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.
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.
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.
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 +.
Working in the dense loop representation, we use the planar Temperley–Lieb algebra to build integrable lattice models called logarithmic minimal models . Specifically, we construct Yang–Baxter integrable Temperley–Lieb models on the strip acting on link states and consider their associated Hamiltonian limits. These models and their associated representations of the Temperley–Lieb algebra are inherently non-local and not (time-reversal) symmetric. We argue that, in the continuum scaling limit, they yield logarithmic conformal field theories with central charges c = 1−(6( p − p ') 2 / pp '), where p , p ' = 1, 2, ... are coprime. The first few members of the principal series are critical dense polymers ( m = 1, c = −2), critical percolation ( m = 2, c = 0) and the logarithmic Ising model ( m = 3, c = 1/2). For the principal series, we find an infinite family of integrable and conformal boundary conditions organized in an extended Kac table with conformal weights Δ r , s = ((( m +1) r − ms ) 2 −1)/4 m ( m +1), r , s = 1, 2, .... The associated conformal partition functions are given in terms of Virasoro characters of highest-weight representations. Individually, these characters decompose into a finite number of characters of irreducible representations. We show with examples how indecomposable representations arise from fusion.
Solvable critical dense polymers is a Yang-Baxter integrable model of polymers on the square lattice. It is the first member LM(1,2) of the family of logarithmic minimal models LM(p,p'). The associated logarithmic conformal field theory admits an infinite family of Kac representations labelled by the Kac labels r,s=1,2,.... In this paper, we explicitly construct the conjugate boundary conditions on the strip. The boundary operators are labelled by the Kac fusion labels (r,s)=(r,1) x (1,s) and involve a boundary field xi. Tuning the field xi appropriately, we solve exactly for the transfer matrix eigenvalues on arbitrary finite-width strips and obtain the conformal spectra using the Euler-Maclaurin formula. The key to the solution is an inversion identity satisfied by the commuting double-row transfer matrices. The transfer matrix eigenvalues are classified by the physical combinatorics of the patterns of zeros in the complex spectral-parameter plane. This yields selection rules for the physically relevant solutions to the inversion identity which takes the form of a decomposition into irreducible blocks corresponding combinatorially to finitized characters given by generalized q-Catalan polynomials. This decomposition is in accord with the decomposition of the Kac characters into irreducible characters. In the scaling limit, we confirm the central charge c=-2 and the Kac formula for the conformal weights Delta_{r,s}=((2r-s)^2-1)/8 for r,s=1,2,3,... in the infinitely extended Kac table.
Solvable critical dense polymers is a Yang-Baxter integrable model of polymers on the square lattice. It is the first member LM(1,2) of the family of logarithmic minimal models LM(p,p'). The associated logarithmic conformal field theory admits an infinite family of Kac representations labelled by the Kac labels r,s=1,2,.... In this paper, we explicitly construct the conjugate boundary conditions on the strip. The boundary operators are labelled by the Kac fusion labels (r,s)=(r,1) x (1,s) and involve a boundary field xi. Tuning the field xi appropriately, we solve exactly for the transfer matrix eigenvalues on arbitrary finite-width strips and obtain the conformal spectra using the Euler-Maclaurin formula. The key to the solution is an inversion identity satisfied by the commuting double-row transfer matrices. The transfer matrix eigenvalues are classified by the physical combinatorics of the patterns of zeros in the complex spectral-parameter plane. This yields selection rules for the physically relevant solutions to the inversion identity which takes the form of a decomposition into irreducible blocks corresponding combinatorially to finitized characters given by generalized q-Catalan polynomials. This decomposition is in accord with the decomposition of the Kac characters into irreducible characters. In the scaling limit, we confirm the central charge c=-2 and the Kac formula for the conformal weights Delta_{r,s}=((2r-s)^2-1)/8 for r,s=1,2,3,... in the infinitely extended Kac table.
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.