Entanglement entropy of two disjoint intervals in conformal field theory II

TL;DR

The paper develops a systematic framework to compute entanglement entropy for two disjoint intervals in 1+1D CFTs, extending to the Ising universality class. It derives a full short-interval (small-x) expansion of Trρ_A^n, expresses the Ising scaling function via Riemann-Siegel theta functions, and provides analytic continuation to the Von Neumann case. Key results include explicit Ising formulas for F_n(x), invariance under x→1−x, and detailed leading and subleading contributions from stress-energy, two- and multi-point functions. The work also connects to the compactified boson, verifies consistency with numerical data, and establishes analytic continuation techniques (Carlson’s theorem) to obtain the VN entropy from replica indices.

Abstract

We continue the study of the entanglement entropy of two disjoint intervals in conformal field theories that we started in J. Stat. Mech. (2009) P11001. We compute Trρ_A^n for any integer n for the Ising universality class and the final result is expressed as a sum of Riemann-Siegel theta functions. These predictions are checked against existing numerical data. We provide a systematic method that gives the full asymptotic expansion of the scaling function for small four-point ratio (i.e. short intervals). These formulas are compared with the direct expansion of the full results for free compactified boson and Ising model. We finally provide the analytic continuation of the first term in this expansion in a completely analytic form.

Figures (4)

• Figure 1: ${\cal F}_{3,4}(x)$: The extrapolated numerical data of Ref. fc-10 are compared with our prediction. Only $x\leq 0.5$ are reported because of the symmetry $x\to1-x$.
• Figure 2: This figure shows the importance of the Carlson's theorem to choose the correct analytic continuation. For two values of $\alpha$ ($0.1$ and $0.4$), we report $s_n/(n-1)$ as function of $n$, as obtained by summing with the function $f(k)$ and $g(k)$ (i.e. using Eq. (\ref{['s2wrong']}) or Eq. (\ref{['s2final']}) respectively). It is evident that when using $f(k)$ an oscillatory behavior (with period $1$) is superimposed to a smooth curve, while for $g(k)$ the resulting $s_2(n)$ is smooth. For integer numbers $\geq2$ they have the same value, as they should. The limits for $n\to1$ are very different and the one obtained from $g(k)$ is the correct one.
• Figure 3: The cut plane represented on the left can be mapped through a conformal transformation to a cylinder of length $L$ and circumference $W$ (right) and so modular parameter $q=e^{-2\pi L/W}$. The upper and lower edges of the cuts are mapped into semicircular arcs $(S_{1L},S_{1R})$ and $(S_{2L},S_{2R})$ at each end of the open cylinder.
• Figure 4: The equivalent of the $n$-sheeted Riemann surface (with $n=4$) for the cylinder geometry. $O_j$ represents the operator "propagating" in the cylinder $j$.