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.
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.
In this thesis we present a first study of the multifractal properties of the ground state of the one-dimensional Bose-Hubbard model in Fock space using two different bases, Fock and momentum basis. In the beginning we give a brief overview of the properties of the Bose-Hubbard model, introduce the concept of multifractality, and discuss exact diagonalisation and quantum Monte Carlo, the numerical methods we employ throughout the present work. Then we treat the limits of vanishing and infinite bosonic interaction strengths analytically, and their vicinities by using perturbation theory. We find that the limit of vanishing interaction exhibits non-trivial multifractality in the Fock basis, and that the generalised fractal dimensions exhibit (dominantly) logarithmic finite-size corrections in this limit. Our observations for other values of the interaction strength suggest that this form of the corrections might be equally valid. In order to get access to the multifractal properties at arbitrary values of the interaction strength, we use exact diagonalisation (for system sizes up to L = 10 at unit filling) and quantum Monte Carlo simulations (which enable us to reach L = 30 in certain cases, corresponding to a Hilbert space of size N = 6*10^16, and therefore to one of the largest ever considered for a multifractal analysis). While we see that the generalised fractal dimensions in the Fock basis decrease monotonously when the interaction strength is increased, we observe an inverted behaviour in the momentum basis, i.e. a monotonous increase with increasing interaction strength. Furthermore, our results suggest the existence of non-trivial multifractality in the ground state for a large range of interaction values. Finally, we investigate whether it is possible to characterise the superfluid to Mott insulator transition with the help of multifractality. We find that an analysis of the generalised fractal dimensions for different densities exposes qualitatively the transition. We furthermore consider different ideas, originating from our results in combination with other approaches used in the literature, to quantitatively characterise the transition. While the investigated methods carry great potential, we confirmed that a quantitatively accurate location of the transition using these techniques, if possible, requires access to even larger system sizes.
We implement a recursive Green's function method to extract the Fock space (FS) propagator and associated self-energy across the many-body localization (MBL) transition, for one-dimensional interacting fermions in a random on-site potential. We show that the typical value of the imaginary part of the local FS self-energy, ${\mathrm{\ensuremath{\Delta}}}_{t}$, related to the decay rate of an initially localized state, acts as a probabilistic order parameter for the thermal to MBL phase transition and can be used to characterize critical properties of the transition as well as the multifractal nature of MBL states as a function of disorder strength W. In particular, we show that a fractal dimension Ds extracted from ${\mathrm{\ensuremath{\Delta}}}_{t}$ jumps discontinuously across the transition, from Ds<1 in the MBL phase to Ds=1 in the thermal phase. Moreover, ${\mathrm{\ensuremath{\Delta}}}_{t}$ follows an asymmetrical finite-size scaling form across the thermal-MBL transition, where a nonergodic volume in the thermal phase diverges with a Kosterlitz-Thouless--like essential singularity at the critical point Wc and controls the continuous vanishing of ${\mathrm{\ensuremath{\Delta}}}_{t}$ as Wc is approached. In contrast, a correlation length ($\ensuremath{\xi}$) extracted from ${\mathrm{\ensuremath{\Delta}}}_{t}$ exhibits a power-law divergence on approaching Wc from the MBL phase.
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.
The study of interacting quantum many-body systems is constrained by the exponential growth of the Hilbert space with system size, which limits both numerical simulations and theoretical understanding. This thesis explores how the interplay between disorder and interactions shapes localization and ergodicity in quantum many-body systems, using the exact diagonalization method, and how machine learning can serve as an alternative approach for studying these systems. We first investigate the many-body localization problem in a one-dimensional SU(2)-invariant random antiferromagnetic Heisenberg chain using exact and sparse matrix diagonalization methods. By analyzing energy level gap-ratio statistics, participation ratio, entanglement entropy and the entanglement spectral parameter as a function of disorder strength under di!erent random coupling distributions, we identify clear qualitative di!erences from the widely studied random field Heisenberg chain. For system sizes up to L = 24, signatures of localized behaviour are much less pronounced than in the case of the random field model. Motivated by the limitations of conventional numerical methods, we examine machine learning as a scalable paradigm for many-body physics. Using the Bose- Hubbard model in one and two dimensions as a benchmark, we test HUBBARDNET, a deep neural network architecture, by replacing standard stochastic gradient descent with adaptive moment estimation (ADAM), adjusting the learning rate for improved optimization, and introducing physics-informed output activations that can resolve extremely small wave-function amplitudes. This approach yields ground-state energy errors reduced by orders of magnitude and wave function fidelities exceeding 99%, remaining robust across four decades of interaction strength and over superfluid, Mott-insulating, and quantum chaotic regimes. Beyond energy prediction, the trained models accurately reproduce generalized inverse participation ratios and multifractal dimensions for ground and excited states, capturing localization, delocalization, and multifractality across the spectrum, with excited-state training explicitly guided by fractal dimensions that encode the multifractal properties of the wave functions.
We study the local density of the Bose-Hubbard model in the presence of on-site disorder near the Bose-glass transition using multifractal, typical medium, and percolation theories. At incommensurate filling our findings support the scenario of percolating superfluid clusters enhancing Anderson localization. Scaling analysis of the superfluid density at the incommensurate filling of $\ensuremath{\rho}=1.1$ and on-site interaction U=80t predicts a superfluid-Bose-glass transition at disorder strength of ${\mathrm{\ensuremath{\Delta}}}_{c}\ensuremath{\approx}30t$. At this filling the local-density distribution becomes more skew with increasing disorder strength. Multifractal analysis suggests a multifractal behavior resembling that of the Anderson localization. In the Bose-glass phase the mode of the local-density distribution approaches an integer value as expected from typical medium theory for the Anderson localization. Percolation analysis points to a phase transition of percolating noninteger filled sites around the same value of disorder. On the other hand, the behavior at commensurate filling is rather different. Close to the tip of the Mott lobe ($\ensuremath{\rho}=1,U=22t$) we find a Mott-insulator-Bose-glass transition at disorder strength of ${\mathrm{\ensuremath{\Delta}}}_{c}\ensuremath{\approx}16t$. An analysis of the local-density distribution shows Gaussian-like behavior for a wide range of disorders above and below the transition.