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arXiv:1509.07576 · Journal of Physics A Mathematical and Theoretical

Fused RSOS lattice models as higher-level nonunitary minimal cosets

Elena Tartaglia, Paul A. Pearce

Published Mar 29, 2016• 6 citations
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Abstract

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 +.

DOI
10.1088/1751-8113/49/18/184002

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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 +.
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 .
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 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.
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 .
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.