Volume Law and Quantum Criticality in the Entanglement Entropy of Excited Eigenstates of the Quantum Ising Model

Much has been learned about universal properties of entanglement entropies in ground states of quantum many-body lattice systems. Here we unveil universal properties of the average bipartite entanglement entropy of eigenstates of the paradigmatic quantum Ising model in one dimension. The leading term exhibits a volume-law scaling that we argue is universal for translationally invariant quadratic models. The subleading term is constant at the critical field for the quantum phase transition and vanishes otherwise (in the thermodynamic limit); i.e., the critical field can be identified from subleading corrections to the average (over all eigenstates) entanglement entropy.

Figure 1

Entanglement entropy in the quantum Ising model, Eq. (2), as a function of the transverse field $h$. The subsystem volume is one-half of that of the system ($f=1/2$). (a) Ground-state entanglement entropy $S_{\mathrm{gs}}$. (b) Average (over all eigenstates) entanglement entropy density $s$, defined in Eq. (5). The horizontal line in (b) depicts the extrapolated result in the thermodynamic limit.

Figure 2

Average entanglement entropy density differences, $s_{\mathrm{qI}}-s_{\mathrm{NI}}$, at two subsystem fractions $f=1/2$ and $1/4$ for (a) $h=0$, (b) $h=1$, and (c) $h=5$. Lines are fits to the results for $L\ge 24$. Note the difference between the $x$ axes in panels (a) and (c), $1/L^2$, vs panel (b), $1/L$. Inset in (b), rescaled entanglement entropy density $\mathrm{\Delta }sL=(s_{\mathrm{qI}}-s_{\mathrm{NI}})L$ vs $1/L$ at $f=1/2$ and $h=1$. The line depicts a linear fit $A+B/L$ to the results for $L\ge 24$, with $A=0.311$ and $B=0.154$.

Figure 3

(a) Rescaled entanglement entropy density $(s_{\mathrm{qI}}-s_{\mathrm{NI}})L$ vs $\mathrm{\Gamma }(h)/(AL)$ for $h<1$. Filled symbols are exact results (namely, the average over entire Hilbert space with $2^L$ eigenstates), shown for $22\le L\le 32$, while open symbols show averages over $10^8$ random eigenstates for $34\le L\le 44$. For a given $h$, the color of the open symbols is identical to that of the filled ones. The solid line is the function $A[1-e^{-\mathrm{\Gamma }(h)/(AL)}]$, where the constant $A=0.311$ (horizontal line). (b) Rescaled entanglement entropy density $(s_{\mathrm{qI}}-s_{\mathrm{NI}})L^2$ vs $1/L$ for $h<1$. The symbol coding is the same as in (a), while the solid lines are linear fits $\mathrm{\Gamma }(h)+\zeta (h)/L$ to the exact results for $L\ge 26$. (Inset) The symbols depict the values of $\mathrm{\Gamma }(h)$, multiplied by ($1-h$), plotted vs ($1-h$). The solid line is a fitting function $\mathrm{\Gamma }(h)(1-h)=\alpha (1-h{)}^2+\beta$, with $\alpha =1.234$ and $\beta =1.294$.

Figure 4

(a) Rescaled entanglement entropy density $(s_{\mathrm{qI}}-s_{\mathrm{NI}})L$ vs $\overline{\mathrm{\Gamma }}(h)/(AL)$ for $h>1$. The system sizes and color coding are identical to the ones used in Fig. 3. The solid line is the function $A[1-e^{-\overline{\mathrm{\Gamma }}(h)/(AL)}]$, where the constant $A=0.311$ (horizontal line). (b) Rescaled entanglement entropy density $(s_{\mathrm{qI}}-s_{\mathrm{NI}})L^2$ vs $1/L$ for $h>1$. The symbol coding is the same as in (a), while solid lines are linear fits $\overline{\mathrm{\Gamma }}(h)+\overline{\zeta }(h)/L$ to the exact results for $L\ge 26$. (Inset) The symbols depict the values of $\overline{\mathrm{\Gamma }}(h)$, multiplied by ($h-1$), plotted vs ($h-1$). The solid line is a fitting function ${[\overline{\mathrm{\Gamma }}(h)(h-1)]}^{-1}=\overline{\alpha }+\overline{\beta }(h-1)e^{-\overline{\eta }/(h-1)}$, with $\overline{\alpha }=1.028$, $\overline{\beta }=0.371$, and $\overline{\eta }=0.629$.

Figure 5

Rescaled entanglement entropy density $s_{\rho }-s_{\mathrm{NI}}$ vs $1/L^2$ for various translationally invariant quadratic models. We replace $u_k$ and $v_k$ in Eq. (3) by $u_k=\cos [\sin (\rho k)]$ and $v_k=\sin [\sin (\rho k)]$ for $\rho \ge 1$, and $u_k=\cos (k)$ and $v_k=\sin (k)$ for $\rho =0$. Solid and dashed lines are linear fits $w_{\rho }/L^2$ to the results for $L\ge 26$, with $w_0=w_1=4.41$ and $w_2=8.96$. The dashed-dotted line is a guide to the eye with $w_3=3w_1$.