We study the ground state phases of Bose-Fermi mixtures in one-dimensional optical lattices with quantum Monte Carlo simulations using the canonical worm algorithm. Depending on the filling of bosons and fermions, and the on-site intra- and interspecies interaction, different kinds of incompressible and superfluid phases appear. On the compressible side, correlations between bosons and fermions can lead to a distinctive behavior of the bosonic superfluid density and the fermionic stiffness, as well as of the equal-time Green functions, which allow one to identify regions where the two species exhibit anticorrelated flow. We present here complete phase diagrams for these systems at different fillings and as a function of the interaction parameters.
In this paper we provide a detailed description of the ground-state phase diagram of interacting, disordered bosons on a lattice. We describe a quantum Monte Carlo algorithm that incorporates in an efficient manner the required bosonic wave-function symmetry. We consider the ordered case, where we evaluate the compressibility gap and show the lowest three Mott insulating lobes. We obtain the critical ratio of interaction strength to hopping at which the onset of superfluidity occurs for the first lobe, and the critical exponents \ensuremath{\nu} and z. For the disordered model we show the effect of randomness on the phase diagram and the superfluid correlations. We also measure the response of the superfluid density, ${\mathrm{\ensuremath{\rho}}}_{\mathit{s}}$, to external perturbations. This provides an unambiguous characterization of the recently observed Bose and Anderson glass phases.
I argue that the system of interacting bosons at zero temperature and in random external potential possesses a simple critical point that describes the proliferation of disorder-induced defects in the superfluid ground state, and that is located at weak disorder close to and above one dimension. This makes it possible to address the critical behavior at the superfluid-Bose glass transition in dirty boson systems by expanding around the lower critical dimension of
d=1. Within the formulated renormalization procedure near
d=1, the dynamical critical exponent is obtained exactly and the correlation length critical exponent is calculated as a Laurent series in the parameter $\sqrt{\ensuremath{\epsilon}},$ with $\ensuremath{\epsilon}=d\ensuremath{-}1:$
z=d, $\ensuremath{\nu}=1/\sqrt{3\ensuremath{\epsilon}}+O(\sqrt{\ensuremath{\epsilon}})$ for the short-range, and
z=1, $\ensuremath{\nu}=\sqrt{2/3\ensuremath{\epsilon}}+O(\sqrt{\ensuremath{\epsilon}}),$ for the long-range Coulomb interaction between bosons. The identified critical point should be stable against the residual perturbations in the effective action for the superfluid, at least in dimensions
1< d< 2, for both short-range and Coulomb interactions. For the superfluid-Mott insulator transition in the system in a periodic potential and at a commensurate density of bosons, I find $\ensuremath{\nu}=(\frac{1}{2}\sqrt{\ensuremath{\epsilon}})+\frac{1}{4}+O(\sqrt{\ensuremath{\epsilon}}),$ which yields a result in reasonable agreement with the known
XY critical exponent in
d=2. The critical behavior of the superfluid density, phonon velocity, and the compressibility in the system with short-range interactions is discussed.
The phase diagrams and phase transitions of bosons with short-ranged repulsive interactions moving in periodic and/or random external potentials at zero temperature are investigated with emphasis on the superfluid-insulator transition induced by varying a parameter such as the density. Bosons in periodic potentials (e.g., on a lattice) at T=0 exhibit two types of phases: a superfluid phase and Mott insulating phases characterized by integer (or commensurate) boson densities, by the existence of a gap for particle-hole excitations, and by zero compressibility. Generically, the superfluid onset transition in d dimensions from a Mott insulator to superfluidity is ``ideal,'' or mean field in character, but at special multicritical points with particle-hole symmetry it is in the universality class of the (d+1)-dimensional XY model. In the presence of disorder, a third, ``Bose glass'' phase exists. This phase is insulating because of the localization effects of the randomness and analogous to the Fermi glass phase of interacting fermions in a strongly disordered potential.The Bose glass phase is characterized by a finite compressibility, no gap, but an infinite superfluid susceptibility. In the presence of disorder the transition to superfluidity is argued to occur only from the Bose glass phase, and never directly from the Mott insulator. This zero-temperature superfluid-insulator transition is studied via generalizations of the Josephson scaling relation for the superfluid density at the ordinary \ensuremath{\lambda} transition, highlighting the crucial role of quantum fluctuations. The transition is found to have a dynamic critical exponent z exactly equal to d and correlation length and order-parameter correlation exponents \ensuremath{\nu} and \ensuremath{\eta} which satisfy the bounds \ensuremath{\nu}\ensuremath{\ge}2/d and \ensuremath{\eta}\ensuremath{\le}2-d, respectively. It is argued that the superfluid-insulator transition in the presence of disorder may have an upper critical dimension
dc which is infinite, but a perturbative renormalization-group calculation wherein the critical exponents have mean-field values for weak disorder above d=4 is also discussed. Many of these conclusions are verified by explicit calculations on a model of one-dimensional bosons in the presence of both random and periodic potentials. The general results are applied to experiments on
4He absorbed in porous media such as Vycor. Some measurable properties of the superfluid onset are predicted exactly [e.g., the exponent x relating the \ensuremath{\lambda} transition temperature to the zero-temperature superfluid density is found to be d/2(d-1)], while stringent bounds are placed on others. Analysis of preliminary data is consistent with these predictions.
The values of the insulator gap Δ in one-dimensional systems of interacting bosons described by the Hubbard Hamiltonian are calculated at low temperatures by the quantum world-line Monte Carlo algorithm. The dependence of Δ on the size of the system, the temperature, and the parameters of the model is investigated. It is shown that a chain with N a =50 sites is already sufficient to estimate the thermodynamic value of the critical quantity ( t/U ) c for which a transition from the insulator into the superfluid state occurs in a commensurate system. To within the computational error, this value, ( t/U ) c =0.300±0.005, agrees with the value ( t/U ) c =0.304±0.002 obtained previously by the combined “exact diagonalization + renormalization-group analysis” method. The characteristic Kosterlitz-Thouless behavior of the insulator gap is demonstrated near the critical region: Δ∼exp[− b (1− t/t c ) −1/2 ].
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.