Entanglement entropy of two disjoint intervals in conformal field theory

TL;DR

This work analyzes the entanglement entropy of two disjoint intervals in the CFT of a Luttinger liquid described by a free compactified boson. It expresses $\mathrm{Tr}\,\rho_A^n$ as a four-point twist-field correlator on an $n$-sheeted Riemann surface and derives a compact representation in terms of Riemann-Siegel theta functions, including distinct quantum and classical contributions. In the decompactification limit, the authors perform an analytic continuation to real $n$ and extract the entanglement entropy and mutual information, finding good agreement with XXZ numerical data in the appropriate regime. They also provide explicit forms for special $n$ (2,3,4), generalize to multiple radii, and discuss small-$x$ behavior, highlighting the open problem of the full analytic continuation to arbitrary $n$ and its implications for broader quantum-quench and multi-interval entanglement studies.

Abstract

We study the entanglement of two disjoint intervals in the conformal field theory of the Luttinger liquid (free compactified boson). Trρ_A^n for any integer n is calculated as the four-point function of a particular type of twist fields and the final result is expressed in a compact form in terms of the Riemann-Siegel theta functions. In the decompactification limit we provide the analytic continuation valid for all model parameters and from this we extract the entanglement entropy. These predictions are checked against existing numerical data.

Figures (6)

• Figure 1:
• Figure 2: Mutual information $I_{A_1:A_2}^{(1)}$ for the XXZ model. All numerical data are extracted from Ref. fps-09. Left: $I_{A_1:A_2}^{(1)}$ for $x=1/2$ (top curve) and $x=1/4$ (bottom curve) as function of $\eta$. The continuous curve is the decompactification result Eq. (\ref{['Idec']}). Right: $I_{A_1:A_2}^{(1)}$ at fixed $\eta$ as function of $x$. The top curve corresponds to $\eta=0.295$ small enough to agree for all considered $x$ with Eq. (\ref{['Idec']}). The other curves correspond to higher values of $\eta$, when the small $\eta$ approximation looses validity.
• Figure 3: $I_n$ in Eq. (\ref{['In']}) as function of $\eta$ for $x$ constant $x=1/2$ (left panel) and $x=1/4$ (right panel). In each plot the three curves corresponds to different values of $n=2,3,4$ (from bottom to top).
• Figure 4: $I_n$ in Eq. (\ref{['In']}) as function of $x$ at constant $\eta=0.3$ (left panel) and $\eta=0.5$ (right panel). In each plot the three curves correspond to different values of $n=2,3,4$ (from bottom to top).
• Figure 5: The two closed loops ${\cal C}_1$ and ${\cal C}_2$ we considered as basis. The different sheets are indicated as solid vs dashed lines.