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

Abstract

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 $\lambda =\frac{(p^{\prime }-p)\pi }{p^{\prime }}$ a rational fraction of $\pi$. The $1\times n$ transfer matrices of the dense loop model analogs, namely the logarithmic minimal models $\mathcal{LM}(p,p^{\prime })$, 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 $n=p^{\prime }-1$, in terms of the product $Q^{+}(u)Q^{-}(u)$ or $Q(u{)}^2$. It follows that the zeros of $T^{p^{\prime }-1}(u+\lambda )$ and $D^{p^{\prime }-1}(u+\lambda )$ are comprised of the Bethe roots and complete $p^{\prime }$ 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.

arXiv comment38 pages

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• Rational Q-systems at Root of Unity I. Closed Chains
• Fusion hierarchies, $T$-systems and $Y$-systems for the dilute $A_2^{(2)}$ loop models on a strip
• On the Q operator and the spectrum of the XXZ model at root of unity