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Europhysics Letters (EPL)

Operator Content of the Solvable Lattice Models Associated with the Affine Lie Algebras D L +2 (1)

Je-Young Choi, Kyung-Hoon Kwon, Doochul Kim

Published Dec 15, 1989• 0 citations
Publisher

Abstract

We consider solvable lattice models D L +2 (1) (λ) associated with the Dynkin diagrams of the affine Lie algebras D L +2 (1) and parametrized by the crossing parameter λ (0 < λ < 2π/3). The transfer matrix spectra for the strip widths ⩽ 10 are classified according to the irreducible representations of the symmetry groups under several boundary conditions. All L - and λ-dependences of scaling dimensions are embodied in the thermal scaling dimension X T = 2/{ L 2 (1 - λ/π)}. The operator content of the model is the same as that of the Ashkin-Teller model with the same thermal scaling dimension. This generalizes the results of Pasquier (λ = 0) and of the magnetic hard square model ( L = 3).

DOI
10.1209/0295-5075/10/8/001

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We consider solvable lattice models D L +2 (1) (λ) associated with the Dynkin diagrams of the affine Lie algebras D L +2 (1) and parametrized by the crossing parameter λ (0 < λ < 2π/3). The transfer matrix spectra for the strip widths ⩽ 10 are classified according to the irreducible representations of the symmetry groups under several boundary conditions. All L - and λ-dependences of scaling dimensions are embodied in the thermal scaling dimension X T = 2/{ L 2 (1 - λ/π)}. The operator content of the model is the same as that of the Ashkin-Teller model with the same thermal scaling dimension. This generalizes the results of Pasquier (λ = 0) and of the magnetic hard square model ( L = 3).
Transfer matrix spectra can be used to establish a connection between two dimensional critical lattice models and conformal field theories. Affine–D models are a class of spin models with (L+3) state per site (L=3, 4, ...) generalizing the multicritical line of the magnetic hard square model. There are two continuous parameters characterizing the models; crossing parameter λ with which critical exponents vary continuously and spectral parameter u which controls anisotropy of the interactions. Their transfer matrix spectra combined with finite‐size scaling theory show that the models in the positive u regime belong to the universality class of the Ashkin–Teller model and are related to the c=1 conformal field theory by r={2L2(1−λ/π)}−1/2, where r is the orbifold radius. The models in the negative u regime are found to be related to those in the positive u regime by a certain transformation and also belong to the Ashkin–Teller universality class with the correspondence r={2L2λ/π}−1/2.
An equivalence between generalized restricted solid-on-solid models, associated with sets of graphs, and multi-colour loop models is established. As an application the authors consider solvable loop models and, in this way, obtain new solvable families of critical RSOS models. These families can all be classified by the Dynkin diagrams of the simply laced Lie algebras. For one of the RSOS models, labelled by the Lie algebra pair (A L ,A L ) and related to the C 2 (1) vertex model, they give an off-critical extension, which breaks the Z 2 symmetry of the Dynkin diagrams.
Based on numerical analysis, the authors conjecture the operator content of the finite-size limit of the spectra of the Ashkin-Teller model with free boundary conditions. The same operator content is obtained from a Hamiltonian with a four-fermion interaction and a U(1) Kac-Moody Sugawara structure. For some special values of the coupling constant the model exhibits N=2 superconformal and Zamolodchikov-Fateev invariance. The operator content in these cases is expressed in terms of irreducible representations of the corresponding algebras.
Integrable open quantum spin-chain transfer matrices constructed from trigonometric R-matrices associated to affine Lie algebras g ˆ , and from certain K-matrices (reflection matrices) depending on a discrete parameter p , were recently considered in arXiv:1802.04864 and arXiv:1805.10144 . It was shown there that these transfer matrices have quantum group symmetry corresponding to removing the p th node from the g ˆ Dynkin diagram. Here we determine the spectrum of these transfer matrices by using analytical Bethe ansatz, and we determine the dependence of the corresponding Bethe equations on p . We propose formulas for the Dynkin labels of the Bethe states in terms of the numbers of Bethe roots of each type. We also briefly study how duality transformations are implemented on the Bethe ansatz solutions.
The commuting row-to-row transfer matrices of magnetic hard squares on the multicritical T manifold are shown to satisfy a special functional equation called an inversion identity.For strip widths up to N=32, these equations are solved numerically for the transfer matrix eigenvalues. The central charge or conformal anomaly c=1 is obtained from 1/N 2 corrections to the exact bulk free energy and scaling dimensions are obtained from the eigenvalue gaps. The magnetic scaling dimension is x m =1/8 and the sublattice density difference scaling dimension is x e =1/9(2-y), where y=2 lambda / pi and the interaction-dependent crossing parameter lambda varies between 0 and 2 pi /3. Scaling dimensions for further operators fall in the sequence x n =n 2 x e where n=1,2,3,4. The critical exponents also vary continuously along the multicritical line and are given by alpha =(14-9y)/(16-9y), beta m =9(2-y)/16(16-9y), beta e =1/2(16-9y), nu =9(2-y)/2(16-9y) and delta =15. These exponents are simply related to the exponents of the critical eight-vertex and Ashkin-Teller models, which also exhibit Z 2 *Z 2 symmetry, and the authors argue that these models lie in the same universality class.