Extended T-systems, Q matrices and T-Q relations for models at roots of unity

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.