The emergence of state collapse and thermalization in the geometric quantum state of two sites in an extended Hubbard chain undergoing unitary evolution

Geometric quantum mechanics (GQM) is a differential-geometric formalism for quantum mechanics. Its notion of a Geometric Quantum State (GQS) allows the description of ensembles of quantum states conditioned on the state of the environment, making apparent a phase-space like geometric structure and retaining information about correlations between system and environment. In this work we study the GQS of one and two sites of a 1D extended Hubbard model and report the resulting phenomenology. Remarkably, the one-body GQS of all sites exhibits the emergence of classicality, while the two-body GQS of any two sites exhibits the emergence and coexistence of two distinct phases, occurring in separate subspaces: a classical and a thermalizing one. The former points out that GQM offers new insights into the description of measurement-induced collapse, resulting from an entirely unitary dynamics. The latter points to GQM’s relevance for the quantum foundation for thermodynamics.