We consider the φ 1,3 off-critical perturbation 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 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.
We consider the φ 1,3 off-critical perturbation 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 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.
We consider the integrable minimal models ℳ( m , m ′; t ), corresponding to the φ 1,3 perturbation off-criticality, in the logarithmic limit m , m ′ → ∞, m / m ′ → p / p ′ where p , p ′ are coprime and the limit is taken through coprime values of m , m ′. We view these off-critical minimal models ℳ( m , m ′; t ) as the continuum scaling limit of the Forrester–Baxter restricted solid-on-solid (RSOS) models on the square lattice. Applying corner transfer matrices to the Forrester–Baxter RSOS models in regime III, we argue that taking first the thermodynamic limit and second the logarithmic limit yields off-critical logarithmic minimal models ℒℳ( p , p ′; t ) corresponding to the φ 1,3 perturbation of the critical logarithmic minimal models ℒℳ( p , p ′). Specifically, in accord with the Kyoto correspondence principle, we show that the logarithmic limit of the one-dimensional configurational sums yields finitized quasi-rational characters of the Kac representations of the critical logarithmic minimal models ℒℳ( p , p ′). We also calculate the logarithmic limit of certain off-critical observables 𝒪 r , s related to one-point functions and show that the associated critical exponents produce all conformal dimensions in the infinitely extended Kac table. The corresponding Kac labels ( r , s ) satisfy ( p s − p ′ r ) 2 < 8 p ( p ′ − p ). The exponent 2 − α = p ′/2( p ′ − p ) is obtained from the logarithmic limit of the free energy giving the conformal dimension for the perturbing field t . As befits a non-unitary theory, some observables 𝒪 r , s diverge at criticality.
Abstract We consider the Forrester–Baxter RSOS lattice models with crossing parameter λ = ( m ′− m ) π / m ′ in Regime III. In the continuum scaling limit, these models are described by the minimal models . We conjecture that, for λ < π / n , the n × n fused RSOS models with are described by the higher-level coset at fractional level k = nM /( M ′− M )−2 with . To support this conjecture, we investigate the one-dimensional sums arising from Baxter’s off-critical corner transfer matrices. In unitary cases ( m = m ′−1) it is known that, up to leading powers of q , these coincide with the branching functions . For general nonunitary cases ( m < m ′−1), we identify the ground state one-dimensional RSOS paths and relate them to the quantum numbers ( r , s , ℓ ) in the various sectors. For n = 1, 2, 3, we obtain the local energy functions H ( a , b , c ) in a suitable gauge and verify that the associated one-dimensional sums produce finitized forms that converge, as N becomes large, to the fractional level branching functions . Extending the work of Schilling, we also conjecture finitized bosonic branching functions for general n and check that these agree with the one-dimensional sums for n = 1, 2, 3 out to system sizes N = 14. Lastly, the finitized Kac characters of the n × n fused logarithmic minimal models are obtained by taking the logarithmic limit with m ′/ m → p ′/ p +.
We consider the critical nonunitary minimal model M (3, 5) with integrable boundaries and analyze the patterns of zeros of the eigenvalues of the transfer matrix and then determine the spectrum of the critical theory using the thermodynamic Bethe ansatz ( TBA ) equations. Solving the TBA functional equation satisfied by the transfer matrices of the associated A 4 restricted solid-on-solid Forrester–Baxter lattice model in regime III in the continuum scaling limit, we derive the integral TBA equations for all excitations in the ( r , s ) = (1, 1) sector and then determine their corresponding energies. We classify the excitations in terms of ( m , n ) systems .
In the present paper we study scaling field theories related to the class of exactly solvable noncritical models of statistical mechanics which are known as RSOS (restricted solid-on-solid) or IRF (interaction round a face) models.These models are related to simple Lie algebras and they generalize the integrable version of RSOS model proposed by Andrews, Baxter and Forrester. 1 > The models related to gl(n) were found by Jimbo, Miwa and Okado, 2 > the models related to other classical Lie algebras were obtained by the same authors in Ref. 3) and by Pasquier in Ref. 4) for Dn-case with a special coupling constant.Critical properties of JMO model were found in Ref. 5) where the authors obtain the explicit formulas for magnetization in the model and compute its critical behavior.It was established there that the critical behavior of these models was described by coset conformal field theories. 6>. 7 > Therefore it is natural to expect that corresponding scaling field theories will be integrable perturbations of these conformal field theories.s>In this paper we describe the thermodynamics of the noncritical magnetic chain.Then we find the ground state, excitations over the ground state and the scattering amplitudes of excitations.By getting the scaling limit over the ground state of the model we compute the central charges of corresponding field theories, the dimensions of perturbations, the spectrum and scattering amplitudes of excitations.Under certain assumptions we generalize these results for any Lie algebra of A-D-E type.In certain simplest cases this gives known S-matrices like Es, E1, Es scalar S-matrices. 9> In certain other models matrix elements of the S-matrix are given by Pasquier's solutions 4 > of the Yang-Baxter equation.We do not discuss here the relation of RSOS models to integrable models of vertex type. 23>' 25 > Partially the relation was explained in Refs.4) and 26).The relation between the spectrum of row-to-row transfer matrices in RSOS and in vertex models will be given in a separate publication.For some of the scaling models the same answer was obtained by Bernard and
In the present paper we study scaling field theories related to the class of exactly solvable noncritical models of statistical mechanics which are known as RSOS (restricted solid-on-solid) or IRF (interaction round a face) models. These models are related to simple Lie algebras and they generalize the integrable version of RSOS model proposed by Andrews, Baxter and Forrester. 1> The models related to gl(n) were found by Jimbo, Miwa and Okado,2> the models related to other classical Lie algebras were obtained by the same authors in Ref. 3) and by Pasquier in Ref. 4) for Dn-case with a special coupling constant. Critical properties of JMO model were found in Ref. 5) where the authors obtain the explicit formulas for magnetization in the model and compute its critical behavior. It was established there that the critical behavior of these models was described by coset conformal field theories. 6>.7> Therefore it is natural to expect that corre sponding scaling field theories will be integrable perturbations of these conformal field theories. s> In this paper we describe the thermodynamics of the noncritical magnetic chain. Then we find the ground state, excitations over the ground state and the scattering amplitudes of excitations. By getting the scaling limit over the ground state of the model we compute the central charges of corresponding field theories, the dimensions of perturbations, the spectrum and scattering amplitudes of excitations. Under certain assumptions we generalize these results for any Lie algebra of A-D-E type. In certain simplest cases this gives known S-matrices like Es, E1, Es scalar S-matrices.9> In certain other models matrix elements of the S-matrix are given by Pasquier's solutions 4> of the Yang-Baxter equation. We do not discuss here the relation of RSOS models to integrable models of vertex type. 23>' 25> Partially the relation was explained in Refs. 4) and 26). The relation between the spectrum of row-to-row transfer matrices in RSOS and in vertex models will be given in a separate publication. For some of the scaling models the same answer was obtained by Bernard and
Abstract 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 (Bianchini et al 2015 J. Stat. Mech. P03010 ) 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 (Bianchini et al 2015 J. Phys. A: Math. Theor. 48 04FT01 ), 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 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 (2012 J. Stat. Mech. P09014 ) are put forward.