We consider the Y -systems satisfied by the A(1)1, A(1)2, A(2)2 vertex and loop models at roots of unity with twisted boundary conditions on the cylinder. The vertex models are the 6-, 15- and Izergin-Korepin 19-vertex models respectively. The corresponding loop models are the dense, fully packed and dilute Temperley-Lieb loop models respectively. For all three models, our focus is on roots of unity values of eiλ with the crossing parameter λ corresponding to the principal and dual series of these models. Converting the known functional equations to nonlinear integral equations in the form of thermodynamic Bethe ansatz equations, we solve the Y -systems for the finite-size 1/N corrections to the groundstate eigenvalue following the methods of Klümper and Pearce. The resulting expressions for c. 24Δ, where c is the central charge and Δ is the conformal weight associated with the groundstate, are simplified using various dilogarithm identities. Our analytic results are in agreement with previous results obtained by different methods.
We consider the Y -systems satisfied by the A(1)1, A(1)2, A(2)2 vertex and loop models at roots of unity with twisted boundary conditions on the cylinder. The vertex models are the 6-, 15- and Izergin-Korepin 19-vertex models respectively. The corresponding loop models are the dense, fully packed and dilute Temperley-Lieb loop models respectively. For all three models, our focus is on roots of unity values of eiλ with the crossing parameter λ corresponding to the principal and dual series of these models. Converting the known functional equations to nonlinear integral equations in the form of thermodynamic Bethe ansatz equations, we solve the Y -systems for the finite-size 1/N corrections to the groundstate eigenvalue following the methods of Klümper and Pearce. The resulting expressions for c. 24Δ, where c is the central charge and Δ is the conformal weight associated with the groundstate, are simplified using various dilogarithm identities. Our analytic results are in agreement with previous results obtained by different methods.
Abstract We consider the Y -systems satisfied by the A 1 ( 1 ) , A 2 ( 1 ) , A 2 ( 2 ) vertex and loop models at roots of unity with twisted boundary conditions on the cylinder. The vertex models are the 6-, 15- and Izergin–Korepin 19-vertex models respectively. The corresponding loop models are the dense, fully packed and dilute Temperley–Lieb loop models respectively. For all three models, our focus is on roots of unity values of e i λ with the crossing parameter λ corresponding to the principal and dual series of these models. Converting the known functional equations to nonlinear integral equations in the form of thermodynamic Bethe ansatz equations, we solve the Y -systems for the finite-size 1 N corrections to the groundstate eigenvalue following the methods of Klümper and Pearce. The resulting expressions for c − 24Δ, where c is the central charge and Δ is the conformal weight associated with the groundstate, are simplified using various dilogarithm identities. Our analytic results are in agreement with previous results obtained by different methods.
Abstract Using the planar Temperley–Lieb algebra, critical bond percolation on the square lattice can be reformulated as a loop model. In this form, it is incorporated as L M ( 2 , 3 ) in the Yang–Baxter integrable family of logarithmic minimal models L M ( p , p ′ ) . We consider this model of percolation in the presence of boundaries and with periodic boundary conditions. Inspired by Kuniba, Sakai and Suzuki, we rewrite the recently obtained infinite Y -system of functional equations. In this way, we obtain nonlinear integral equations in the form of a closed finite set of TBA equations described by a D 3 Dynkin diagram. Following the methods of Klümper and Pearce, we solve the TBA equations for the conformal finite-size corrections. For the ground states of the standard modules on the strip, these agree with the known central charge c = 0 and conformal weights Δ 1 , s for s ∈ Z ⩾ 1 with Δ r , s = ( ( 3 r − 2 s ) 2 − 1 ) / 24 . For the periodic case, the finite-size corrections agree with the conformal weights Δ 0 , s , Δ 1 , s with s
A new and general analytic method for calculating finite-size corrections and central charges is applied to the 6- and 19-vertex models and their related spin-1/2 and spin-1 XXZ chains with twisted boundary conditions. Nonlinear integral equations are derived from which the central charge c can be extracted in terms of Rogers dilogarithms. For twist angle phi , the central charge is c=3S/S+1 (1-4(S-1) phi 2 / pi ( pi -2S gamma )) where gamma is the crossing parameter or chain anisotropy and spin S=1/2 or 1. For periodic boundary conditions ( phi =0) this reduces to the known results c=1 and c=3/2, respectively.
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 We study the dilute <?CDATA 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?> λ/π rational and thus <?CDATA 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)?> A2(2) loop models with periodic boundary conditions.
Abstract Critical site percolation on the triangular lattice is described by the Yang–Baxter solvable dilute A 2 ( 2 ) loop model with crossing parameter specialized to λ = π 3 , corresponding to the contractible loop fugacity β = − 2 cos 4 λ = 1 . We study the functional relations satisfied by the commuting transfer matrices of this model and the associated Bethe ansatz equations. The single and double row transfer matrices are respectively endowed with strip and periodic boundary conditions, and are elements of the ordinary and periodic dilute Temperley–Lieb algebras. The standard modules for these algebras are labelled by the number of defects d and, in the latter case, also by the twist e i γ . Nonlinear integral equation techniques are used to analytically solve the Bethe ansatz functional equations in the scaling limit for the central charge c = 0 and conformal weights Δ , Δ ˉ . For the ground states, we find Δ = Δ 1 , d + 1 for strip boundary conditions and ( Δ , Δ ˉ ) = ( Δ γ / π , d / 2 , Δ γ / π , − d / 2 ) for periodic boundary conditions, where Δ r , s = 1 24 ( ( 3 r − 2 s ) 2 − 1 ) . We give explicit conjectures for the scaling limit of the trace of the transfer matrix in each standard module. For d ⩽ 8 , these conjectures are supported by numerical solutions of the logarithmic form of the Bethe ansatz equations for the leading 20 or more c