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Structure of chaotic eigenstates and their entanglement entropy

Chaitanya Murthy, Mark Srednicki

TL;DR

This work analyzes entanglement properties of energy eigenstates in chaotic, ETH-satisfying systems partitioned into two subsystems. By refining the Deutsch–Lu–Grover ansatz to specify the envelope of the coefficient matrix $M_{iJ}$ with a Gaussian window of width $\Delta$ tied to the interaction, it derives the reduced density matrix and shows a universal square-root correction to the entanglement entropy when the subsystems are of equal volume: $S_{\text{ent}} = \frac{1}{2} S(E) - \sqrt{\frac{C}{2\pi}} + O(A)$. The correction is expressed entirely in terms of thermodynamic quantities, and generalizes to multiple conserved quantities via a capacity matrix. The paper also provides the corresponding corrections to Rényi entropies, and demonstrates that the width $\Delta$ scales with boundary area in higher dimensions, linking entanglement structure directly to thermodynamic response. Overall, it casts Vidmar–Rigol’s observation as a generic feature of ETH-chaotic systems and clarifies the relationship between entanglement, energy density, and thermodynamics across dimensions and conserved quantities.

Abstract

We consider a chaotic many-body system (i.e., one that satisfies the eigenstate thermalization hypothesis) that is split into two subsystems, with an interaction along their mutual boundary, and study the entanglement properties of an energy eigenstate with nonzero energy density. When the two subsystems have nearly equal volumes, we find a universal correction to the entanglement entropy that is proportional to the square root of the system's heat capacity (or a sum of capacities, if there are conserved quantities in addition to energy). This phenomenon was first noted by Vidmar and Rigol in a specific system; our analysis shows that it is generic, and expresses it in terms of thermodynamic properties of the system. Our conclusions are based on a refined version of a model of a chaotic eigenstate originally due to Deutsch, and analyzed more recently by Lu and Grover.

Structure of chaotic eigenstates and their entanglement entropy

TL;DR

This work analyzes entanglement properties of energy eigenstates in chaotic, ETH-satisfying systems partitioned into two subsystems. By refining the Deutsch–Lu–Grover ansatz to specify the envelope of the coefficient matrix with a Gaussian window of width tied to the interaction, it derives the reduced density matrix and shows a universal square-root correction to the entanglement entropy when the subsystems are of equal volume: . The correction is expressed entirely in terms of thermodynamic quantities, and generalizes to multiple conserved quantities via a capacity matrix. The paper also provides the corresponding corrections to Rényi entropies, and demonstrates that the width scales with boundary area in higher dimensions, linking entanglement structure directly to thermodynamic response. Overall, it casts Vidmar–Rigol’s observation as a generic feature of ETH-chaotic systems and clarifies the relationship between entanglement, energy density, and thermodynamics across dimensions and conserved quantities.

Abstract

We consider a chaotic many-body system (i.e., one that satisfies the eigenstate thermalization hypothesis) that is split into two subsystems, with an interaction along their mutual boundary, and study the entanglement properties of an energy eigenstate with nonzero energy density. When the two subsystems have nearly equal volumes, we find a universal correction to the entanglement entropy that is proportional to the square root of the system's heat capacity (or a sum of capacities, if there are conserved quantities in addition to energy). This phenomenon was first noted by Vidmar and Rigol in a specific system; our analysis shows that it is generic, and expresses it in terms of thermodynamic properties of the system. Our conclusions are based on a refined version of a model of a chaotic eigenstate originally due to Deutsch, and analyzed more recently by Lu and Grover.

Paper Structure

This paper contains 5 sections, 76 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Envelope function of the coefficient matrix
  3. Reduced density matrix
  4. Corrections to the entanglement and Rényi entropies
  5. Conclusions

Figures (1)

  • Figure 1: A plot of the function $\Phi(x)$ defined in Eq. (\ref{['eq:Phi']}), which parameterizes the correction to the entanglement entropy, Eq. (\ref{['eq:Sent_form']}).