Mutual Information and Boson Radius in a $c=1$ Critical System in One Dimension

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.

Figure 1

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 (4).

Figure 2

Mutual information $I_{A\mathrm{\text{:}}B}$ 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 $\Delta$’s. Filled (empty) symbols correspond to $L=28(24)$.

Figure 3

The deviation of the “Rényi” mutual information $I_{A\mathrm{\text{:}}B}^{(n)}$ (with $n\to 1$ and $n=2$, 3, 4) from the CC result $I_{A\mathrm{\text{:}}B}^{\mathrm{CC}(n)}$ for divisions ($r$, $\frac{L}{2}-r$, $r$, $\frac{L}{2}-r$).

Figure 4

$I_{A\mathrm{\text{:}}B}-I_{A\mathrm{\text{:}}B}^{\mathrm{CC}}$ versus the cross ratio $x$ given in (12). 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 (or gray) symbols correspond to $L=28$ and 24, respectively.

Figure 5

$n=2$ “Rényi” mutual information for the $1:1:1:1$ division versus $\eta =2\pi R^2$. The same symbols as in Fig. 1 are used.