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Acta Physica Polonica A

Fock Space Localization of Many-Body States in the Tilted Bose-Hubbard Model

Mathias Schneider, Alberto Rodríguez, Andreas Buchleitner

Published Nov 1, 2019• 0 citations
Publisher

Abstract

We analyze the eigenstates' Fock space structure in the tilted Bose-Hubbard Hamiltonian.Fock space localization is quantified in terms of finite-size generalized fractal dimensions.This approach allows us to monitor the parametric evolution of localization across avoided crossings in the energy spectrum.We identify a manifold of strongly localized eigenstates, both in Fock and configuration space, which persists in regimes where all system parameters (hopping, interaction, and tilt strengths) are comparable.For a fixed number of bosons, the localization properties of that manifold are independent of the lattice size.The influence of avoided crossings on the structure of the localized manifold is discussed in detail.

DOI
10.12693/aphyspola.136.834

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We analyze the eigenstates' Fock space structure in the tilted Bose-Hubbard Hamiltonian.Fock space localization is quantified in terms of finite-size generalized fractal dimensions.This approach allows us to monitor the parametric evolution of localization across avoided crossings in the energy spectrum.We identify a manifold of strongly localized eigenstates, both in Fock and configuration space, which persists in regimes where all system parameters (hopping, interaction, and tilt strengths) are comparable.For a fixed number of bosons, the localization properties of that manifold are independent of the lattice size.The influence of avoided crossings on the structure of the localized manifold is discussed in detail.
We identify the chaotic phase of the Bose-Hubbard Hamiltonian by the energy-resolved correlation between spectral features and structural changes of the associated eigenstates as exposed by their generalized fractal dimensions. The eigenvectors are shown to become ergodic in the thermodynamic limit, in the configuration space Fock basis, in which random matrix theory offers a remarkable description of their typical structure. The distributions of the generalized fractal dimensions, however, are ever more distinguishable from random matrix theory as the Hilbert space dimension grows.
We demonstrate many-body multifractality of the Bose-Hubbard Hamiltonian's ground state in Fock space, for arbitrary values of the interparticle interaction. Generalized fractal dimensions unambiguously signal, even for small system sizes, the emergence of a Mott insulator that cannot, however, be naively identified with a localized phase in Fock space. We show that the scaling of the derivative of any generalized fractal dimension with respect to the interaction strength encodes the critical point of the superfluid to the Mott insulator transition, and provides an efficient way to accurately estimate its position. We further establish that the transition can be quantitatively characterized by one single wave function amplitude from the exponentially large Fock space.
The chaotic phase of the tilted Bose-Hubbard model is identified as a function of energy, tilt strength, and particle interaction, from the eigenstate structure and the statistical features of the energy spectrum. Our analysis reveals that the chaotic phase of the bare Bose-Hubbard Hamiltonian can be enhanced by the presence of a moderate tilt. We further unveil the development and scaling of the chaotic regime from the perspective of a homogeneous density configuration typically used in cold atom experiments, providing a valuable phase diagram for future theoretical and experimental studies of this system.
Abstract We investigate the chaotic phase of the Bose–Hubbard model (Pausch et al 2021 Phys. Rev. Lett. 126 150601) in relation to the bosonic embedded random matrix ensemble, which mirrors the dominant few-body nature of many-particle interactions, and hence the Fock space sparsity of quantum many-body systems. The energy dependence of the chaotic regime is well described by the bosonic embedded ensemble, which also reproduces the Bose–Hubbard chaotic eigenvector features, quantified by the expectation value and eigenstate-to-eigenstate fluctuations of fractal dimensions. Despite this agreement, in terms of the fractal dimension distribution, these two models depart from each other and from the Gaussian orthogonal ensemble as Hilbert space grows. These results provide further evidence of a way to discriminate among different many-body Hamiltonians in the chaotic regime.
In the presence of strong enough disorder one-dimensional systems of interacting spinless fermions at nonzero filling factor are known to be in a many-body localized phase. When represented in `Fock space,' the Hamiltonian of such a system looks like that of a single `particle' hopping on a Fock lattice in the presence of a random disordered potential. The coordination number of the Fock lattice increases linearly with the system size LLL in one dimension. Thus in the thermodynamic limit $L\ensuremath{\rightarrow}\ensuremath{\infty}$, the disordered interacting problem in one dimension maps on to an Anderson model with infinite coordination number. Despite this, this system displays localization which appears counterintuitive. A close observation of the on-site disorder potentials on the Fock lattice reveals a large degree of correlation among them as they are derived from an exponentially smaller number of on-site disorder potentials in real space. This indicates that the correlations between the on-site disorder potentials on a Fock lattice has a strong effect on the localization properties of the corresponding many-body system. This intuition is also consistent with studies of quantum random energy model where the typical mid-spectrum states are ergodic and the on-site potentials in Fock space are completely uncorrelated. In this work we perform a systematic quantitative exploration of the nature of correlations of the Fock space potential required for localization. We study different functional variations of the disorder correlation in Fock lattice by analyzing the eigenspectrum obtained through exact diagonalization. Without changing the typical strength of the on-site disorder potential in Fock lattice we show that changing the correlation strength can induce thermalization or localization in systems. From among the various forms of correlations we study, we find that only the linear variation of correlations with Hamming distance in Fock space is able to drive a thermal-MBL phase transition where the transition is driven by the correlation strength. Systems with the other forms of correlations we study are found to be ergodic.