Statistical Spectroscopy of Quantum Many-Body Systems

A central theme in the field of quantum many-body physics is the characterization of thermal properties, which can be accessed, for instance, through the analysis of the Hamiltonian spectrum. This thesis approaches the problem from both numerical and analytical viewpoints, treating the spectrum as a statistical object whose correlations encode predictive information about the physics of the underlying system. In the first part of this thesis, we review the relation between thermalization, quantum integrability and chaos in one-dimensional quantum many-body systems. We focus on the statistics of energy levels and establish whether quantum integrability can be assessed from a statistical viewpoint. We then consider the problem of intermediate statistics between purely integrable and non-integrable quantum many-body systems. The first contribution of this thesis is an answer to the problem of discriminating between the spectrum of a genuine integrable system and a statistical mixture of, possibly exponentially many, non-ergodic sectors. We show that the spectral decimation algorithm can capture both qualitative and quantitative features of a `characteristic symmetry sector' from an unbiased analysis of numerically available spectral data. When applied to physically relevant systems, the algorithm is able to identify traces of Hilbert space fragmentation and disorder-induced many-body localization. The second part of this thesis is dedicated to a class of Hamiltonians of particular interest to statistical spectroscopy. Such Hamiltonians involve only the permutation of sites without the presence of external interactions. Nonetheless, they generate a highly tunable and numerically implementable class of models. For instance, it is easy to engineer integrability and its breaking while preserving essential structures such as the block-diagonal decomposition. These Hamiltonians are naturally suited to generating unitary random walks on the permutation group, which can, for instance, be compared with their classical counterparts. After a detailed discussion of the numerical implementation of these Hamiltonians and their spectral properties, we depart from the analysis of the system's static properties to answer a genuine out-of-equilibrium question: do quantum coherences accelerate mixing in the computational basis? We consider an exactly solvable class of quantum stochastic walks which, in the absence of quantum coherences, admit a limit coinciding with a classical continuous-time Markov chain. While the trace distance of the associated density matrices from the maximally mixed state remains identical to that in the classical limit, we show that the probability distance in the presence of quantum coherences, which defines mixing in the computational basis, is upper bounded by the classical limit. An analysis of the slowest mode identifies a necessary condition under which quantum coherences and classical mixing become of the same order, thereby defining a slow-mode time scale. Finally, a numerical analysis of quantum stochastic walks based on random transpositions corroborates the analytical findings, in particular the accelerated mixing. We also identify a closed-form expression for the ratio of quantum to classical mixing times, which yields a collapse over numerically accessible system sizes. Overall, this thesis presents a modern approach to the theory of statistical spectroscopy by proposing analytically grounded and numerically implementable protocols to ascertain the thermal properties of a one-dimensional quantum many-body system from the analysis of the spectrum within numerically accessible volumes. It also shows that the analysis of the statistical properties of Hamiltonians can be of practical aid in the study of out-of-equilibrium properties of quantum many-body systems, such as mixing in quantum stochastic walks.