We examine the Bose-Hubbard model in the Penrose lattice based on inhomogeneous mean-field theory. Since the averaged coordination number in the Penrose lattice is four, the mean-field phase diagram consisting of the Mott insulator (MI) and superfluid (SF) phase is similar to that of the square lattice. However, the spatial distribution of the Bose condensate in the SF phase is significantly different from uniform distribution in the square lattice. We find a fractal structure in its distribution near the MI-SF phase boundary. The emergence of the fractal structure is a consequence of the cooperative effect between quasiperiodicity in the Penrose lattice and criticality at the phase transition.
We examine the Bose-Hubbard model in the Penrose lattice based on inhomogeneous mean-field theory. Since the averaged coordination number in the Penrose lattice is four, the mean-field phase diagram consisting of the Mott insulator (MI) and superfluid (SF) phase is similar to that of the square lattice. However, the spatial distribution of the Bose condensate in the SF phase is significantly different from uniform distribution in the square lattice. We find a fractal structure in its distribution near the MI-SF phase boundary. The emergence of the fractal structure is a consequence of the cooperative effect between quasiperiodicity in the Penrose lattice and criticality at the phase transition.
We investigate the temperature-dependent behavior emerging in the vicinity of the superfluid (SF) to Mott insulator (MI) transition of interacting bosons in a two-dimensional optical lattice, described by the Bose-Hubbard model. The equilibrium phase diagram at finite temperatures is computed by means of the cluster mean-field theory (CMF) where the effect of non-local correlations is analyzed systematically by finite-size scaling of the cluster size. The phase diagram exhibits a rich structure including a transition and a crossover of the SF and MI phases respectively to a normal fluid (NF) state at finite temperature. In order to characterize these phases, and the NF transition and crossover scales, we calculate, in addition to the condensate amplitude, the superfluid fraction, sound velocity and compressibility. The phase boundaries obtained by CMF with finite-size scaling agree quantitatively with quantum Monte Carlo (QMC) results as well as with experiments. The von Neumann entanglement entropy of a cluster exhibits critical enhancement near the SF-MI quantum critical point (QCP). We also discuss the behavior of the transition lines near this QCP at the particle-hole symmetric point located at the tip of a Mott lobe as well as away from particle-hole symmetry.
Abstract We develop an inhomogeneous mean‐field theory for the extended Bose‐Hubbard model with a quadratic, confining potential. In the absence of this potential, our mean‐field theory yields the phase diagram of the homogeneous extended Bose‐Hubbard model. This phase diagram shows a superfluid (SF) phase and lobes of Mott‐insulator (MI), density‐wave (DW), and supersolid (SS) phases in the plane of the chemical potential µ and on‐site repulsion U; we present phase diagrams for representative values of V, the repulsive energy for bosons on nearest‐neighbor sites. We demonstrate that, when the confining potential is present, superfluid and density‐wave order parameters are nonuniform; in particular, we obtain, for a few representative values of parameters, spherical shells of SF, MI, DW, and SS phases. We explore the implications of our study for experiments on cold‐atom dipolar condensates in optical lattices in a confining potential.
We present an extensive study of Mott insulator (MI) and superfluid (SF) shells in Bose-Hubbard (BH) models for bosons in optical lattices with harmonic traps. For this we apply the inhomogeneous mean-field theory developed by Sheshadri et al. [Phys. Rev. Lett. 75, 4075 (1995)]. Our results for the BH model with one type of spinless bosons agree quantitatively with quantum Monte Carlo simulations. Our approach is numerically less intensive than such simulations, so we are able to perform calculations on experimentally realistic, large three-dimensional systems, explore a wide range of parameter values, and make direct contact with a variety of experimental measurements. We also extend our inhomogeneous mean-field theory to study BH models with harmonic traps and (a) two species of bosons or (b) spin-1 bosons. With two species of bosons, we obtain rich phase diagrams with a variety of SF and MI phases and associated shells when we include a quadratic confining potential. For the spin-1 BH model, we show, in a representative case, that the system can display alternating shells of polar SF and MI phases, and we make interesting predictions for experiments in such systems.
We numerically investigate the two-dimensional Bose-Hubbard model with local onsite disorder, where the competition between disorder and short-range interactions leads to the emergence of a Bose glass (BG) phase between the Mott insulator (MI) and superfluid (SF) phases. In order to analyze the inhomogeneous system we employ real-space bosonic dynamical mean-field theory (RBDMFT) and perform an ensemble average over disorder realizations. To distinguish the MI from the BG phase, we compare the Edwards-Anderson order parameter and the compressibility with the energy-gap condition. To identify the insulator to SF transition, we apply a percolation analysis to the condensate order parameter. In qualitative accordance with the theorem of inclusions we always find an intermediate BG phase between the SF and MI. However, the quantitative comparison indicates significant deviations between the MI to BG phase boundary expected in the thermodynamic limit and the one obtained for a finite system size. Additionally, RBMDFT is capable of reliably calculating spectral information throughout the phase diagram. Analyzing the spectral function reveals evidence for analytically predicted damped localized modes in the dispersion relation in the strong-coupling regime.
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.
The Bose glass (BG) phase is the Griffiths region of the disordered Bose–Hubbard model (BHM), characterized by finite, quasi-superfluid clusters within a Mott insulating background. We propose to utilize this characterization to identify the complete zero-temperature phase diagram of the disordered BHM in d ⩾ 2 dimensions by analysing the geometric properties of what we call superfluid (SF) clusters, which are defined to be clusters of sites with non-integer expectation values for the local boson occupation number. The Mott insulator phase then is the region in the phase diagram where no SF clusters exist, and the SF phase the region where SF clusters percolate—the BG phase is inbetween: SF clusters exist, but do not percolate. This definition is particularly useful in the context of local mean field (LMF) or Gutzwiller–Ansatz calculations, where we show that an identification of the phases on the basis of global quantities such as the averaged SF order parameter and the compressibility is misleading. We apply the SF cluster analysis to the LMF ground states of the two-dimensional disordered BHM to produce its phase diagram and find (a) an excellent agreement with the phase diagram predicted on the basis of quantum Monte Carlo simulations for the commensurate density n = 1 and (b) large differences to stochastic mean field and other mean field predictions for fixed disorder strength. The relation of the percolation transition of the SF clusters with the onset of non-vanishing SF stiffness indicating the BG to SF transition is discussed.