Finite-Temperature Critical Behavior of Mutual Information

We study mutual information for Renyi entropy of arbitrary index $n$, in interacting quantum systems at finite-temperature critical points, using high-temperature expansion, quantum Monte Carlo simulations and scaling theory. We find that, for $n>1$, the critical behavior is manifest at two temperatures $T_c$ and $n{T}_c$. For the $XXZ$ model with Ising anisotropy, the coefficient of the area law has a $t\ln t$ singularity, whereas the subleading correction from corners has a logarithmic divergence, with a coefficient related to the exact results of Cardy and Peschel. For $T<n{T}_c$ there is a constant term associated with broken symmetries that jumps at both $T_c$ and $n{T}_c$, which can be understood in terms of a scaling function analogous to the boundary entropy of Affleck and Ludwig.

Figure 1

In the HTE (left), we consider partitions of the infinite square-plane where region $A$ could be a half-plane such as $a\cup b$ or a quadrant such as $b$ or $c$. In QMC simulations (right) the $N$-site real space lattice is a torus with periodic boundaries, or a cylinder with the dashed boundaries open. In the torus, region $A$ can be $e$ (“square”) or $e\cup f$ (“strip”), while for the cylindrical we use a strip ($e\cup f$) region $A$.

Figure 2

A comparison of MI calculated with HTE, and QMC on $32\times 32$ simulation cells on toroid and cylindrical geometries (see Fig. 1). Inset: MI plotted against $L$ at $\beta =0.204$, showing excellent fit to linear form.

Figure 3

MI divided by boundary length for $I_1$ from Eq. (4), compared to ED on a $4\times 4$ system with $3\times 3$ region $A$; and $I_4$, compared to QMC for $10\times 10$ and $20\times 20$. A sharp change in slope in $I_4/L$ occurs at $1/4T_c$, where $T_c=2.24$, obtained from the crossing of the fourth-order Binder cumulant for the staggered magnetization. Inset: the constant scaling term $d_4$ extracted from a linear fit to the QMC data.