Gauge invariant quantum thermodynamics: foundations and applications to critical systems

Abstract

In this formalism, thermodynamic quantities are described as functionals of the density operator and emerge from a coarse-graining process over all unitary transformations that preserve the system’s energy - a set we refer to as the thermodynamic group. This approach intrinsically incorporates informational access limitations to the system, constrained to energy basis measurements. Our contribution generalizes the expressions for invariant heat and work beyond the treatment presented in [3], explicitly including the consideration of dynamic degenerate energy levels. More than a formal extension, we establish a rigorous physical interpretation of these quantities through the non-unitary dynamics of the effective state obtained by the coarse-graining induced by the thermodynamic group. We demonstrate that the fundamental restriction to energy basis measurements implies the inaccessibility of part of the system’s energy, interpreted as an effective heat associated with the production of coherences in the energy basis. Consequently, the derived work corresponds to the system’s effective work, with its energetic contribution reduced by the portion allocated as heat. Within the second law context, we introduce the gauge-invariant entropy, proving it satisfies all required properties for a thermodynamic entropy. This entropy exhibits an explicit dependence on the thermodynamic group, manifested through an entropic contribution directly linked to the Hamiltonian’s spectral degeneracies. Crucially, we identify that the origin of entropy production coincides with that of heat in closed systems, both emerging from the inherent non-unitarity in the coarse-grained state’s dynamics. As a final application, we implement our formalism in the quantum dynamics of spin systems undergoing quenches, revealing that coherent heat captures signatures of irreversibility. In critical systems, the quantities derived within this theory prove fundamental for understanding modifications in the system’s symmetry structure relative to the internal symmetries of the thermodynamic gauge group