Generalized mutual information of quantum critical chains

We study the generalized mutual information ${\tilde{I}}_n$ of the ground state of different critical quantum chains. The generalized mutual information definition that we use is based on the well established concept of the Rényi divergence. We calculate this quantity numerically for several distinct quantum chains having either discrete $Z\left(Q\right)$ symmetries ($Q$-state Potts model with $Q=2,3,4$ and $Z\left(Q\right)$ parafermionic models with $Q=5,6,7,8$ and also Ashkin-Teller model with different anisotropies) or the $U\left(1\right)$ continuous symmetries (Klein-Gordon field theory, $XXZ$ and spin-1 Fateev-Zamolodchikov quantum chains with different anisotropies). For the spin chains these calculations were done by expressing the ground-state wave functions in two special bases. Our results indicate some general behavior for particular ranges of values of the parameter $n$ that defines ${\tilde{I}}_n$. For a system, with total size $L$ and subsystem sizes $\ell$ and $L-\ell$, the ${\tilde{I}}_n$ has a logarithmic leading behavior given by $\frac{{\tilde{c}}_n}{4}log[\frac{L}{\pi }sin\left(\frac{\pi \ell }{L}\right)]$ where the coefficient ${\tilde{c}}_n$ is linearly dependent on the central charge $c$ of the underlying conformal field theory describing the system's critical properties.

Figure 1

The generalized mutual information ${\tilde{I}}_n(\ell ,L-\ell )$ of the $L=28$ sites periodic Ising quantum chain, as a function of $ln[Lsin\left(\frac{\pi \ell }{L}\right)]/4$. The ground-state wave function is in the basis where the matrices $S_i$ are diagonal ($S$ bases).

Figure 2

The generalized mutual information ${\tilde{I}}_n(\ell ,L-\ell )$ of the $L=28$ sites periodic Ising quantum chain, as a function of $ln[Lsin\left(\frac{\pi \ell }{L}\right)]/4$. The ground-state wave function is in the basis where the matrices $R_i$ are diagonal ($R$ bases).

Figure 3

The ratio ${\tilde{c}}_n/c$ of the coefficient of the logarithm in Eq. (8) and the central charge $c$ for the $Q$-state Potts model with $Q=2,3$, and 4, and for the Ashkin-Teller model (A-T) with different anisotropies $\Delta$. The Ashkin-Teller model at the isotropic point ($\Delta =1$) is equivalent to the four-state Potts model. The ground-state wave functions are in the bases where the $S_i$ matrices are diagonal. The lattice sizes of the models are shown and the coefficients ${\tilde{c}}_n$ were estimated by using the subsystem sizes $\ell =3,5,...,\text{Int}[L/2]$.

Figure 4

Same as Fig. 3, but with the ground-state wave functions of the quantum spin Hamiltonians expressed in the bases where the matrices $R_i$ are diagonal. The lattice sizes of the models are shown in the figure, as well as the subsystems sizes $\ell$ used to estimate ${\tilde{c}}_n$.

Figure 5

The values of the ratios ${\tilde{c}}_n/c$ of Figs. 3 and 4 for the $Q=2,3$ and four-state Potts model are shown in the same figure, for comparison.

Figure 6

The values of ${\tilde{c}}_n/c$ obtained from the data of Figs. 1 and 2 for the Ising quantum chain with $L=28$ sites and eigenfunction expressed in $S$ and $R$ basis.

Figure 7

The generalized mutual information ${\tilde{I}}_n(\ell ,L-\ell )$ of the $L=10$ sites periodic $Z\left(7\right)$-parafermionic quantum chain, as a function of $ln[Lsin\left(\frac{\pi \ell }{L}\right)]/4$. The ground-state wave function is in the basis where the $S_i$ matrices are diagonal ($S$ bases).

Figure 8

The ratio ${\tilde{c}}_n/c$ of the coefficient of the logarithm in Eq. (8) and the central charge $c$ for the $Z\left(Q\right)$-parafermionic models with $Q=5,6,7$, and 8. The ground states are in the bases where the $S_i$ matrices are diagonal. The lattice sizes of the models are shown in the figure and the coefficients ${\tilde{c}}_n$ were estimated by using the subsystem sizes $\ell =3,5,...,\text{Int}[L/2]$.

Figure 9

Same as Fig. 8, but with the ground-state wave function of the quantum spin Hamiltonians expressed in the bases where the matrices $R_i$ are diagonal. The lattice sizes of the models, as well as the subsystems sizes $\ell$ used to estimate ${\tilde{c}}_n$ are shown.

Figure 10

The second term in Eq. (22), ${\tilde{M}}_n(\ell ,L-\ell )$, as a function of $ln[Lsin\left(\frac{\pi \ell }{L}\right)]$ for periodic quantum harmonic chain with $L=120$ sites.

Figure 11

The coefficient of the logarithm ${\tilde{c}}_n$ in Eq. (8). The lattice size $L=120$ and the coefficients ${\tilde{c}}_n$ were estimated by using the subsystem sizes $\ell =3,5,...\text{Int}[L/2]$. The red line is given by Eq. (24).

Figure 12

The generalized mutual information ${\tilde{I}}_n(\ell ,L-\ell )$ of the periodic XXZ quantum chain with anisotropy $\Delta =-1/2$, as a function of $ln[sin\left(\frac{\pi \ell }{L}\right)]/4$. The ground-state wave function is in the basis where the ${\sigma }_i^z$ matrices are diagonal (${\sigma }^z$ basis). The results are for lattice sizes $L=28$ and $L=30$ and give an idea of the finite-size corrections.

Figure 13

The ratio ${\tilde{c}}_n/c$ of the coefficient of the logarithm in Eq. (8) with the central charge $c$ for the $XXZ$ and for the spin-1 Fateev-Zamolodchikov quantum chains (F-Z). The $XXZ$ (Fateev-Zamolodchikov) ground-state wave function are in the ${\sigma }^z$ ($S^z$) basis. The results for the XXZ are for the anisotropies $\Delta =0,-1/2$ and in the case of the Fateev-Zamolodchikov model their are for the couplings $\gamma =\pi /3,\pi /4$. The lattice sizes of the models are shown and the coefficients ${\tilde{c}}_n$ were estimated by using the subsystem sizes $\ell =4,5,...,L/2$.