Entanglement Entropy of Non Unitary Conformal Field Theory

Abstract

In this letter we show that the Rényi 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 $\Delta \ne 0$ 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.

5 pages, 2 figures. Revised version including new derivation of the EE of logarithmic CFT. To appear in J. Phys. A. (fast track communications)

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