Abstract Tartaglia and Pearce have argued that the nonunitary fused Forrester–Baxter models are described, in the continuum scaling limit, by the minimal models constructed as the higher-level conformal cosets at integer fusion level and fractional level with . These results rely on Yang–Baxter integrability and are valid in Regime III for models determined by the crossing parameter in the interval . Combinatorially, Baxter’s one-dimensional sums generate the finitized branching functions as weighted walks on the Dynkin diagram. The ground state walks terminate within shaded n -bands, consisting of n contiguous shaded 1-bands. The shaded 1-bands occur between heights where , . These results do not extend to the interval since, for these models, there are no shaded n bands to support the ground states. Here we consider the models in the interval and investigate the associated one-dimensional sums. In this interval, we verify that the one-dimensional sums produce new finitized Virasoro characters of the minimal models with . We further conjecture finitized bosonic forms and check that these agree with the ground state one-dimensional sums out to system sizes N = 12. The models thus realize new Yang–Baxter integrable models in the universality classes of the minimal models . For the series with , the spin-1 one-dimensional sums were previously analysed by Jacob and Mathieu without the underlying Yang–Baxter structure. Finitized Kac characters for the logarithmic minimal models are also obtained for by taking the logarithmic limit with .
Abstract Tartaglia and Pearce have argued that the nonunitary fused Forrester–Baxter models are described, in the continuum scaling limit, by the minimal models constructed as the higher-level conformal cosets at integer fusion level and fractional level with . These results rely on Yang–Baxter integrability and are valid in Regime III for models determined by the crossing parameter in the interval . Combinatorially, Baxter’s one-dimensional sums generate the finitized branching functions as weighted walks on the Dynkin diagram. The ground state walks terminate within shaded n -bands, consisting of n contiguous shaded 1-bands. The shaded 1-bands occur between heights where , . These results do not extend to the interval since, for these models, there are no shaded n bands to support the ground states. Here we consider the models in the interval and investigate the associated one-dimensional sums. In this interval, we verify that the one-dimensional sums produce new finitized Virasoro characters of the minimal models with . We further conjecture finitized bosonic forms and check that these agree with the ground state one-dimensional sums out to system sizes N = 12. The models thus realize new Yang–Baxter integrable models in the universality classes of the minimal models . For the series with , the spin-1 one-dimensional sums were previously analysed by Jacob and Mathieu without the underlying Yang–Baxter structure. Finitized Kac characters for the logarithmic minimal models are also obtained for by taking the logarithmic limit with .
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.
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 unitary and nonunitary ( A , G ) coset minimal models on the cylinder with G = A , D , E . Elementary topological defects are implemented as non-contractible Verlinde, Pasquier and Ocneanu lines around the cylinder. The decomposition of compound defects, formed by fusing elementary defects together, is described by the Verlinde, Pasquier and Ocneanu graph fusion algebras. The action of the compound defects on the left- or right-vacuum boundary state builds the known conformal cylinder boundary conditions. Fusing these defects, in the presence of vacuum boundary conditions, reproduces the known general formulas for the conformal cylinder partition functions. We define the coset graph A ⊗ G / Z 2 , argue that it is a universal object and show that it encodes (i) the coset graph fusion algebra, (ii) the Affleck–Ludwig boundary g -factors, (iii) the defect g -factors given by quantum dimensions and (iv) the relative symmetry resolved Affleck–Ludwig entropies. Additionally, it is shown that the boundary and defect g -factors are related to the asymptotic counting of fusion paths on the coset graph. On the lattice, the defects are constructed as Yang–Baxter integrable seams including special braid and graph automorphism transfer matrices. Remarkably, many of the boundary conformal field theory (CFT) discrete structures, such as fusion matrices, modular matrices, quantum dimensions and defects, appear at the level of lattice Yang–Baxter integrable A - D - E restricted solid-on-solid models and these structures and their properties carry over to the CFT in the continuum scaling limit. Importantly, in the continuum scaling limit, the lattice transfer matrix T - and Y -system functional equations carry over to produce the coset graph fusion algebra for the defect lines. Moreover, the effective central charges and conformal weights are expressed in terms of dilogarithms of the braid and bulk asymptotics of the Y -system expressed in terms of the quantum dimensions.
Abstract The dimer model on a strip is considered as a Yang–Baxter integrable six vertex model at the free-fermion point with crossing parameter and quantum group invariant boundary conditions. A one-to-many mapping of vertex onto dimer configurations allows for the solution of the free-fermion model to be applied to the anisotropic dimer model on a square lattice where the dimers are rotated by compared to their usual orientation. In a suitable gauge, the dimer model is described by the Temperley–Lieb algebra with loop fugacity . It follows that the model is exactly solvable in geometries of arbitrary finite size. We establish and solve transfer matrix inversion identities on the strip with arbitrary finite width N . In the continuum scaling limit, in sectors with magnetization S z , we obtain the conformal weights where . We further show that the corresponding finitized characters decompose into sums of q -Narayana numbers or, equivalently, skew q -binomials. In the particle representation, the local face tile operators give a representation of the fermion algebra and the fermion particle trajectories play the role of nonlocal degrees of freedom. We argue that, in the continuum scaling limit, there exist nontrivial Jordan blocks of rank 2 in the Virasoro dilatation operator L 0 . This confirms that, with quantum group invariant boundary conditions, the dimer model gives rise to a logarithmic conformal field theory with central charge c = −2, minimal conformal weight and effective central charge . Our analysis of the structure of the ensuing rank 2 modules indicates that the familiar staggered c = −2 modules appear as submodules.
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.