Rényi entropy of a line in two-dimensional Ising models

We consider the two-dimensional Ising model on an infinitely long cylinder and study the probabilities $p_i$ to observe a given spin configuration $i$ along a circular section of the cylinder. These probabilities also occur as eigenvalues of reduced density matrices in some Rokhsar-Kivelson wave functions. We analyze the subleading constant to the Rényi entropy $R_n=1/\left(1-n\right)\text{ln}\left({\sum }_i{p}_i^n\right)$ and discuss its scaling properties at the critical point. Studying three different microscopic realizations, we provide numerical evidence that it is universal and behaves in a steplike fashion as a function of $n$ with a discontinuity at the Shannon point $n=1$. As a consequence, a field theoretical argument based on the replica trick would fail to give the correct value at this point. We nevertheless compute it numerically with high precision. Two other values of the Rényi parameter are of special interest: $n=1/2$ and $n=\infty$ are related in a simple way to the Affleck-Ludwig boundary entropies associated to free and fixed boundary conditions, respectively.

Figure 1

(Color online) Cylinder geometry with $L_y⪢L$. A probability $p_i$ is associated to each spin configuration $i$ of the boundary (red circle) between $A$ and $B$.

Figure 2

Shannon entropy $R_1\left(T_c,L\right)$ of the Ising models at the critical point, plotted as a function of $L$ (a linear term, $-0.41L$, has been subtracted for clarity). Data for the square and triangular lattices and for the Ising chain in transverse field are compared. The data are well reproduced by $R_1\left(L\right)\simeq aL+r_1+b/L$ and the subleading constant $r_1$ is evaluated using the three largest sizes. Each line represents the leading term and the constant, $aL+r_1$. The subleading term appears to be the same $r_1\simeq 0.254$ for the three microscopic models.

Figure 3

(Color online) Subleading constant $r_1$ of the Shannon entropy of the Ising chain as a function of $\mu$. The critical point corresponds to $\mu =1$. Inset: $r_1\left(\mu \right)$ for different system sizes, plotted as a function of $\left(\mu -1\right)L$.

Figure 4

(Color online) $2n$ Ising models glued together at their boundary (“Ising book”). In our case, each “page” has periodic boundary conditions along the horizontal axis and is semi-infinite in the vertical direction. Figure 1 corresponds to two pages $\left(n=1\right)$.

Figure 5

(Color online) Subleading constant $r_n$ of the Rényi entropy of the Ising chain (ICTF, at $\mu =1$) and classical Ising models (triangular and square lattice, at $T=T_c$) as a function of the Rényi parameter $n$. The (slow) convergence toward a step function can be observed. Inset: when plotted as a function of $\left(n-1\right)L^{0.25}$, the data collapse reasonably well onto a single curve. For each value of $n$, $r_n$ is obtained by fitting the data for $R_n\left(L\right)$ to $\simeq aL+r_n+b{L}^{-1}+c{L}^{-2}$ using four system size: $L$, $L-2$, $L-4$, and $L-6$, as indicated.

Figure 6

(Color online) Subleading constant $r_n$ of the Rényi entropy of the Ising chain (at the critical point $\mu =1$), in the vicinity of $n=0.5$. Inset: subleading constant multiplied by $L^{0.6}$ as a function of $\left(n-0.5\right)$.