Fusion hierarchies, TT-systems and YY-systems for the dilute A2(2)A_2^{(2)} loop models

The fusion hierarchy, $T$-system and $Y$-system of functional equations are the key to integrability for 2d lattice models. We derive these equations for the generic dilute $A_2^{(2)}$ loop models. The fused transfer matrices are associated with nodes of the infinite dominant integral weight lattice of $s\ell(3)$. For generic values of the crossing parameter $\lambda$, the $T$- and $Y$-systems do not truncate. For the case $\frac{\lambda}{\pi}=\frac{(2p'-p)}{4p'}$ rational so that $x=\mathrm{e}^{\mathrm{i}\lambda}$ is a root of unity, we find explicit closure relations and derive closed finite $T$- and $Y$-systems. The TBA diagrams of the $Y$-systems and associated Thermodynamic Bethe Ansatz (TBA) integral equations are not of simple Dynkin type. They involve $p'+2$ nodes if $p$ is even and $2p'+2$ nodes if $p$ is odd and are related to the TBA diagrams of $A_2^{(1)}$ models at roots of unity by a ${\Bbb Z}_2$ folding which originates from the addition of crossing symmetry. In an appropriate regime, the known central charges are $c=1-\frac{6(p-p')^2}{pp'}$. Prototypical examples of the $A_2^{(2)}$ loop models, at roots of unity, include critical dense polymers ${\cal DLM}(1,2)$ with central charge $c=-2$, $\lambda=\frac{3\pi}{8}$ and loop fugacity $\beta=0$ and critical site percolation on the triangular lattice ${\cal DLM}(2,3)$ with $c=0$, $\lambda=\frac{\pi}{3}$ and $\beta=1$. Solving the TBA equations for the conformal data will determine whether these models lie in the same universality classes as their $A_1^{(1)}$ 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.Comment: 34 page

Fusion hierarchies, TT-systems and YY-systems for the A2(1)A_2^{(1)} models

The family of $A^{(1)}_2$ models on the square lattice includes a dilute loop model, a $15$-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 $s\ell(3)$-type fusion hierarchies. We use these to derive explicit $T$- and $Y$-systems of functional equations. At roots of unity, we further derive closure identities for the functional relations and show that the universal $Y$-system closes finitely. The $A^{(1)}_2$ RSOS models are shown to satisfy the same functional and closure identities but with finite truncation.Comment: 36 page

Fusion hierarchies, T-systems and Y-systems of logarithmic minimal models

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 0<p<p'. For these special values, additional symmetries allow for particular degeneracies in the spectra that account for the logarithmic nature of these theories. For critical dense polymers LM(1,2), D(u) and T(u) are known to satisfy inversion identities that allow us to obtain exact eigenvalues in any representation and for all system sizes N. The generalisation for p'>2 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.Comment: 77 page

A homomorphism between link and XXZ modules over the periodic Temperley-Lieb algebra

We study finite loop models on a lattice wrapped around a cylinder. A section of the cylinder has N sites. We use a family of link modules over the periodic Temperley-Lieb algebra EPTL_N(\beta, \alpha) introduced by Martin and Saleur, and Graham and Lehrer. These are labeled by the numbers of sites N and of defects d, and extend the standard modules of the original Temperley-Lieb algebra. Beside the defining parameters \beta=u^2+u^{-2} with u=e^{i\lambda/2} (weight of contractible loops) and \alpha (weight of non-contractible loops), this family also depends on a twist parameter v that keeps track of how the defects wind around the cylinder. The transfer matrix T_N(\lambda, \nu) depends on the anisotropy \nu and the spectral parameter \lambda that fixes the model. (The thermodynamic limit of T_N is believed to describe a conformal field theory of central charge c=1-6\lambda^2/(\pi(\lambda-\pi)).) The family of periodic XXZ Hamiltonians is extended to depend on this new parameter v and the relationship between this family and the loop models is established. The Gram determinant for the natural bilinear form on these link modules is shown to factorize in terms of an intertwiner i_N^d between these link representations and the eigenspaces of S^z of the XXZ models. This map is shown to be an isomorphism for generic values of u and v and the critical curves in the plane of these parameters for which i_N^d fails to be an isomorphism are given.Comment: Replacement of "The Gram matrix as a connection between periodic loop models and XXZ Hamiltonians", 31 page

Fusion in the periodic Temperley-Lieb algebra and connectivity operators of loop models

In two-dimensional loop models, the scaling properties of critical random curves are encoded in the correlators of connectivity operators. In the dense O($n$) loop model, any such operator is naturally associated to a standard module of the periodic Temperley-Lieb algebra. We introduce a new family of representations of this algebra, with connectivity states that have two marked points, and argue that they define the fusion of two standard modules. We obtain their decomposition on the standard modules for generic values of the parameters, which in turn yields the structure of the operator product expansion of connectivity operators

Validity of the Adiabatic Approximation

We analyze the validity of the adiabatic approximation, and in particular the reliability of what has been called the "standard criterion" for validity of this approximation. Recently, this criterion has been found to be insufficient. We will argue that the criterion is sufficient only when it agrees with the intuitive notion of slowness of evolution of the Hamiltonian. However, it can be insufficient in cases where the Hamiltonian varies rapidly but only by a small amount. We also emphasize the distinction between the adiabatic {\em theorem} and the adiabatic {\em approximation}, two quite different although closely related ideas.Comment: 4 pages, 1 figur

Groundstate finite-size corrections and dilogarithm identities for the twisted A1(1)A_1^{(1)}, A2(1)A_2^{(1)} and A2(2)A_2^{(2)} models

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 $e^{i\lambda}$ with the crossing parameter $\lambda$ 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 (TBA) equations, we solve the $Y$-systems for the finite-size $\frac 1N$ corrections to the groundstate eigenvalue following the methods of Kl\"umper and Pearce. The resulting expressions for $c-24\Delta$, where $c$ is the central charge and $\Delta$ 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 and are new for the dual series of the $A_2^{(1)}$ model

Critical site percolation on the triangular lattice: From integrability to conformal partition functions

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 $\lambda=\frac\pi3$, corresponding to the contractible loop fugacity $\beta=-2\cos4\lambda=1$. 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 labeled by the number of defects $d$ and, in the latter case, also by the twist $e^{i\gamma}$. 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 $\Delta,\bar\Delta$. For the groundstates, we find $\Delta=\Delta_{1,d+1}$ for strip boundary conditions and $(\Delta,\bar\Delta)=(\Delta_{\gamma/\pi,d/2},\Delta_{\gamma/\pi,-d/2})$ for periodic boundary conditions, where $\Delta_{r,s}=\frac1{24}((3r-2s)^2-1)$. We give explicit conjectures for the scaling limit of the trace of the transfer matrix in each standard module. For $d\le8$, these conjectures are supported by numerical solutions of the logarithmic form of the Bethe ansatz equations for the leading $20$ or more conformal eigenenergies. With these conjectures, we apply the Markov traces to obtain the conformal partition functions on the cylinder and torus. These precisely coincide with our previous results for critical bond percolation on the square lattice described by the dense $A_1^{(1)}$ loop model with $\lambda=\frac\pi3$. The concurrence of all this conformal data provides compelling evidence supporting a strong form of universality between these two stochastic models as logarithmic CFTs.Comment: 81 page

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

The mutually commuting $1\times n$ fused single and double-row transfer matrices of the critical six-vertex model are considered at roots of unity $q=e^{i\lambda}$ with crossing parameter $\lambda=\frac{(p'-p)\pi}{p'}$ a rational fraction of $\pi$. The $1\times n$ transfer matrices of the dense loop model analogs, namely the logarithmic minimal models ${\cal LM}(p,p')$, are similarly considered. For these $s\ell(2)$ 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 $U_q(s\ell(2))$ 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-Mangazeev decomposition of the fused transfer matrices $T^{n}(u+\lambda)$ and $D^{n}(u+\lambda)$, at fusion level $n=p'-1$, in terms of the product $Q^+(u)Q^-(u)$ or $Q(u)^2$. It follows that the zeros of $T^{p'-1}(u+\lambda)$ and $D^{p'-1}(u+\lambda)$ are comprised of the Bethe roots and complete $p'$ 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 $n\to\infty$ 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.Comment: 38 page