# The equilibrium landscape of the Heisenberg spin chain

## TL;DR

This paper characterizes the full set of local equilibrium states in the integrable Heisenberg spin chain using two frameworks, revealing a hidden structure and clarifying subtle features of equilibrium ensembles.

## Contribution

It establishes the equivalence of the Thermodynamic Bethe Ansatz and transfer matrix approaches for describing equilibrium states in the Heisenberg chain, and uncovers a hidden structure in the Y-system.

## Key findings

- Demonstrates equivalence of two frameworks for equilibrium states
- Clarifies the breakdown of the canonical Y-system
- Reveals hidden structures in equilibrium ensemble parametrization

## Abstract

We characterise the equilibrium landscape, the entire manifold of local equilibrium states, of an interacting integrable quantum model. Focusing on the isotropic Heisenberg spin chain, we describe in full generality two complementary frameworks for addressing equilibrium ensembles: the functional integral Thermodynamic Bethe Ansatz approach, and the lattice regularisation transfer matrix approach. We demonstrate the equivalence between the two, and in doing so clarify several subtle features of generic equilibrium states. In particular we explain the breakdown of the canonical Y-system, which reflects a hidden structure in the parametrisation of equilibrium ensembles.