We present a first study of the multifractal properties of the ground state of the Bose-Hubbard Hamiltonian in Fock space.Numerical simulations of system sizes up to L = 10 at unit filling suggest that multifractality is present for all values of the bosonic interaction strength.Moreover, the analysis of the generalised fractal dimensions for different densities exposes qualitatively the superfluid to Mott insulator phase transition.
We present a first study of the multifractal properties of the ground state of the Bose-Hubbard Hamiltonian in Fock space.Numerical simulations of system sizes up to L = 10 at unit filling suggest that multifractality is present for all values of the bosonic interaction strength.Moreover, the analysis of the generalised fractal dimensions for different densities exposes qualitatively the superfluid to Mott insulator phase transition.
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 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 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.
Clarifying similarities and differences in physical properties between crystalline and quasicrystalline systems is one of central issues in studying quasicrystals. To contribute to this, we apply multifractal and hyperuniform analyses to nonuniform spatial patterns in the Bose-Hubbard model on the Penrose and Ammann-Beenker tilings. Based on the mean-field approximation, we obtain real-space distributions of local superfluid amplitude and boson density. In both Mott insulating and superfluid phases, the distributions are hyperuniform. Analyzing the order metric that quantifies the complexity of nonuniform spatial patterns, we find that both quasicrystals show a significant increase of the order metric at a phase boundary between the Mott insulating and superfluid phases, in stark contrast to the case of a periodic square lattice. Our results suggest that hyperuniformity is a useful concept to differentiate between crystalline and quasicrystalline bosonic systems. The order metric clarifies if the distribution of a physical quantity reflects the point distribution or not, and quantifies how complex the distribution is in comparison with the point distribution. Moreover, we introduce on-site random potentials into these quasicrystalline Bose-Hubbard models, leading to a Bose glass phase. Contrary to the Mott insulating and superfluid phases, we find that the Bose glass phase is multifractal. The same multifractality appears on a Bose glass phase in the periodic square lattice. Therefore, multifractality is common in a Bose glass phase irrespective of the periodicity of systems.
We establish the full ground state phase diagram of the disordered Bose-Hubbard model in two dimensions at a unity filling factor via quantum Monte Carlo simulations. Similarly to the three-dimensional case we observe extended superfluid regions persisting up to extremely large values of disorder and interaction strength which, however, have small superfluid fractions and thus low transition temperatures. In the vicinity of the superfluid-insulator transition of the pure system, we observe an unexpectedly weak--almost not resolvable--sensitivity of the critical interaction to the strength of (weak) disorder.
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.