Universal Behavior of the Shannon Mutual Information of Critical Quantum Chains

We consider the Shannon mutual information of subsystems of critical quantum chains in their ground states. Our results indicate a universal leading behavior for large subsystem sizes. Moreover, as happens with the entanglement entropy, its finite-size behavior yields the conformal anomaly $c$ of the underlying conformal field theory governing the long-distance physics of the quantum chain. We study analytically a chain of coupled harmonic oscillators and numerically the $Q$-state Potts models ($Q=2$, 3, and 4), the $XXZ$ quantum chain, and the spin-1 Fateev-Zamolodchikov model. The Shannon mutual information is a quantity easily computed, and our results indicate that for relatively small lattice sizes, its finite-size behavior already detects the universality class of quantum critical behavior.

Figure 1

$I(\ell ,L)-I[(L/2),L]$ as a function of the subsystem size for the $Q=2$, 3, and 4 state Potts models in the $R_i$ and $S_i$ bases.

Figure 2

Mutual information for the $Q=2$, 3, and 4 Potts models with different lattice sizes $L$. The dashed straight lines are obtained from the fitting by using the largest lattice size shown for each model. (The ground states were taken in the $R_i$ basis.)

Figure 3

Mutual information as a function of the subsystem size for the $XXZ$ model with different values of the anisotropy $\Delta$ and lattice sizes $L$.

Figure 4

Mutual information dependence with the subsystem size for the Fateev-Zamolodchikov model with different values of $\gamma$ and lattice sizes.

Figure 5

Mutual information for some lattice sizes for the $XXZ$ chain with anisotropies $\Delta =-1/2$ and $\Delta =0$, the ferromagnetic spin-1 Fateev-Zamolodchikov model with anisotropies $\gamma =\pi /3$ and $\gamma =\pi /4$, and for the four-state Potts model. All these models are ruled by a CFT with conformal anomaly $c=1$.