Renyi Entropy of Chaotic Eigenstates

TL;DR

This work derives a universal analytical form for the Renyi entropies of chaotic many-body eigenstates as a function of the subsystem fraction, showing that, unlike von Neumann entropy, Renyi entropies with index $n\neq 1$ generally do not reproduce the thermal Page curve when the subsystem is a finite fraction of the total system. Using ergodicity-based arguments—the many-body Berry conjecture and the ergodic bipartition model—the authors express the averaged Renyi entropy $\bar{S}_n$ through saddle-point combinations of the subsystem and complement densities of states, $s(u)$, yielding $\bar{S}_n = \frac{V}{1-n} [ f s(u_A^*) + n(1-f)s(u_{ar A}^*) - n s(u) ]$ with energy-conservation-determined $u_A^*,u_{ar A}^*$. They show that $\frac{\rmd^2 \bar{S}_n}{\rmd f^2}$ is positive for $n>1$ and negative for $n<1$, producing convexity/concavity in $f$ and a cusp at $f=1/2$, and that von Neumann entropy ($n\to1$) remains linear in $V_A$ (Page curve) while $S_n$ for $n\neq1$ depends on the full density of states. Comparisons with Exact Diagonalization on spin chains validate the predictions, demonstrating that chaotic eigenstates mimic ergodic ensembles in their global entanglement structure and that finite-size and boundary effects shape deviations from the thermodynamic-limit behavior. The results have bearing on distinguishing pure from thermal states and on interpreting information in highly excited quantum many-body systems.

Abstract

Using arguments built on ergodicity, we derive an analytical expression for the Renyi entanglement entropies corresponding to the finite-energy density eigenstates of chaotic many-body Hamiltonians. The expression is a universal function of the density of states and is valid even when the subsystem is a finite fraction of the total system - a regime in which the reduced density matrix is not thermal. We find that in the thermodynamic limit, only the von Neumann entropy density is independent of the subsystem to the total system ratio $V_A/V$, while the Renyi entropy densities depend non-linearly on $V_A/V$. Surprisingly, Renyi entropies $S_n$ for $n > 1$ are convex functions of the subsystem size, with a volume law coefficient that depends on $V_A/V$, and exceeds that of a thermal mixed state at the same energy density. We provide two different arguments to support our results: the first one relies on a many-body version of Berry's formula for chaotic quantum mechanical systems, and is closely related to eigenstate thermalization hypothesis. The second argument relies on the assumption that for a fixed energy in a subsystem, all states in its complement allowed by the energy conservation are equally likely. We perform Exact Diagonalization study on quantum spin-chain Hamiltonians to test our analytical predictions, and find good agreement.

Figures (14)

• Figure 1: The curvature dependence of the Renyi entropy $S_n$ corresponding to chaotic eigenstates derived in the main text (solid lines): in the thermodynamic limit, $S_n$ is a convex (concave) function of $V_A/V$ for $n>1$ ($n<1$) with a cusp singularity at $V_A/V = 1/2$. The dashed lines correspond to the Renyi entropies of the thermal density matrix $\rho^A_{th}(\beta)=\exp(-\beta H_A)/Z$. $S_n$ of an chaotic eigenstate equals the thermal counterpart for $V_A/V<1/2$ at $n=1$, while for $n\neq 1$ equals the thermal counterpart only as $V_A/V\to 0$.
• Figure 2: Probability distribution of the bipartite amplitudes $C_{ij}$ (Eq.\ref{['eq:ergodic']}) when $\left\lvert{E}\right\rangle$ corresponds to a single eigenstate of the Hamiltonian in Eq.\ref{['eq:spin_model']}. The energy window $\Delta$ that appears in Eq.\ref{['eq:ergodic']} is chosen to be 2. The Gaussian distribution is obtained by least square fitting.
• Figure 3: Probability distribution of the amplitudes $C_{\alpha}$ (Eq.\ref{['eq:random_state']}) for a single eigenstate of the Hamiltonian $H=-\sum_{i=1}^N Z_i +\epsilon H_1$, where $H_1$ is a real hermitian random matrix. The Gaussian distribution is obtained by least square fitting.
• Figure 4: The Renyi entropies $S_2$ (top) and $S_{1/2}$ (bottom) for a system with Gaussian density of states (Eq.\ref{['eq:sn_gaussian']}).
• Figure 5: Comparison of the three different ways to average over the random ensembles discussed in the text to obtain the second Renyi entropy. Triangles: $S^A_2(\overline{ \rho_A})$. Crosses: $S^A_2(\overline{\mathop{\mathrm{tr}}\nolimits \rho^2_A})$. Open circles: $S^A_{2,\textrm{avg}} = - \overline{\log \left( \mathop{\mathrm{tr}}\nolimits \left(\rho^2_A \right) \right)}$. Note that $S^A_2(\overline{\mathop{\mathrm{tr}}\nolimits \rho^2_A})$ and $S^A_{2,\textrm{avg}}$ are essentially identical, as they should be (see Appendix \ref{['sec:equiv']} ). The Hamiltonian is $H=-\sum_{i=1}^N Z_i +\epsilon H_1$, where $H_1$ is a real hermitian random matrix.