Distinct finite-temperature phase diagrams of non-invertible Kennedy--Tasaki duals

Abstract

Do two Hamiltonians related by a non-invertible transformation necessarily share the same finite-temperature phase diagram? For a cluster-model interpolation $H(s)$ and its Kennedy--Tasaki dual $\tilde H(s)$, the map is a partial isometry and equates only their all-plus-sector partition functions. In one dimension, thermal order is forbidden on both sides, leaving only the common zero-temperature transition at $s=\tfrac12$. In three dimensions, however, the inequivalence is exact already at $s=0$: the cluster model has an analytic paramagnetic free energy, whereas its dual $\mathbb Z_2$ gauge theory has a deconfinement transition at $T_c\approx1.31$. Quantum Monte Carlo shows that the mismatch occupies a finite window of the interpolation, marked by a self-dual frozen wedge on the cluster side and a deconfined dome on the gauge side; once both close the two phase diagrams agree again, coinciding over the entire remaining interval up to the shared trivial endpoint $s=1$.