The resource theory of thermal operations, an established model for small-scale thermodynamics, provides an extension of equilibrium thermodynamics to nonequilibrium situations. On a lattice of any dimension with any translation-invariant local Hamiltonian, we identify a large set of translation-invariant states that can be reversibly converted to and from the thermal state with thermal operations and a small amount of coherence. These are the spatially ergodic states, i.e., states that have sharp statistics for any translation-invariant observable, and mixtures of such states with the same thermodynamic potential. As an intermediate result, we show for a general state that if the min- and the max-relative entropy to the thermal state coincide approximately, this implies the approximately reversible interconvertibility to and from the thermal state with thermal operations and a small source of coherence. Our results provide a strong link between the abstract resource theory of thermodynamics and more realistic physical systems, as we achieve a robust and operational characterization of the emergence of a thermodynamic potential in translation-invariant lattice systems.
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