Universal behavior of the Shannon and Rényi mutual information of quantum critical chains

We study the Shannon and Rényi mutual information (MI) in the ground state (GS) of different critical quantum spin chains. Despite the apparent basis dependence of these quantities we show the existence of some particular basis (we will call them conformal basis) whose finite-size scaling function is related to the central charge $c$ of the underlying conformal field theory of the model. In particular, we verified that for large index $n$, the MI of a subsystem of size $\ell$ in a periodic chain with $L$ sites behaves as $\frac{c}{4}\frac{n}{n-1}ln[\frac{L}{\pi }sin\left(\frac{\pi \ell }{L}\right)]$, when the ground-state wave function is expressed in these special conformal basis. This is in agreement with recent predictions. For generic local basis, we will show that, although in some cases $b_{n}ln[\frac{L}{\pi }sin\left(\frac{\pi \ell }{L}\right)]$ is a good fit to our numerical data, in general, there is no direct relation between $b_n$ and the central charge of the system. We will support our findings with detailed numerical calculations for the transverse field Ising model, $Q=3,4$ quantum Potts chain, quantum Ashkin-Teller chain, and the XXZ quantum chain. We will also present some additional results of the Shannon mutual information ($n=1$), for the parafermionic $Z_Q$ quantum chains with $Q=5,6,7$, and 8.

Figure 1

Coefficient of the logarithmic term of the Rényi MI in the Ising model in the ${\sigma }^z$ and ${\sigma }^x$ bases. The coefficients were found by restricting the fitting of (11) to the subsystem sizes $\ell =4,5,\cdots ,L/2$. The dashed straight lines are guidelines for $n=1$ and for the central charge $c=0.5$.

Figure 2

Finite-size data of $c_n\left(L\right)$, for $L=12,14,...,30$, for the GS Ising model in ${\sigma }^x$ basis. The coefficients were calculated by conditioning the fitting to the subsystem sizes $\ell =4,5,\cdots ,L/2$.

Figure 3

Coefficient of the logarithmic term of the Rényi MI in the Ising model in the ${\sigma }^y$ basis [$(\theta ,\varphi ,\alpha )=(\frac{\pi }{4},0,0)$] and the $B$ basis [$(\theta ,\varphi ,\alpha )=(\frac{\pi }{3},\pi ,\frac{\pi }{5})$]. The coefficients were found by conditioning the fitting to the subsystem sizes $\ell =4,5,\cdots ,L/2$. The dashed straight lines are guidelines for $n=1$ and for the central charge $c=0.5$.

Figure 4

Coefficient of the logarithmic term of the Rényi MI in the $Q=3$ Potts model in the $R$ and $S$ basis [26]. The coefficients were found by restricting the fitting of (11) to the subsystem sizes $\ell =4,5,\cdots ,\text{Int}[L/2]$. The dashed straight lines are guidelines for $n=1$ and for the central charge $c=0.8$.

Figure 5

Rényi MI with respect to $ln[\frac{L}{\pi }sin\left(\frac{\pi \ell }{L}\right)]$ in the $Q=3$ Potts model in the $R$ and $C$ basis ($\theta ,\varphi )=(\frac{\pi }{2},\frac{\pi }{4}$). In the $R$ basis, the data show a good fit for all values of $n$. In the $C$ basis (except at $n=1$), the fitting is reasonable only if we take just the last five or six points. Notice also that, in the large $n$ limit, the linear coefficient of the fitting that give $c_n$, are very different in the two basis.

Figure 6

Coefficient of the logarithmic term of the Rényi MI in the $Q=4$ Potts model in the $R$ and $S$ bases [26]. The coefficients were found by conditioning the fitting to the subsystem sizes $\ell =4,5,\cdots ,\text{Int}[L/2]$. The dashed straight lines are guidelines for $n=1$ and for the central charge $c=1$.

Figure 7

Coefficient of the logarithmic term of the Rényi MI in the Ashkin-Teller model with $\Delta =0$ in the conformal $R$ and $S$ basis and in the $F$ basis specified by the angles $(\theta ,\varphi )=(\frac{\pi }{4},\frac{\pi }{4})$ in (22). The coefficients [26] were found by conditioning the fitting to the subsystem sizes $\ell =4,5,\cdots ,\text{Int}[L/2]$. The dashed straight lines are guidelines for $n=1$ and for the central charge $c=1$.

Figure 8

Coefficient of the logarithmic term of the Rényi MI in the Ashkin-Teller model with $\Delta =\frac{1}{2}$ in the conformal $R$ and $S$ bases and in the $F$ basis specified by the angles $(\theta ,\varphi )=(\frac{\pi }{4},\frac{\pi }{4})$ in (22). The coefficients [26] were found by restricting the fitting to the subsystem sizes $\ell =4,5,\cdots ,\text{Int}[L/2]$. The dashed straight lines are guidelines for $n=1$ and for the central charge $c=1$.

Figure 9

Coefficient of the logarithmic term of the Rényi MI in the XXZ model with $L=28$ sites and with different anisotropy parameter $\Delta$ in the ${\sigma }^z$ basis. The coefficients [26] were estimated by the average of the fittings obtained by restricting the subsystem sizes to $\ell =4,5,...,14$ and to $\ell =5,6,...,14$. The arrows indicate the predicted critical value $n_c$, where the asymptotic behavior begins. The dashed straight lines are guidelines for $n=1$ and for the central charge $c=1$.

Figure 10

Coefficient of the logarithmic term of the Rényi MI in the XXZ with $L=28$ sites and with different anisotropy parameter $\Delta$ in the ${\sigma }^x$ basis. The coefficients [26] were estimated by the average of the fittings obtained by restricting the subsystem sizes to $\ell =4,5,...,14$ and to $\ell =5,6,...,14$. The coefficients were found by restricting the fitting to the subsystem sizes $\ell =4,5,\cdots ,L/2$. The arrows indicate the predicted critical value $n_c$, where the asymptotic behavior begins. The dashed straight lines are guidelines for $n=1$ and for the central charge $c=1$.

Figure 11

Coefficient of the logarithmic term of the Rényi MI in the XXZ model with $\Delta =-\frac{1}{2}$ in the ${\sigma }^z$, ${\sigma }^x$, $D$ and $E$ basis. The nonconformal bases $D$ and $E$ are obtained by setting in (13) $(\theta ,\pi ,\alpha )=(\frac{\pi }{3},\pi ,\frac{\pi }{5})$ and $(\theta ,\pi ,\alpha )=(\frac{\pi }{2.3},\frac{\pi }{4.5},\frac{\pi }{8.2})$, respectively. The coefficients were found by conditioning the fitting to the subsystem sizes $\ell =5,6,\cdots ,L/2$. The dashed straight lines are guidelines for $n=1$ and for the central charge $c=1$.

Figure 12

The Shannon mutual information $I_1(\ell ,L)$ for the $Z_5$, $Z_6$, $Z_7$, and $Z_8$ parafermionic quantum chains with Hamiltonian given in (33). The results were obtained for lattice sizes $L$ and in the basis where $S$ or $R$ is diagonal.