Quantum thermalization and average entropy of a subsystem

TL;DR

This work analyzes the average von Neumann entropy of a subsystem when the total quantum system is restricted to a narrow energy shell, rather than sampled Haar-randomly. It derives a leading term $\overline{S}_{1} \simeq \ln d_{1}$ with $d_{1} \approx D_{1}^{\gamma}$, where $\gamma = \frac{\ln d_E}{\ln D}$ for nonintegrable (chaotic) systems and $\gamma$ is smaller for integrable systems, linking the entropy to the density of states (DOS). The entropy follows a DOS-dependent volume law, with $\overline{S}_{1} \simeq \frac{1}{2}\ln\text{DOS}$ when $l_1 \sim l_2$ and more generally $\overline{S}_{1} \simeq \ln d_{1}$ for other regime limits, and numerical results on a spin-1/2 chain corroborate the scaling and its dependence on integrability. The approach provides a practical diagnostic for chaotic versus integrable dynamics and offers a scalable, energy-shell-based method to study quantum thermalization using limited subspace information, potentially reducing computational effort via techniques like shift-and-square to access the relevant energy window.

Abstract

Page's seminal result on the average von Neumann (VN) entropy does not immediately apply to realistic many-body systems which are restricted to physically relevant smaller subspaces. We investigate here the VN entropy averaged over the pure states in the subspace $\mathcal{H}_E$ corresponding to a narrow energy shell centered at energy $E$. We find that the average entropy is $\overline{S}_{1} \simeq \ln d_1$, where $d_1$ represents first subsystem's effective number of states relevant to the energy scale $E$. If $d_E = \dim{(\mathcal{H}_E)}$ and $D$ ($D_1$) is the Hilbert space dimension of the full system (first subsystem), we estimate that $d_1 \simeq D_1^γ$, where $γ= \ln (d_E) / \ln (D)$ for nonintegrable (chaotic) systems and $γ< \ln (d_E) / \ln (D)$ for integrable systems. This result can be reinterpreted as a volume-law of entropy, where the volume-law coefficient depends on the density-of-states for nonintegrable systems, and remains below the maximal possible value for integrable systems. We numerically analyze a spin model to substantiate our main results.

Figures (3)

• Figure 1: The subsystem VN entropy ($S_1$) for the individual eigenkets and $\ln$(DOS) are plotted against the energy of the full system (described by the Hamiltonian of Eq. \ref{['spn_ham']}). Calculations are performed with full system size $N=16$ and subsystem size $l_1=6$.
• Figure 2: The average VN entropy ($\overline{S}_1$) of subsystem 1 is plotted against $\ln$(DOS) across the energy spectrum of the full system. The plots are shown separately for the left and the right halves of the spectrum. Calculations are performed with full system size $N=16$ and subsystem size $l_1=6$.
• Figure 3: The subsystem average VN entropy ($\overline{S}_1$) is plotted against the subsystem size ($l_1$) for a fixed energy (corresponding to the highest DOS).