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arXiv:1809.07868 · Journal of Statistical Mechanics Theory and Experiment

Fusion hierarchies, T -systems and Y -systems for the A 2 ( 1 ) models

Alexi Morin-Duchesne, Paul A Pearce, Jørgen Rasmussen

Published Jan 1, 2019• 6 citations
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Abstract

Abstract The family of models on the square lattice includes a dilute loop model, a -vertex model and, at roots of unity, a family of RSOS models. The fused transfer matrices of the general loop and vertex models are shown to satisfy -type fusion hierarchies. We use these to derive explicit - and -systems of functional equations. At roots of unity, we further derive closure identities for the functional relations and show that the universal -system closes finitely. The RSOS models are shown to satisfy the same functional and closure identities but with finite truncation.

DOI
10.1088/1742-5468/aaf632

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FUSION HIERARCHY AND FINITE-SIZE CORRECTIONS OF Uq[sl(2)] INVARIANT VERTEX MODELS WITH OPEN BOUNDARIES

The fused six-vertex models with open boundary conditions are studied. The Bethe ansatz solution given by Sklyanin has been generalized to the transfer matrices of the fused models. We have shown that the eigenvalues of transfer matrices satisfy a group of functional relations, which are the su(2) fusion rule held by the transfer matrices of the fused models. The fused transfer matrices form a commuting family and also commute with the quantum group Uq[sl(2)]. In the case of the parameter qh = −1 (h = 4,5, · · ·) the functional relations in the limit of spectral parameter u → i ∞ are truncated. This shows that the su(2) fusion rule with finite level appears for the six vertex model with the open boundary conditions. We have solved the functional relations to obtain the finite-size corrections of the fused transfer matrices for low-lying excitations. From the corrections the central charges and conformal weights of underlying conformal field theory are extracted. To see different boundary conditions we also have studied the six-vertex model with a twisted boundary condition. hep-th/9502053; to appear in NPB. 1

•Jan 1, 1995•0 citations
Abstract The family of models on the square lattice includes a dilute loop model, a -vertex model and, at roots of unity, a family of RSOS models. The fused transfer matrices of the general loop and vertex models are shown to satisfy -type fusion hierarchies. We use these to derive explicit - and -systems of functional equations. At roots of unity, we further derive closure identities for the functional relations and show that the universal -system closes finitely. The RSOS models are shown to satisfy the same functional and closure identities but with finite truncation.
Abstract The fusion hierarchy, T -system and Y -system of functional equations are the key to exact solvability for 2d lattice models. We derive these equations for the generic dilute loop models. The fused transfer matrices are associated with nodes of the infinite dominant integral weight lattice of . For generic values of the crossing parameter , the T - and Y -systems do not truncate. For the case rational so that is a root of unity, we find explicit closure relations and derive closed finite T - and Y -systems. The thermodynamics Bethe ansatz (TBA) integral equations and the associated TBA diagrams of the Y-systems are not of simple Dynkin type. They involve nodes if p is even and nodes if p is odd and are related to the TBA diagrams of models at roots of unity by a folding which originates from the addition of crossing symmetry. In an appropriate regime, the known central charges are . Prototypical examples of the loop models, at roots of unity, include critical dense polymers with central charge c = −2, and loop fugacity and critical site percolation on the triangular lattice with c = 0, and . Solving the TBA equations for the conformal data will determine whether these models lie in the same universality classes as their counterparts. More specifically, it will confirm the extent to which bond and site percolation lie in the same universality class as logarithmic conformal field theories.
Abstract The mutually commuting fused single and double-row transfer matrices of the critical six-vertex model are considered at roots of unity with crossing parameter a rational fraction of . The transfer matrices of the dense loop model analogs, namely the logarithmic minimal models , are similarly considered. For these models, we find explicit closure relations for the T -system functional equations and obtain extended sets of bilinear T -system identities. We also define extended Q matrices as linear combinations of the fused transfer matrices and obtain extended matrix T - Q relations. These results hold for diagonal twisted boundary conditions on the cylinder as well as invariant/Kac vacuum and off-diagonal/Robin vacuum boundary conditions on the strip. Using our extended T -system and extended T - Q relations for eigenvalues, we deduce the usual scalar Baxter T - Q relation and the Bazhanov–Lukyanov–Zamolodchikov decomposition of the fused transfer matrices and , at fusion level , in terms of the product or Q ( u ) 2 . It follows that the zeros of and are comprised of the Bethe roots and complete strings. We also clarify the formal observations of Pronko and Yang–Nepomechie–Zhang and establish, under favourable conditions, the existence of an infinite fusion limit in the auxiliary space of the fused transfer matrices. Despite this connection, the infinite-dimensional oscillator representations are not needed at roots of unity due to finite closure of the functional equations.
Abstract We study the dilute <?CDATA A2(2)A_{2}^{(2)}A2(2)​?> A2(2) loop models on the geometry of a strip of widthN. Two families of boundary conditions are known to satisfy the boundary Yang–Baxter equation. Fixing the boundary condition on the two ends of the strip leads to four models. We construct the fusion hierarchy of commuting transfer matrices for the model as well as itsT- andY-systems, for these four boundary conditions and with a generic crossing parameterλ. For <?CDATA λ/π\lambda/\piλ/π?> λ/π rational and thus <?CDATA q=−e4iλq = -\mathsf{e}^{4\mathrm{i}\lambda}q=−e4iλ?> q=−e4iλ a root of unity, we prove a linear relation satisfied by the fused transfer matrices that closes the fusion hierarchy into a finite system. The fusion relations allow us to compute the two leading terms in the large-Nexpansion of the free energy, namely the bulk and boundary free energies. These are found to be in agreement with numerical data obtained for smallN. The present work complements a previous study (Morin-Duchesne and Pearce 2019J. Stat. Mech.1909) that investigated the dilute <?CDATA A2(2)A_{2}^{(2)}A2(2)​?> A2(2) loop models with periodic boundary conditions.
We study a one-dimensional lattice model of interacting spinless fermions. This model is integrable for both periodic and open boundary conditions; the latter case includes the presence of Grassmann valued non-diagonal boundary fields breaking the bulk U (1) symmetry of the model. Starting from the embedding of this model into a graded Yang–Baxter algebra, an infinite hierarchy of commuting transfer matrices is constructed by means of a fusion procedure. For certain values of the coupling constant related to anisotropies of the underlying vertex model taken at roots of unity, this hierarchy is shown to truncate giving a finite set of functional equations for the spectrum of the transfer matrices. For generic coupling constants, the spectral problem is formulated in terms of a functional (or TQ-)equation which can be solved by Bethe ansatz methods for periodic and diagonal open boundary conditions. Possible approaches for the solution of the model with generic non-diagonal boundary fields are discussed.
A Temperley-Lieb (TL) loop model is a Yang-Baxter integrable lattice model with nonlocal degrees of freedom. On a strip of width N, the evolution operator is the double-row transfer tangle D(u), an element of the TL algebra TL_N(beta) with loop fugacity beta=2cos(lambda). Similarly on a cylinder, the single-row transfer tangle T(u) is an element of the enlarged periodic TL algebra. The logarithmic minimal models LM(p,p') comprise a subfamily of the TL loop models for which the crossing parameter lambda=(p'-p)pi/p' is parameterised by coprime integers 02 takes the form of functional relations for D(u) and T(u) of polynomial degree p'. These derive from fusion hierarchies of commuting transfer tangles D^{m,n}(u) and T^{m,n}(u) where D(u)=D^{1,1}(u) and T(u)=T^{1,1}(u). The fused transfer tangles are constructed from (m,n)-fused face operators involving Wenzl-Jones projectors P_k on k=m or k=n nodes. Some projectors P_k are singular for k>p'-1, but we argue that D^{m,n}(u) and T^{m,n}(u) are well defined for all m,n. For generic lambda, we derive the fusion hierarchies and the associated T- and Y-systems. For the logarithmic theories, the closure of the fusion hierarchies at n=p' translates into functional relations of polynomial degree p' for D^{m,1}(u) and T^{m,1}(u). We also derive the closure of the Y-systems for the logarithmic theories. The T- and Y-systems are the key to exact integrability and we observe that the underlying structure of these functional equations relate to Dynkin diagrams of affine Lie algebras.
The fused six-vertex models with open boundary conditions are studied. The Bethe ansatz solution given by Sklyanin has been generalized to the transfer matrices of the fused models. We have shown that the eigenvalues of transfer matrices satisfy a group of functional relations, which are the su(2) fusion rule held by the transfer matrices of the fused models. The fused transfer matrices form a commuting family and also commute with the quantum group Uq[sl(2)]. In the case of the parameter qh = −1 (h = 4,5, · · ·) the functional relations in the limit of spectral parameter u → i ∞ are truncated. This shows that the su(2) fusion rule with finite level appears for the six vertex model with the open boundary conditions. We have solved the functional relations to obtain the finite-size corrections of the fused transfer matrices for low-lying excitations. From the corrections the central charges and conformal weights of underlying conformal field theory are extracted. To see different boundary conditions we also have studied the six-vertex model with a twisted boundary condition. hep-th/9502053; to appear in NPB. 1