Mutual Information and Boson Radius in c=1 Critical Systems in One Dimension

TL;DR

The present study provides a new way to determine the boson radius, and furthermore demonstrates the power of the mutual information to extract more refined information of conformal field theory than the central charge.

Abstract

We study the generic scaling properties of the mutual information between two disjoint intervals, in a class of one-dimensional quantum critical systems described by the c=1 bosonic field theory. A numerical analysis of a spin-chain model reveals that the mutual information is scale-invariant and depends directly on the boson radius. We interpret the results in terms of correlation functions of branch-point twist fields. The present study provides a new way to determine the boson radius, and furthermore demonstrates the power of the mutual information to extract more refined information of conformal field theory than the central charge.

Figures (5)

• Figure 1: (color online) The mutual information for fixed divisions $r_A$:$r_C$:$r_B$:$r_D$=1:1:1:1 and 1:2:1:2, versus $\eta=2\pi R^2$. We set the magnetization at $M=\frac{k}{L}$ with $k=0,1,\dots,\frac{L}{2}-3$ for $-1<\Delta\le 1$ and with $k=1,\dots,\frac{L}{2}-3$ for $1<\Delta$, so that the system is inside the critical phase. Black and green points correspond to the larger ($L=28,30$) and smaller ($L=24$) systems, respectively. Horizontal red lines indicate the Calabrese-Cardy result \ref{['eq:MI_CC']}.
• Figure 2: (color online) Mutual information $I_{A:B}$ as a function of $\frac{r}{L}$ for divisions $(r_A,r_C,r_B,r_D)=(r,\frac{L}{2}-r,r,\frac{L}{2}-r)$. We set $h=0$, and symbols with different shapes correspond to different $\Delta=-0.8,-0.6,0,1$. Filled and empty symbols correspond to $L=28$ and $24$, respectively.
• Figure 3: (color online) The deviation of the "Rényi" mutual information $I_{A:B}^{(n)}$ from the CC result $I_{A:B}^{\mathrm{CC}(n)}$ for divisions $(r,\frac{L}{2}-r,r,\frac{L}{2}-r)$. Different symbols correspond to $n=1,2,3,4$.
• Figure 4: (color online) $I_{A:B}-I_{A:B}^{\mathrm{CC}}$ versus the cross ratio $x$ given in \ref{['eq:cr_L']}. All the divisions $(r_A,r_C,r_B,r_D)$ with $3\le r_A \le r_B$ and $3\le r_C \le r_D$ are examined. For $L=28~(24)$, there are totally $305~(152)$ possibilities of such divisions. Black and Green symbols correspond to $L=28$ and $24$, respectively.
• Figure 5: (color online) $n=2$ "Rényi" mutual information for the 1:1:1:1 division versus $\eta=2\pi R^2$. The same symbols as in Fig. \ref{['fig:MI_R']} are used.