Renyi entropy of chaotic eigenstates

Using arguments built on ergodicity, we derive an analytical expression for the Renyi entanglement entropies which, we conjecture, applies 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 nonlinearly 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 the 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 an exact diagonalization study on quantum spin-chain Hamiltonians to test our analytical predictions.

Figure 1

The curvature dependence of the Renyi entropy $S_n$ derived in the main text (solid lines) using arguments built on ergodicity: 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 ${\rho }_{th}^A\left(\beta \right)=exp(-\beta H_A)/Z$. $S_n$ derived built on ergodicity equals the thermal counterpart for $V_A/V<1/2$ at $n=1$, while for $n\ne 1$ it equals the thermal counterpart only as $V_A/V\to 0$. For each line style, the lines shown in descending order from the top correspond to $n<1$, $n=1$, and $n>1$ respectively.

Figure 2

The Renyi entropies $S_2$ (top) and $S_{1/2}$ (bottom) for a system of volume $V=1000$ with Gaussian density of states [Eq. (27)]. The various lines plotted in descending order from the top correspond to $\beta =0.0,0.2,0.4,0.6,0.8$ respectively.

Figure 3

Comparison of the three different ways to average over the random ensembles discussed in the text to obtain the second Renyi entropy. Triangles: $S_2^A\left(\overline{{\rho }_A}\right)$. Crosses: $S_2^A\left(\overline{\text{tr}{\rho }_A^2}\right)$. Open circles: $S_{2,\mathrm{avg}}^A=-\overline{\text{ln}\left[\text{tr}\left({\rho }_A^2\right)\right]}$. Note that $S_2^A\left(\overline{\text{tr}{\rho }_A^2}\right)$ and $S_{2,\mathrm{avg}}^A$ are essentially identical, as they should be (see Appendix pp2). The Hamiltonian is $H=-{\sum }_{i=1}^L{Z}_i+\varepsilon H_1$ for $L=12$ and $\varepsilon =0.1$, where $H_1$ is a real Hermitian random matrix. For each marker style, the data in descending order from the top correspond to $\beta =0.0,0.17,0.35,0.55,0.8$.

Figure 4

Probability amplitudes for $C_{\alpha }$ [Eq. (8)] corresponding to the Hamiltonian in Eq. (31) with $L=18$ when $H_0$ and $H_1$ are given by Eq. (32) for various values of the perturbation $\varepsilon$: (a) $\varepsilon =0.01$, (b) $\varepsilon =0.204$, (c) $\varepsilon =0.404$, and (d) $\varepsilon =0.724$. Blue dots: exact results from eigenstates. Red dashed lines: Gaussian fitting. We choose eigenstates at energy $E=0$ and the width of energy window is $\mathrm{\Delta }=1$. The Gaussian distribution is obtained by least square fitting.

Figure 5

Comparison of the second Renyi entropy $S_2$ for a Berry state, with those obtained from the exact diagonalization for the Hamiltonian $H=H_0+\varepsilon H_1$ for $L=20$ and $\varepsilon =0.204$ where $H_0$ and $H_1$ are given by Eq. (32). The solid dots correspond to $S_2$ of eigenstates averaged over an energy window of width $\mathrm{\Delta }E=2$, and the vertical bars denote the standard deviation in $S_2$ in this energy window. Solid lines correspond to $S_2$ for the Berry state using Eq. (11). For each marker (line) style, the data in descending order from the top correspond to $\beta =0.0,0.13,0.27,0.39$.

Figure 6

Probability distribution of the amplitudes $C_{\alpha }$ [Eq. (8)] for a single eigenstate corresponding to inverse temperature $\beta =0.26$ of the Hamiltonian $H=-{\sum }_{i=1}^L{Z}_i+\varepsilon H_1$, where $H_1$ is a real Hermitian random matrix. We choose $L=16$ and $\varepsilon =0.1$. The Gaussian distribution is obtained by least square fitting.

Figure 7

Comparison of the second Renyi entropy $S_2$ of one-dimensional (1D) spin model with Hamiltonian $H=-{\sum }_{i=1}^L{Z}_i+\varepsilon H_1$ where $H_1$ is a random matrix. We choose $L=16$ and $\varepsilon =0.1$. The three plotted quantities correspond to Renyi entropy $S_2$ of eigenstates (solid dots), Berry states [Eq. (11)] (solid lines), and CTPQ states [Eq. (29)] (dashed lines). For each marker (line) style, the data in descending order from the top correspond to $\beta =0.26,0.39,0.55$.

Figure 8

Finite size scaling of the difference between the third Renyi entropy $S_3$ obtained from analytical expressions and exact diagonalization results for the eigenstates of a 1D spin model with Hamiltonian $H=-{\sum }_{i=1}^L{Z}_i+\varepsilon H_1$ where $H_1$ is a random matrix. Here $\varepsilon =0.1$. A data point is obtained by averaging $\mathrm{\Delta }{S}_3$ for eigenstates in the range $0<\beta <0.5$. $L_A$ is chosen to be $L/3$.

Figure 9

Probability distribution of the bipartite amplitudes $C_{ij}$ [Eq. (5)] when $|E\rangle$ corresponds to a single eigenstate of the Hamiltonian in Eq. (35) with inverse temperature $\beta =0.18$. We choose $L=16$ and the subsystem size $L_A=8$. The energy window $\mathrm{\Delta }$ that appears in Eq. (5) is chosen to be 2. The Gaussian distribution is obtained by least square fitting.

Figure 10

Comparison of the second Renyi entropy $S_2$ and the third Renyi entropy $S_3$ obtained by different methods for the eigenstates of 1D nonintegrable Hamiltonian in Eq. (35) for $L=20$. Top: Exact diagonalization (solid dots), ergodic bipartition states, Eq. (11) (solid lines), and CTPQ states, Eq. (29) (dashed lines). For each marker (line) style, the data in descending order from the top correspond to $\beta =0.12,0.18,0.24,0.3$. Bottom: Exact diagonalization (solid dots), exact expression for ergodic bipartition states, Eq. (14) (solid lines), leading order expression for ergodic bipartition states, Eq. (15) (dash-dot lines). For each marker (line) style, the data in descending order from the top correspond to $\beta =0.06,0.18,0.24$.

Figure 11

Finite size scaling for of $\mathrm{\Delta }{S}_n/L_A$ for ergodic bipartition (EB) states and CTPQ states where $\mathrm{\Delta }{S}_n$ is defined as the difference between the analytical expressions for the corresponding states (EB or CTPQ) and the exact diagonalization results. The Hamiltonian is given by Eq. (35) and eigenstates correspond to $\beta =0.06$. $L_A$ is chosen to be $L/3$ for all $L$. Note that in the middle panel, “EB (leading order)” refers to the expression in Eq. (15) while “EB (exact)” refers to Eq. (14). In the top panel we use the exact expression for EB states [Eq. (11)] while in the bottom panel, we use the leading order result [Eq. (15)] for EB states.

Figure 12

Evolution of the shape of the Renyi entropy $S_2$ as the total system is increased for a system with Gaussian density of states [Eq. (24)] for inverse temperature $\beta =0.6$. The various curves in ascending order correspond to $V=10,20,30,50,500$ respectively. Note that as $V\to \infty$, the Renyi entropy is a convex function for all $f$, and has a cusp singularity at $f=1/2$.

Figure 13

The density of states as a function of the temperature of the eigenstates for an 18 site spin chain. The temperature for individual eigenstates $|E_n\rangle$ is evaluated by solving the equation $\frac{\text{tr}\left(H{e}^{-\beta \left(E_n\right)H}\right)}{\text{tr}\left(e^{-\beta \left(E_n\right)H}\right)}=E_n$.

Figure 14

Allowed relative positions of the energies $u,u_A^{*}$ and $u_{\overline{A}}^{*}$ that solve Eq. (18). Note that the concavity of $s\left(u\right)$ curve, imposed by the non-negative value of specific heat, plays a crucial role.