In this letter we show that the Rényi entanglement entropy of a region of large size 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 ∼ c eff (n+1) 6n log , where c eff = c − 24∆ > 0 is the effective central charge, c (which may be negative) is the central charge of the conformal field theory and ∆ = 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 = ∆I +N with N nilpotent), then there is an additional term proportional to log(log ). 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.

In this paper we study the simplest massive 1 + 1 dimensional integrable quantum field theory which can be described as a perturbation of a non-unitary minimal conformal field theory: the Lee-Yang model. We are particularly interested in the features of the bi-partite entanglement entropy for this model and on building blocks thereof, namely twist field form factors. Non-unitarity selects out a new type of twist field as the operator whose two-point function (appropriately normalized) yields the entanglement entropy. We compute this two-point function both from a form factor expansion and by means of perturbed conformal field theory. We find good agreement with CFT predictions put forward in a recent work involving the present authors. In particular, our results are consistent with a scaling of the entanglement entropy given by c eff 3 log where c eff is the effective central charge of the theory (a positive number related to the central charge) and is the size of the region. Furthermore the form factor expansion of twist fields allows us to explore the large region limit of the entanglement entropy and find the next-to-leading order correction to saturation. We find that this correction is very different from its counterpart in unitary models. Whereas in the latter case, it had a form depending only on few parameters of the model (the particle spectrum), it appears to be much more model-dependent for non-unitary models.

Well-known measures of entanglement in one-dimensional many body quantum systems, such as the entanglement entropy and the logarithmic negativity, may be expressed in terms of the correlation functions of local fields known as branch point twist fields in a replica quantum field theory. In this "replica" approach the computation of measures of entanglement generally involves a mathematically non-trivial analytic continuation in the number of replicas. In this paper we consider two-point functions of twist fields and their analytic continuation in the 1+1 dimensional massive (non-compactified) free Boson theory. This is one of the few theories for which all matrix elements of twist fields are known so that we may hope to compute correlation functions very precisely. We study two particular two-point functions which are related to the logarithmic negativity of semi-infinite disjoint intervals and to the entanglement entropy of one interval. We show that our prescription for the analytic continuation yields results which are in full agreement with conformal field theory predictions in the short-distance limit. We provide numerical estimates of universal quantities and their ratios, both in the massless (twist field structure constants) and the massive (expectation values of twist fields) theory. We find that particular ratios are given by divergent form factor expansions. We propose such divergences stem from the presence of logarithmic factors in addition to the expected power-law behaviour of two-point functions at short-distances. Surprisingly, at criticality these corrections give rise to a log(log ) correction to the entanglement entropy of one interval of length . This hitherto overlooked result is in agreement with results by Calabrese, Cardy and Tonni and has been independently derived by Blondeau-Fournier and Doyon (in preparation).

We consider the ϕ 1,3 off-critical perturbation M(m, m ′ ; t) of the general non-unitary minimal models where 2 ≤ m ≤ m ′ and m, m ′ are coprime and t measures the departure from criticality corresponding to the ϕ 1,3 integrable perturbation. We view these models as the continuum scaling limit in the ferromagnetic Regime III of the Forrester-Baxter Restricted Solid-On-Solid (RSOS) models on the square lattice. We also consider the RSOS models in the antiferromagnetic Regime II related in the continuum scaling limit to Z n parfermions with n = m ′ − 2. Using an elliptic Yang-Baxter algebra of planar tiles encoding the allowed face configurations, we obtain the Hamiltonians of the associated quantum chains defined as the logarithmic derivative of the transfer matrices with periodic boundary conditions. The transfer matrices and Hamiltonians act on a vector space of paths on the A m ′ −1 Dynkin diagram whose dimension is counted by generalized Fibonacci numbers.1

Using a Corner Transfer Matrix approach, we compute the bipartite entanglement Rényi entropy in the off-critical perturbations of non-unitary conformal minimal models realised by lattice spin chains Hamiltonians related to the Forrester Baxter RSOS models [30] in regime III. This allows to show on a set of explicit examples that the Rényi entropies for non-unitary theories rescale near criticality as the logarithm of the correlation length with a coefficient proportional to the effective central charge. This complements a similar result, recently established for the size rescaling at the critical point [23], showing the expected agreement of the two behaviours. We also compute the first subleading unusual correction to the scaling behaviour, showing that it is expressible in terms of expansions of various fractional powers of the correlation length, related to the differences ∆ − ∆ min between the conformal dimensions of fields in the theory and the minimal conformal dimension. Finally, a few observations on the limit leading to the off-critical logarithmic minimal models of Pearce and Seaton [49] are put forward.