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arXiv:2504.14675 · Physical review. B./Physical review. B

Page curve like dynamics in interacting quantum systems

Tamoghna Ray, Abhishek Dhar, Manas Kulkarni

Published Nov 12, 2025• 1 citation
arXivHTMLPDF

Abstract

We study the dynamics of entanglement in a one-dimensional $XXZ$ spin-$1/2$ 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.

DOI
10.1103/z3sj-gn7t

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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.
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.
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.
Abstract We consider time dynamics of entanglement entropy (EE) between a filled fermionic system and an empty reservoir. We employ two different kinds of quantum trajectory approaches, namely stochastic unitary process and quantum state diffusion that mimics space-time white noise and quantum measurement at each local site of the lattice, respectively. For both protocols, we observe and characterize the full page curve-like dynamics for the EE. Depending on the protocol and the setup, we observe very distinct characteristics of the page curve and the associated Page time and page value. We also compute the number of fermions leaking to the reservoir and the associated current and shed light on their plausible connections with EE. Our approach is versatile, extendable to other setups and trajectory protocols, and potentially implementable in a variety of engineered quantum platforms.
We present a controllable quantum spin-chain model that reproduces the Page curve (the rise-and-fall of bipartite entanglement expected in black-hole evaporation), using only local interactions and a kinematic reduction of the subsystem size. Two transverse-field Ising chains are coupled to form a pure bipartite state; Hawking-like evaporation is implemented by dynamically shrinking the 'system' chain and enlarging the 'environment' chain, while unitary real-time evolution is simulated with matrix product state (MPS) tensor networks. The characteristic Page curve profile emerges robustly under this controlled subsystem resizing and notably persists even when the explicit Hamiltonian coupling across the boundary is set to zero, demonstrating that shrinking Hilbert-space dimension alone can generate Page curve behaviour. We show that the detailed shape of the curve depends on the internal information dynamics: operation at criticality yields a smooth profile, whereas moving away from criticality distorts entanglement growth and decay. These results position locally interacting spin chains as a realistic platform for probing black-hole-inspired information dynamics on current quantum hardware.
Quantum many-body systems are commonly considered as quantum chaotic if their spectral statistics, such as the level spacing distribution, agree with those of random matrix theory (RMT). Using the example of the kicked Ising chain we demonstrate that even if both level spacing distribution and eigenvector statistics agree well with random matrix predictions, the entanglement entropy deviates from the expected RMT behavior, i.e., the Page curve. To explain this observation we propose a quantity that is based on the effective Hamiltonian of the kicked system. Specifically, we analyze the distribution of the strengths of the effective spin interactions and compare them with analytical results that we obtain for circular ensembles. Thereby we group the effective spin interactions corresponding to the number k of spins which contribute to the interaction. By this the deviations of the entanglement entropy can be attributed to significantly different behavior of the k-spin interactions compared with RMT.