Entanglement entropy of non-unitary conformal field theory

Here we show that the Rényi entanglement entropy of a region of large size ℓ 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}∼ \frac{{{c}_eff}(n+1)}{6n}log \ell , where {{c}_eff}=c-24Δ \gt 0 is the effective central charge, c (which may be negative) is the central charge of the conformal field theory and Δ \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}=Δ 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.