Entanglement Entropy of Eigenstates of Quadratic Fermionic Hamiltonians

In a seminal paper [D. N. Page, Phys. Rev. Lett. 71, 1291 (1993)], Page proved that the average entanglement entropy of subsystems of random pure states is $S_{\text{ave}}\simeq \ln {\mathcal{D}}_{\mathrm{A}}-(1/2){\mathcal{D}}_{\mathrm{A}}^2/\mathcal{D}$ for $1\ll {\mathcal{D}}_{\mathrm{A}}\le \sqrt{\mathcal{D}}$, where ${\mathcal{D}}_{\mathrm{A}}$ and $\mathcal{D}$ are the Hilbert space dimensions of the subsystem and the system, respectively. Hence, typical pure states are (nearly) maximally entangled. We develop tools to compute the average entanglement entropy $⟨S⟩$ of all eigenstates of quadratic fermionic Hamiltonians. In particular, we derive exact bounds for the most general translationally invariant models $\ln {\mathcal{D}}_{\mathrm{A}}-(\ln {\mathcal{D}}_{\mathrm{A}}{)}^2/\ln \mathcal{D}\le ⟨S⟩\le \ln {\mathcal{D}}_{\mathrm{A}}-[1/(2\ln 2)](\ln {\mathcal{D}}_{\mathrm{A}}{)}^2/\ln \mathcal{D}$. Consequently, we prove that (i) if the subsystem size is a finite fraction of the system size, then $⟨S⟩<\ln {\mathcal{D}}_{\mathrm{A}}$ in the thermodynamic limit; i.e., the average over eigenstates of the Hamiltonian departs from the result for typical pure states, and (ii) in the limit in which the subsystem size is a vanishing fraction of the system size, the average entanglement entropy is maximal; i.e., typical eigenstates of such Hamiltonians exhibit eigenstate thermalization.

Figure 1

Entanglement entropy of eigenstates of noninteracting spinless fermions in a periodic chain with $L=36$ sites. Results are plotted as a function of the linear subsystem size $L_A$. (Lower line) average entanglement entropy $⟨S⟩$ of all eigenstates and (upper line) upper bound $S_{\max }=\ln {\mathcal{D}}_A=L_A\ln 2$ for $L_A\le L/2$ [$S_{\max }=(L-L_A)\ln 2$ for $L_A>L/2$]. Each pixel denotes the weight of eigenstates $|m⟩$, with target entropy $S$, defined as ${\mathcal{P}}_S={\mathcal{D}}^{-1}{\sum }_{0\le (S-S_m)<\ln 2}$.

Figure 2

Entanglement entropy mean, bounds, and variance for free fermions on a periodic chain. (a) Upper bounds ($S_1^{+}$, $S_2^{+}$) and lower bounds ($S_1^{-}$, $S_2^{-}$), given by Eqs. (9) and (11), for the average entanglement entropy. The upper (magenta) line is the maximal entanglement entropy $S_{\max }$, and the thick black line is the average entanglement entropy $⟨S⟩$ on a lattice with $L=36$ sites (same results as in Fig. 1). (b) ${\mathrm{\Sigma }}_S$ for $L_A=60$ in ensembles of $10^6$ randomly sampled eigenstates and (solid line) the prediction from Eq. (14). (c) ${\mathrm{\Sigma }}_S$ for $L_A/L=1/2$, calculated using all eigenstates in lattices with $L\le 38$. The solid line is a single-parameter fit to ${\mathrm{\Sigma }}_S=a/\sqrt{L}$ for $L\ge 30$, with $a=0.410$.

Figure 3

Distribution of eigenvalues of the restricted complex structure for vanishing subsystem fraction. The overlapping solid lines depict ${\mathcal{P}}_{\lambda }$, which are averages of the discrete distribution $p_{\lambda }=\sum _{|{\lambda }_j-\lambda |<\delta \lambda /2}$ over $10^6$ random eigenstates, where ${\lambda }_j$ are eigenvalues of $[iJ{]}_A$, and we take $\delta \lambda ={10}^{-2}$. The symbols depict ${\mathcal{P}}_{\lambda }^{(\mathrm{TGE})}$, which are averages of the discrete distribution $p_{\lambda }^{(\mathrm{TGE})}=\sum _{|{\lambda }_j-\lambda |<\delta \lambda /2}$ over $10^6$ realizations of the Toeplitz Gaussian ensemble (TGE), where ${\lambda }_j$ are the eigenvalues of the TGE, and we take $\delta \lambda ={10}^{-2}$. Both axes are renormalized to show data collapse. (Inset) scaling of $\overline{\mathcal{S}}$ [see Eq. (16)]. The symbols show numerical results of an average over $10^6$ random eigenstates. The solid line shows the results of Eq. (16).