Canonical formulation for the thermodynamics of $sl_n$-invariant integrable spin chains

Integrable quantum spin chains display distinctive physical properties making them a laboratory to test and assess different states of matter. The study of the finite temperature properties is possible by use of the thermodynamic Bethe ansatz, however at the expense of dealing with non-linear integral equations for, in general, infinitely many auxiliary functions. The definition of an alternative finite set of auxiliary functions allowing for the complete description of their thermodynamic properties at finite temperature and fields has been elusive. Indeed, in the context of $sl_n$-invariant models satisfactory auxiliary functions have been established only for $n \leq 4$. In this paper we take a step further by proposing a systematic approach to generate finite sets of auxiliary functions for $sl_n$-invariant models. We refer to this construction as the canonical formulation. The numerical efficiency is illustrated for $n=5$, for which we present some of the thermodynamic properties of the corresponding spin chain.