Entanglement Entropy of Eigenstates of Quantum Chaotic Hamiltonians

TL;DR

The paper investigates how close the bipartite entanglement entropy of eigenstates of quantum-chaotic Hamiltonians is to its maximal value, focusing on systems away from half filling and partitioned in two equal halves. It decomposes the entropy of random pure states with fixed particle number into a mean-field term and a fluctuation term, deriving an upper bound on the fluctuation contribution that scales as the square root of the subsystem volume at f=1/2. Numerical results for random canonical states and highly excited eigenstates indicate this bound is saturated with increasing system size, showing a subleading sqrt-volume correction to the Page-like maximal entanglement at half filling. The study also extends the analysis to random states without fixed particle number, confirming similar scaling behavior and highlighting that high-energy eigenstates at non-half filling exhibit subextensive deviations from maximal entanglement.

Abstract

In quantum statistical mechanics, it is of fundamental interest to understand how close the bipartite entanglement entropy of eigenstates of quantum chaotic Hamiltonians is to maximal. For random pure states in the Hilbert space, the average entanglement entropy is known to be nearly maximal, with a deviation that is, at most, a constant. Here we prove that, in a system that is away from half filling and divided in two equal halves, an upper bound for the average entanglement entropy of random pure states with a fixed particle number and normally distributed real coefficients exhibits a deviation from the maximal value that grows with the square root of the volume of the system. Exact numerical results for highly excited eigenstates of a particle number conserving quantum chaotic model indicate that the bound is saturated with increasing system size.

Figures (5)

• Figure 1: Average entanglement entropy in eigenstates of a quantum chaotic Hamiltonian, Eq. (\ref{['def_Ham']}), and in random canonical states, Eq. (\ref{['def_psiN']}). (Main panel) $\bar{S}/[L/2]$ vs the subsystem fraction $f=L_A/L$ for $n=1/2$, 1/3, 1/4 and 1/6 ($L=22$, 24, 24, and 30, respectively). Solid (dashed) lines show the mean-field entanglement entropy $S_{\rm MF}$ ($S_{\rm MF}^*$) from Eq. (\ref{['Smf_long']}) [Eq. (\ref{['Smf_short']})]. (Inset) $\bar{S} - S_{\rm MF}$ vs $1/L_A$ for $f=n=1/2$.
• Figure 2: Fluctuation contribution to the average entanglement entropy when $f=1/2$, for average site occupations $n=1/3$ (a), 1/4 (b), and $1/6$ (c). The circles display results for random canonical states, Eq. (\ref{['def_psiN']}) (see Ref. suppmat for details on the numerical calculations), and the dashed lines are linear fits to those results. The squares display the results for eigenstates of Hamiltonian (\ref{['def_Ham']}). The diamonds are the upper bound $S^{\rm bound}_{\rm fluct}$ from Eq. (\ref{['def_Sbound']}) and the solid lines are $S^{\rm bound*}_{\rm fluct}$ from Eq. (\ref{['def_Sstar']}). The zigzag structure of $S^{\rm bound}_{\rm fluct}$ in (b) and (c) is a finite-size effect suppmat.
• Figure 3: Entanglement entropy of random pure states when $f=1/2$, for average site occupations $n=1/3$ (a) and $n=1/4$ in (b). The circles display results for random canonical states, Eq. (\ref{['def_psiN']}). [Circles and dashed lines are identical to the ones in Figs. \ref{['fig2']}(a) and \ref{['fig2']}(b).] The pentagons display results for random states without a fixed particle number, Eq. (\ref{['def_psi_Nall']}), where $S_{\rm MF} \to S_{\rm MF}'$. Solid lines are linear fits in which the slope is chosen to be identical to the one for the dashed lines (see Ref. suppmat for details on the numerical calculations).
• Figure S1: Scaling of $S_{\rm fluct}^{\rm bound} / \sqrt{L_A}$ with the inverse subsystem size $1/L_A$ for $f=1/2$ and $n=1/6$. Diamonds are exact numerical results, the thick solid line is $S_{\rm fluct}^{\rm bound*}/\sqrt{L_A}$ from Eq. (\ref{['def_Sstar2']}), and the thin solid line is $\tilde{S}_{\rm fluct}^{\rm bound}/\sqrt{L_A}$. [We obtain $\tilde{S}_{\rm fluct}^{\rm bound}$ from $S_{\rm fluct}^{\rm bound}$ upon replacing the binomial coefficients in Eq. (\ref{['def_Sbound2']}) with Stirling's approximation.]
• Figure S2: Scaling of $-\tilde{S}_{\rm fluct}^{\rm bound}$ vs $L$. (a) Subsystem volumes $L_A = f L$ for $f=1/2$ (dashed lines) and $f=9/20$ (solid lines). Results are shown for $n=1/10$ (thick red lines) and for $n=1/5$ (thin black lines). (b) Subsystem volumes $L_A = L/2 - \gamma L^\alpha$ and $n=1/6$, for different values of $\alpha$. For $\alpha = 0$, we set $\gamma = 0$, while for $\alpha>0$ we set $\gamma=1/4$.