Entanglement Entropy of Non Unitary Conformal Field Theory

TL;DR

This paper investigates the behavior of entanglement entropy in non-unitary conformal field theories, revealing a generalized logarithmic scaling with system size and introducing new algebraic techniques and results for non-unitary models.

Contribution

It provides the first comprehensive analysis of entanglement entropy in non-unitary CFTs, including new formulas, proofs, and numerical evidence, extending known results from unitary models.

Findings

Entanglement entropy scales as (c_eff(n+1)/6n) log(ell) in non-unitary CFTs.

Additional log(log(ell)) term appears for models with logarithmic conformal eigenspaces.

Numerical and analytical evidence supports the theoretical predictions.

Abstract

In this letter we show that the R\'enyi entanglement entropy of a region of large size $\ell$ in a one-dimensional critical model whose ground state breaks conformal invariance (such as in those described by non-unitary conformal field theories), behaves as $S_n \sim \frac{c_{\mathrm{eff}}(n+1)}{6n} \log \ell$, where $c_{\mathrm{eff}}=c-24\Delta>0$ is the effective central charge, $c$ (which may be negative) is the central charge of the conformal field theory and $\Delta\neq 0$ is the lowest holomorphic conformal dimension in the theory. We also obtain results for models with boundaries, and with a large but finite correlation length, and we show that if the lowest conformal eigenspace is logarithmic ($L_0 = \Delta I + N$ with $N$ nilpotent), then there is an additional term proportional to $\log(\log \ell)$. These results generalize the well known expressions for unitary models. We provide a general proof, and report on numerical evidence for a non-unitary spin chain and an analytical computation using the corner transfer matrix method for a non-unitary lattice model. We use a new algebraic technique for studying the branching that arises within the replica approach, and find a new expression for the entanglement entropy in terms of correlation functions of twist fields for non-unitary models.