Geometry, Topology, and Dynamics of Many-Body Systems: Quantum and Classical Perspectives

In the present thesis, we study various instances of many-body systems, both quantum and classical, adopting an overall geometrical and topological perspective. Systems with many degrees of freedom can exhibit a great variety of emergent phenomena, from metastability to dynamical or thermodynamic phase transitions, and their intricate properties often elude our full understanding. Through careful inspection of the underlying geometrical structures behind these complex objects, and using diverse mathematical tools, some of which are borrowed from the field of Riemannian geometry and topology, we try to characterize their most prominent aspects. The first part of this thesis focuses on the field of quantum information, exploring in particular the notions of entanglement and correlations in discrete quantum systems, first in pure and then in mixed states. We propose a novel measure of entanglement for pure quantum states, the entanglement distance, and discuss and test it using various examples. We then study maximally entangled pure states, revealing in part their internal structures, using the intuitive notion of correlations and projective measurements. Doing so, we are able to show simple relations between pre-measurement correlations and post-measurement expectation values and provide an upper bound to the persistency of entanglement of such states. We further demonstrate how the connectivity properties of a paradigmatic model for quantum computing, the celebrated graph state, can be probed through the proper use of quantum correlators. Our study is then extended to the framework of mixed quantum states. We infer from the pure state entanglement distance, the induced measures of quantum correlations, on the one hand, and of mixed state entanglement, on the other hand. We emphasize the strengths and limitations of the latter, in particular the heavy optimization procedure it implies, for which we propose a workaround. Finally, we investigate the superradiant transition present in the Tavis–Cummings model and reveal that it is accompanied by a jump in the quantum correlation and entanglement between the atoms. In the second part of our work, we tackle two classical models, both possessing ergodicity-breaking and strongly non-linear behaviours. We first present a comprehensive numerical and analytical investigation of a toy model, which is a prototypical example of a long-range interacting, strongly non-additive model: the Hamiltonian Mean Field (HMF). At low energy in the microcanonical ensemble, a metastable state may arise, coined as a bicluster. By inspecting the dynamics of a macroscopic quantity, the magnetization, and by observing the occurrence of two distinct timescales in the system, we provide a quite intuitive and self-consistent scenario accounting for the great stability of biclusters, which sustain themselves for extensive amounts of time. Finally, we present the results of an extensive numerical study, applying the topological theory of phase transition to a model of glass-forming material. Such systems are notoriously difficult to simulate and equilibrate, because of the very slow dynamics, characteristic of the glass phase, yielding a tendency to remain stuck in small regions of phase space. Within a microcanonical framework, a Monte Carlo algorithm was developed, for which various numerical methods were implemented, such as parallel tempering and particle swapping. Our results, though preliminary, are encouraging: the heat capacity is found to exhibit clear peaks for two values of the energy, indicating a two-step second order transition, in correspondence with topological changes of the equipotential level sets on which the system is confined. We conclude by a thorough discussion of our findings, and by a proposition of a promising follow-up research, that would top off the bridge between our diverse finding: namely, applying the topological theory to quantum phase transitions, possibly drawing a further link with the creation of entanglement in these processes.