A review of Monte Carlo simulations for the Bose–Hubbard model with diagonal disorder

We review the physics of the Bose–Hubbard model with disorder in the chemical potential focusing on recently published analytical arguments in combination with quantum Monte Carlo simulations. Apart from the superfluid and Mott insulator phases that can occur in this system without disorder, disorder allows for an additional phase, called the Bose glass phase. The topology of the phase diagram is subject to strong theorems proving that the Bose Glass phase must intervene between the superfluid and the Mott insulator and implying a Griffiths transition between the Mott insulator and the Bose glass. The full phase diagrams in 3d and 2d are discussed, and we zoom in on the insensitivity of the transition line between the superfluid and the Bose glass in the close vicinity of the tip of the Mott insulator lobe. We briefly comment on the established and remaining questions in the 1d case, and give a short overview of numerical work on related models.     

Quantum glass phases in the disordered Bose-Hubbard model

The phase diagram of the Bose-Hubbard model in the presence of off-diagonal disorder is determined using quantum Monte Carlo simulations. A sequence of quantum glass phases intervene at the interface between the Mott insulating and the superfluid phases of the clean system. In addition to the standard Bose glass phase, the coexistence of gapless and gapped regions close to the Mott insulating phase leads to a novel Mott glass regime which is incompressible yet gapless. Numerical evidence for the properties of these phases is given in terms of global (compressibility, superfluid stiffness) and local (compressibility, momentum distribution) observables.     

Insulating phases and superfluid-insulator transition of disordered boson chains

Using a strong disorder real-space renormalization group, we study the phase diagram of a fully disordered chain of interacting bosons. Since this approach does not suffer from runaway flows, it allows a direct study of the insulating phases, not accessible in a weak disorder perturbative treatment. We find that the universal properties of the insulating phase are determined by the details and symmetries of the on-site chemical-potential disorder. Three insulating phases are possible: (i) an incompressible Mott glass with a finite superfluid susceptibility, (ii) a random-singlet glass with diverging compressibility and superfluid susceptibility, (iii) a Bose glass with a finite compressibility but diverging superfluid susceptibility. In addition to characterizing the insulating phases, we show that the superfluid-insulator transition is always described by Kosterlitz-Thouless-like flows.     

Ultracold atoms in optical lattices with random on-site interactions

We consider the physics of lattice bosons affected by disordered on-site interparticle interactions. Characteristic qualitative changes in the zero-temperature phase diagram are observed when compared to the case of randomness in the chemical potential. The Mott-insulating regions shrink and eventually vanish for any finite disorder strength beyond a sufficiently large filling factor. Furthermore, at low values of the chemical potential both the superfluid and Mott insulator are stable towards formation of a Bose glass leading to a possibly nontrivial tricritical point. We discuss feasible experimental realizations of our scenario in the context of ultracold atoms on optical lattices.     

Phase glass is a Bose metal: a new conducting state in two dimensions

In the quantum rotor model with random exchange interactions having a nonzero mean, three phases, a (i) phase (Bose) glass, (ii) superfluid, and (iii) Mott insulator, meet at a bicritical point. We demonstrate that proximity to the bicritical point and the coupling between the energy landscape and the dissipative degrees of freedom of the phase glass lead to a metallic state at T = 0. Consequently, the phase glass is unique in that it represents a concrete example of a metallic state that is mediated by disorder, even in 2D. We propose that the experimentally observed metallic phase which intervenes between the insulator and the superconductor in a wide range of thin films is in actuality a phase glass.     

Effective area and expansion energy of trapped Bose gas in a combined magnetic-optical potential

In this Letter a conventional method of statistical physics and quantum mechanics is used to calculate the effective area and the expansion energy for trapped Bose gas in a combined optical-magnetic potential. Correction due to the finite number of particles, interatomic interaction and the deepness of the lattice potential are given simultaneously. It is found that the system possess two different phases which are superfluid phase and Mott insulator phase. The critical temperature which separate these two phases is calculated. In the superfluid phase both the effective area and expansion energy is sensitive to the variation of temperature and lattice depth. Mott insulator phase is characterized by vanishing of the condensed fraction and freezing of the effective area at the value which corresponding to BEC transition temperature. So these parameters can serve as a practical thermometer for such system. The expansion energy shows that the lack of expansion in any direction is due to the strong anisotropy of the trapping potential in this direction. The obtained results provide a solid theoretical foundation for the current experiments.     

Superfluid-Mott-Insulator Phase Transition Phases of Bosons in an and Collective Fluctuations in both Optical Lattice

The Bose Hubbard model describing interacting bosons in an optical lattice is reduced to a simple spin-1 XY model with single-ion anisotropy in the vicinity of the Mott phase. In the strong coupling Mott insulating regime, we propose a mean t~eld theory based on a constraint SU（3） pseudo-boson representation on the effective model and discuss the excitation spectra and the phase transition to the superfluid state. Further to the superfluid phase, we use the coherent-state approach to derive the collective excitation modes. It is found that the Mort phase has two degenerate gapped quadratic excitation spectra which graduate into two degenerate gapless linear ones at the transition point, and one gapless linear mode with one gapped quadratic mode in the superfluid phase.     

Noise correlations of fermions and hard core bosons in a quasi-periodic potential

We use Hanbury-Brown-Twiss interferometry (HBTI) to study various quantum phases of hard core bosons (HCBs) and ideal fermions confined in a one-dimensional quasi-periodic (QP) potential. For HCBs, the QP potential induces a cascade of Mott-like band-insulator phases in the extended regime, in addition to the Mott insulator, Bose glass, and superfluid phases. At critical filling factors, the appearance of these insulating phases is heralded by a peak to dip transition in the interferogram, which reflects the fermionic aspect of HCBs. On the other hand, ideal fermions in the extended phase display various complexities of incommensurate structures such as devil’s staircases and Arnold tongues. In the localized phase, the HCB and the fermion correlations are identical except for the sign of the peaks. Finally, we demonstrate that HBTI provides an effective method to distinguish Mott and glassy phases.     

Transition from a strongly interacting 1d superfluid to a Mott insulator

We study 1D trapped Bose gases in the strongly interacting regime. The systems are created in an optical lattice and are subject to a longitudinal periodic potential. Bragg spectroscopy enables us to investigate the excitation spectrum in different regimes. In the superfluid phase a broad continuum of excitations is observed which calls for an interpretation beyond the Bogoliubov spectrum taking into account the effect of strong interactions. In the Mott insulating phase a discrete spectrum is measured. Both phases are compared to the 3D situation and to the crossover regime from 1D to 3D. The coherence length and coherent fraction of the gas are measured in all configurations. We observe signatures for increased fluctuations characteristic for 1D systems. Moreover, the collective oscillations cease near the transition to the Mott insulator phase.     

Superfluid to Mott‐insulator transition in an anisotropic two‐dimensional optical lattice

We study the superfluid to Mott‐insulator transition of bosons in an optical anisotropic lattice by employing the Bose‐Hubbard model living on a two‐dimensional lattice with anisotropy parameter κ. The compressible superfluid state and incompressible Mott‐insulator (MI) lobes are efficiently described analytically, using the quantum U(1) rotor approach. The ground state phase diagram showing the evolution of the MI lobes is quantified for arbitrary values of κ, corresponding to various kind of lattices: from square, through rectangular to almost one‐dimensional.     

Phase transition in a system of one-dimensional bosons with strong disorder

We study one-dimensional disordered bosons at large commensurate filling. Using a real space renormalization group approach, we find a new random fixed point which controls a phase transition from a superfluid to an incompressible Mott glass. The transition can be tuned by changing the disorder distribution even with vanishing interactions. We derive the properties of the transition, which suggest that it is in the Kosterlitz-Thouless universality class.     

Effective Boseben Hubbard interaction with enhanced nonlinearity in an array of coupled cavities

An array of coupled cavities, each of which contains an N four-level atom, is investigated. When cavity fields dispersively interact with the atoms, an effective Bose—Hubbard model can be achieved. By numerically comparing the full Hamiltonian with the effective one, we find that within the parameters region, the effective Hamiltonian can completely account for the Mott-insulator as well as the phase transition from the similar Mott-insulator to superfluid. Through jointly adjusting the classical Rabi frequency and the detuning, the nonlinearity can be improved.     

Atomic Bose and Anderson glasses in optical lattices

An ultracold atomic Bose gas in an optical lattice is shown to provide an ideal system for the controlled analysis of disordered Bose lattice gases. This goal may be easily achieved under the current experimental conditions by introducing a pseudorandom potential created by a second additional lattice or, alternatively, by placing a speckle pattern on the main lattice. We show that, for a noncommensurable filling factor, in the strong-interaction limit, a controlled growing of the disorder drives a dynamical transition from superfluid to Bose-glass phase. Similarly, in the weak interaction limit, a dynamical transition from superfluid to Anderson-glass phase may be observed. In both regimes, we show that even very low-intensity disorder-inducing lasers cause large modifications of the superfluid fraction of the system.     

Quantum spherical model in a random field: coherent state path integral approach

By introducing boson operators, a quantum spherical XY model in the presence of a random field has been studied by the coherent state path integral approach. The phase diagram is obtained, and the effects of the random-field fluctuations on the possibilities of the existence of a ferromagnetic phase are discussed. At the critical point, , the order parameter M describing the ordered ferromagnetic phase disappears as .Since the model is equivalent to a Bose system, we also show that the phase transition at zero temperature between the superfluid and the disordered Mott insulator phases occurs at the chemical potential , where J₀ is the strength of the exchange interaction. As the temperature T goes to zero, the asymptotic behavior of the entropy and the specific heat are and , respectively. Received: 20 May 1997 / Accepted: 20 October 1997     

Phase transition in a two-dimensional dipole-oriented exciton system

A dipole-oriented two-dimensional exciton system in electrically biased GaAs/AlGaAs coupled quantum wells has been studied through photoluminescence. The system has a sample-dependent built-in random potential which traps excitons at low temperature. The average photoluminescence photon energy shows a sudden reduction when the excitation intensity exceeds a critical value at low temperatures. This suggests a phase transition from a Bose glass to superfluid phase.     

Perturbative calculation of critical exponents for the Bose–Hubbard model

We develop a strategy for calculating critical exponents for the Mott insulator-to-superfluid transition shown by the Bose–Hubbard model. Our approach is based on the field-theoretic concept of the effective potential, which provides a natural extension of the Landau theory of phase transitions to quantum critical phenomena. The coefficients of the Landau expansion of that effective potential are obtained by high-order perturbation theory. We counteract the divergency of the weak-coupling perturbation series by including the seldom considered Landau coefficient a ₆ into our analysis. Our preliminary results indicate that the critical exponents for both the condensate density and the superfluid density, as derived from the two-dimensional Bose–Hubbard model, deviate by less than 1 % from the best known estimates computed so far for the three-dimensional XY universality class.     

Disordered Bose-Einstein condensates in quasi-one-dimensional magnetic microtraps

We analyze the effects of a random magnetic potential in a microfabricated waveguide for ultracold atoms. We find that the shape and position fluctuations of a current carrying wire induce a strong Gaussian correlated random potential with a length scale set by the atom-wire separation. The theory is used to explain quantitatively the observed fragmentation of the Bose-Einstein condensates in atomic waveguides. Furthermore, we show that nonlinear dynamics can be used to provide important insights into the nature of the strongly fragmented condensates. We argue that a quantum phase transition from the superfluid to the insulating Bose glass phase may be reached and detected under the realistic experimental conditions.     

Benchmark study of the SF-MI phase transition of the Bose–Hubbard model with density-induced tunneling

The Bose–Hubbard model (BHM) is a standard model which describes the quantum behavior of ultracold bosons in optical lattice. When tuning the model parameters, a quantum phase transition from superfluid (SF) phase to Mott insulating (MI) phase emerges. However, an extra tunneling process – the density-induced tunneling – is usually ignored in the standard BHM. Using process-chain method, we give a thorough study of the phase diagram of the BHM with density-induced tunneling in different particle density regions and spatial dimensions. We find the density-induced tunneling process can affect the SF-MI phase boundary dramatically, by suppressing the MI region and tune the tip of the phase boundary to lower chemical potential. Our unbiased numerical study gives benchmark results of the phase diagram of the BHM with density-induced tunneling.     

Dynamical conductivity of the two-dimensional Bose condensate: Superfluid-insulator transition for a long-range random potential

We calculate the dynamical conductivity of a disordered charged Bose condensate in two dimensions with a long-range random potential due to charged impurities with a large spacer width . Analytical results for the frequency-dependent conductivity for weak disorder are derived. For strong disorder the frequency-dependent conductivity is given in terms of a transcendental equation. The disorder-induced transition from a superfluid phase to an insulator phase is discussed. The density-density relaxation function and the screening properties of the disordered Bose gas are calculated. Experimental results for high-T _c superconductors are discussed.     

Impurity-induced phase transition in the interacting Bose gas: 1. Analytical results

We derive selfconsistency equations for the density relaxation and the longitudinal dynamical conductivity of the interacting Bose gas at temperature zero moving in a random potential. The equations describe a disorder-induced transition from a superfluid phase to an insulating phase, where the density is non-ergodic. The interaction of the bosons is treated in random phase approximation and the coupling to the impurities is calculated within generalized selfconsistent current relaxation theory. Scaling laws are discussed and explicit results are presented for the repulsive Bosgas with neutral impurities and for the charged Bose gas with charged impurities.