Area-Law Entanglement in Quantum Chaotic System

TL;DR

Entanglement entropy is often used to diagnose chaos through volume-law scaling in highly excited states, but this work constructs a Floquet-driven, Rydberg-blockaded chain that is chaotic by COE level statistics and local thermalization while enforcing a strict entanglement bound of $\ln 2$ for all eigenstates. The anomaly arises from blockade-induced subspace structure that fixes the bipartite Schmidt rank to at most 2, and is formalized via a duality to single-particle quantum walks on median graphs derived from 2-SAT problems. A general three-step recipe is provided to design quantum many-body Hamiltonians with bounded entanglement, including a concrete rank-3 example yielding $S_{\max} = \ln 3$, illustrating the broad applicability of the approach. The results demonstrate that entanglement entropy is not a universal chaos diagnostic and reveal how Hilbert-space geometry can govern thermalization, with potential extensions to higher dimensions and a separation between entanglement and thermodynamic entropy.

Abstract

Entanglement entropy is a fundamental diagnostic for quantum chaos, typically exhibiting volume-law scaling in highly excited eigenstates of chaotic many-body systems. In this work, we present a striking counterexample: a Floquet-driven quantum many-body system with Rydberg-like blockade that, despite being fully chaotic as indicated by its Wigner-Dyson level statistics and local thermalization, exhibits a strict area-law entanglement entropy. Specifically, the entanglement entropy of every Floquet eigenstate is bounded by $\ln2$, independent of system size. We trace this anomaly to the specific Hilbert space structure imposed by the blockades, which restricts the Schmidt rank across a bipartition. Furthermore, we generalize this discovery by establishing a duality between constrained many-body Hamiltonians and single-particle quantum walks on median graphs, and we outline a general procedure for constructing systems with an entanglement entropy bounded by a predetermined constant. Our results demonstrate that entanglement entropy alone is an insufficient diagnostic of many-body quantum chaos and highlight the profound impact of Hilbert space geometry on quantum dynamics and thermalization.

Figures (3)

• Figure 1: (a) The product states that span the constraint subspace for $N=6$. $\mathbin{\circ}$ represents an atom in the ground state, while $\mathbin{\bullet}$ represents an atom in the excited state. Two product states are connected by a line when the Hamming distance between them is 1. (b) The Rydberg-like interaction. The circle represents a Rydberg atom. The $V$ inside the circle represents the energy of the atom in the Rydberg state, and the solid lines represent the interactions between two atoms.
• Figure 2: Indications of chaos in our quantum many-body system with $\omega = 0.9071$. (a) Statistics of quasi-energy level spacing for $N=1000$. $P(s)$ represents the probability distribution of the nearest-neighbor quasi-energy level spacing $s$, fitted by a Wigner-Dyson distribution, with the solid line representing the standard COE level-spacing distribution. (b) Expectation value of the spin-flipping operator $\hat{X}_{250}$ at the 250th site in each Floquet eigenstate for $N=1000$, showing $\braket{\hat{X}_{250}} \approx 0$ for all eigenstates. (c, d) Entanglement entropy of the Floquet eigenstates for $N=1000$ (c) and $N=500$ (d), both clearly bounded by $\ln 2$. $\epsilon$ and $S$ refer to the quasi-energy and von Neumann entropy, respectively.
• Figure 3: A median graph with $N=4$. The entanglement entropy of the corresponding quantum system is bounded by $\ln 3$.