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arXiv:1812.01471 · Journal of Physics A Mathematical and Theoretical

Extended T -systems, Q matrices and T - Q relations for s ℓ ( 2 ) models at roots of unity

Holger Frahm, Alexi Morin-Duchesne, Paul A Pearce

Published May 24, 2019• 9 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8121/ab2490

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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 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 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 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.
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.
For general Temperley–Lieb loop models, including the logarithmic minimal models LM ( p , p ′ ) with p , p ′ coprime integers, we construct an infinite family of Robin boundary conditions on the strip as linear combinations of Neumann and Dirichlet boundary conditions. These boundary conditions are Yang–Baxter integrable and allow loop segments to terminate on the boundary. Algebraically, the Robin boundary conditions are described by the one-boundary Temperley–Lieb algebra. Solvable critical dense polymers is the first member LM ( 1 , 2 ) of the family of logarithmic minimal models and has loop fugacity β = 0 and central charge c = − 2 . Specialising to LM ( 1 , 2 ) with our Robin boundary conditions, we solve the model exactly on strips of arbitrary finite size N and extract the finite-size conformal corrections using an Euler–Maclaurin formula. The key to the solution is an inversion identity satisfied by the commuting double row transfer matrices. This inversion identity is established directly in the Temperley–Lieb algebra. We classify the eigenvalues of the double row transfer matrices using the physical combinatorics of the patterns of zeros in the complex spectral parameter plane and obtain finitised characters related to spaces of coinvariants of Z 4 fermions. In the continuum scaling limit, the Robin boundary conditions are associated with irreducible Virasoro Verma modules with conformal weights Δ r , s − 1 2 = 1 32 ( L 2 − 4 ) where L = 2 s − 1 − 4 r , r ∈ Z , s ∈ N . These conformal weights populate a Kac table with half-integer Kac labels. Fusion of the corresponding modules with the generators of the Kac fusion algebra is examined and general fusion rules are proposed.
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.
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