Interchangeability of periodic XXZ limits

Establish that the inverse-temperature limit and thermodynamic limit can be interchanged for the spin-\tfrac12 Heisenberg XXZ chain with periodic boundary conditions in the state \langle\cdot\rangle_{\beta,T_L}, using suitable space-time six-vertex convergence results.

Background

The main theorem proves interchangeability of the limits for several finite-volume states in a specified anisotropy range, but explicitly excludes the periodic-boundary Gibbs state \langle\cdot\rangle_{\beta,T_L}. The paper explains that resolving this would require extending Sheffield’s convergence theory to suitable space-time models and combining it with the paper’s delocalisation result.