden coupling between a system and a bath. The theory we developed can work in a general environment without Markovian approximation. A perturbative formula is given for bosonic environments and fermionic environments, respectively. We find the Rényi entropy response is only related to the spectral functions of the system and the environment, together with a statistical kernel distribution function. We find at2growth/decay in the short time limit and a lineartgrowth/decay in a longer time scale for the second Rényi entropy response. A non-monotonic behavior of Rényi entropy for fermionic systems is found to be quite general when the environmental temperature is the lower one. A Fourier’s law in heat transport is obtained when two systems’ temperatures are close to each other. A consistency check is made for Sachdev-Ye-Kitaev model coupling to free fermions, a Page curve alike dynamics is found in a process dual to black hole evaporation. An oscillation of Rényi entropy is found for an environment with a gapped spectrum.
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.