On Rényi entropies of disjoint intervals in conformal field theory

TL;DR

This paper develops a concrete CFT framework for the Rényi entropies of N disjoint intervals in 1+1D critical systems, focusing on the free compactified boson (c = 1) and the Ising model (c = 1/2).By mapping Tr ρ_A^n to a 2N-point twist-field correlator on a specially constructed Riemann surface, the authors express the result in terms of Riemann theta functions, with the genus fixed at g = (N−1)(n−1) and the full dependence captured by the harmonic ratios x.For the free boson, the Rényi correlator factorizes into quantum and classical parts, F_{N,n}(x) = F^{ ext{qu}}_{N,n} F^{ ext{cl}}_{N,n}(η), enabling decompactification (η → ∞) analyses and direct comparison with lattice harmonic chains; for the Dirac and Ising cases, similar theta-function structures are obtained with appropriate characteristics or sums.On the lattice side, harmonic chains validate the decompactification regime, while MPS computations for Ising confirm the CFT predictions after finite-size scaling and correction analyses, highlighting the role of twist-field correlators as a bridge between continuum and lattice results.

Abstract

We study the Rényi entropies of N disjoint intervals in the conformal field theories given by the free compactified boson and the Ising model. They are computed as the 2N point function of twist fields, by employing the partition function of the model on a particular class of Riemann surfaces. The results are written in terms of Riemann theta functions. The prediction for the free boson in the decompactification regime is checked against exact results for the harmonic chain. For the Ising model, matrix product states computations agree with the conformal field theory result once the finite size corrections have been taken into account.

Figures (34)

• Figure 1: A typical configuration of disjoint intervals in the infinite line. We consider the entanglement between $A=\cup_{i=1}^N A_i$ (in this figure $N=4$) and its complement $B$.
• Figure 2: The path integral representation of $\textrm{Tr}\rho_A^n$ involves a Riemann surface $\mathscr{R}_{N,n}$, which is shown here for $N=3$ and $n=3$.
• Figure 3: The domain $0<x_1<x_2<x_3<1$ of the function $\mathcal{F}_{3,n}(\boldsymbol{x})$. The lines within this domain are the configurations defined in (\ref{['N3 configs def']}).
• Figure 4: The canonical homology basis $\{ a_{\alpha,j}, b_{\alpha,j} \}$ for $N=3$ intervals of equal length and $n=4$ sheets. The sheets are ordered starting from the top. For each cut, the upper part (red) is identified with the lower part (blue) of the corresponding cut on the next sheet in a cyclic way, according to (\ref{['replica boundary conditions']}).
• Figure 5: The Riemann surface $\mathscr{R}_{3,4}$ with the canonical homology basis $\{ a_{\alpha,j}, b_{\alpha,j} \}$, represented also in Fig. \ref{['fig EGmultisheets']}.