Extended T-systems, Q matrices and T-Q relations for $ \newcommand{\e}{{\rm e}} s\ell(2)$ models at roots of unity

Abstract

The mutually commuting 1 × n fused single and double-row transfer matrices of the critical six-vertex model are considered at roots of unity q = e iλ with crossing parameter λ = (p ′ −p)π p ′ a rational fraction of π. The 1 × n transfer matrices of the dense loop model analogs, namely the logarithmic minimal models LM(p, p ′ ), are similarly considered. For these sℓ(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 (sℓ(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-Lukyanov-Zamolodchikov decomposition of the fused transfer matrices T n (u + λ) and D n (u + λ), at fusion level n = p ′ − 1, in terms of the productIt follows that the zeros of T p ′ −1 (u + λ) and D p ′ −1 (u + λ) 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 → ∞ in the auxiliary space of the fused transfer matrices. Despite this connection, the infinitedimensional oscillator representations are not needed at roots of unity due to finite closure of the functional equations.

On the Q operator and the spectrum of the XXZ model at root of unity

“…From the factorised expression (6.10) we see that the zeroes of the eigenvalues ofT s (u, φ) are either zeroes ofQ u + 2s+1 2 η, φ or zeroes ofP u − 2s+1 2 η, φ . Specialising s = ( 2 − 1)/2, T ( 2 −1)/2 = T ( 2 −1)/2 and this agrees with Ref [70]…”

“…Morin-Duchesne and Pearce[70] observed that the zeroes of the eigenvalues of the Q operator appear as part of zeroes of eigenvalues of the transfer matrix whose argument is shifted as T ( 2 −1)/2 (u + 2 η/2, φ) at root of unity. In our framework this is clear too.…”

Fusion hierarchies, T-systems and Y-systems for the dilute $\boldsymbol{A_2^{(2)}}$ loop models

“…Rodney Baxter [1] pioneered the use of functional equations to calculate various physically relevant lattice quantities for solvable lattice models. Nowadays, it is well understood that on an extended lattice, the integrability structures are embodied in the fusion hierarchy [53], T-and Y-systems [12,25,[54][55][56][57] and T-Q functional equations [5][6][7][8][9] satisfied by the transfer matrices. The T-system is well suited to the calculation of non-universal lattice quantities such as free energies using the inversion relation method [58,59].…”

Fusion hierarchies, T-systems and Y-systems for the dilute A2(2) loop models on a strip

“…This in turn ensures the existence of a one-parameter family of commuting transfer matrices. Solutions to the Yang-Baxter equations are often labeled by affine Lie algebras [6,7], with the most studied cases corresponding to the low-rank algebras A 1 loop model and the related six-vertex models, it is fair to say that their sℓ (2) integrability structures are now well understood [2,[8][9][10][11].…”