A Statistical Approach to Quantum Many-Body Eigenstates near Criticality: Multifractality in Hilbert Space

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Certain isolated quantum many-body systems never reach equilibrium (defined via suitable single-particle observables) under their own unitary dynamics. Even at the single-particle level, the system retains information about its initial state encoded in local degrees of freedom for arbitrarily long times, due to that fact that many-particle interactions fail to provide a thermal bath for the system itself. Such behaviour is a reflection of the structure of the many-body excited eigenstates of the system, which exhibit localisation in Hilbert space. This occurs for example in interacting quantum systems subjected to strong disorder (many-body localisation), as well as in certain classically chaotic many-body systems. A deeper understanding of ergodicity, chaos and thermalisation in generic many-body quantum systems requires a detailed study of the nature of many-body wavefunctions and related observables in Hilbert space. In particular, wavefunction multifractality seems to be a generic feature of many-body systems that plays a key role in this phenomenology. Multifractal wavefunctions occupy an infinite volume in the thermodynamic limit -like extended states do-, but at the same time a vanishing fraction of the whole space -inheriting the non-ergodicity of localised states. The structure of many-body wavefunctions, and the quantification of their multifractal fluctuations and how these evolve as the system flows from ergodic to localised is not fully understood, yet this determines crucially the behaviour of physical observables. We propose to tackle the study of multifractality in Hilbert space using a numerical beyond-standard generalised multifractal analysis in combination with finite-size scaling. This powerful technique enlarges the applicability of multifractal analysis to describe regions of extended, multifractal and localised states and the scaling between these. With our innovative approach we aim to: (a) Give a clear perspective of the many-body localisation phenomenology in Hilbert space for disordered interacting fermions. We will provide unambiguous evidence of the existence (or not) of ergodic, truly localised, and multifractal intermediate phases of the many-body eigenstates. This will lead to a high-precision phase diagram of localisation and multifractality for interacting fermions, as a function of energy density, interaction strength and disorder.(b) Unravel multifractality in Fock space for interacting bosons in the absence of disorder. Our preliminary investigations reveal multifractality in the ground state of such a system. We will study the evolution of multifractality as a function of the inter-particle interaction strength, in relation with the Mott to superfluid phase transition. By adding a tilt to the local chemical potential, we will investigate the multifractal properties of robust (parametric solitonic) states in the excitation energy spectrum, and relate them to those of the system's ground state.