Area-law violation for the mutual information in a nonequilibrium steady state

We study the nonequilibrium steady state of an infinite chain of free fermions, resulting from an initial state where the two sides of the system are prepared at different temperatures. The mutual information is calculated between two adjacent segments of the chain and is found to scale logarithmically in the subsystem size. This provides an example of the violation of the area law in a quantum many-body system outside a zero-temperature regime. The prefactor of the logarithm is obtained analytically and, furthermore, the same prefactor is shown to govern the logarithmic increase of mutual information in time, before the system relaxes locally to the steady state.

Figure 1

Contour of the integral yielding the entropies $S_{\alpha }$ in the complex $\lambda$ plane. The large (blue) contour $\Gamma$ is used in Eq. (12), while the two small (red) contours ${\Gamma }_1$ and ${\Gamma }_2$ are associated to the jump symbols, responsible for the $lnL$ contributions.

Figure 2

Prefactor $\sigma$ of the logarithmic term in the steady-state mutual information $I(A:B)$ as a function of ${\beta }_{\ell }$ and for various ${\beta }_r$. The lines represent the analytic result in Eq. (15), while the dots are the results of fitting the numerical data.

Figure 3

Time evolution of the mutual information $I_t(A:B)$ for ${\beta }_{\ell }=5$, ${\beta }_r=0$, and various segment sizes $L$, after connecting the chains at $t=0$. The inset shows the difference from the steady-state value $I(A:B)$, plotted against $ln(t/L)$.

Figure 4

Sketch of the symbol $F\left(q\right)$ for a given ${\beta }_{\ell }$ and ${\beta }_r$.

Figure 5

Final integration contour decomposes into two intervals and four circular contours around the points $a$, $b$, $1-a$, and $1-b$.