We use a large-N renormalization-group (RG) method to study a model of interacting boson system with a quenched random potential. In the absence of impurities, the pure boson system has a critical point that describes the superfluid--Mott-insulator (SF--MI) transition. The SF--MI transition of d-dimensional bosons belongs to the (d+1)-dimensional XY model universality class. In this paper, we study the dirty-boson critical points in the neighborhood of this pure SF--MI critical point. In general, the on-site random potential in the original lattice model gives two types of randomness in the effective-field-theoretic action. One is the randomness in the effective on-site repulsion w(x) and the other is the randomness of the chemical potential u(x). It turns out that d=2 is the critical dimension for both types of disorder but the roles of these two types of disorder are reversed as d=2 is crossed. Applying \ensuremath{\epsilon}=d-2 expansion, we found coupled RG equations for both kinds of randomness which reveal several nontrivial critical points. All the weak random fixed points we found have three or more relevant directions. We conclude that the direct SF--MI transition is unlikely to occur near two dimensions.
We use a large-N renormalization-group (RG) method to study a model of interacting boson system with a quenched random potential. In the absence of impurities, the pure boson system has a critical point that describes the superfluid--Mott-insulator (SF--MI) transition. The SF--MI transition of d-dimensional bosons belongs to the (d+1)-dimensional XY model universality class. In this paper, we study the dirty-boson critical points in the neighborhood of this pure SF--MI critical point. In general, the on-site random potential in the original lattice model gives two types of randomness in the effective-field-theoretic action. One is the randomness in the effective on-site repulsion w(x) and the other is the randomness of the chemical potential u(x). It turns out that d=2 is the critical dimension for both types of disorder but the roles of these two types of disorder are reversed as d=2 is crossed. Applying \ensuremath{\epsilon}=d-2 expansion, we found coupled RG equations for both kinds of randomness which reveal several nontrivial critical points. All the weak random fixed points we found have three or more relevant directions. We conclude that the direct SF--MI transition is unlikely to occur near two dimensions.
The large-N commensurate dirty boson model, in both the weakly and strongly commensurate cases, is considered via a perturbative renormalization-group treatment. In the weakly commensurate case, there exists a fixed line under renormalization-group flow, with varying amounts of disorder along the line. Including 1/N corrections causes the system to flow to strong disorder, indicating that the model does not have a phase transition perturbatively connected to the Mott insulator-superfluid (MI-SF) transition. I discuss the qualitative effects of instantons on the low-energy density of excitations. In the strongly commensurate case, a fixed point found previously is considered and results are obtained for higher moments of the correlation functions. To lowest order, correlation functions have a log-normal distribution. Finally, I prove two interesting theorems for large-N vector models with disorder, relevant to the problem of replica symmetry breaking and frustration in such systems.
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.
We study the zero-temperature superfluid-insulator transition for a two-dimensional model of interacting, lattice bosons in the presence of quenched disorder and particle-hole symmetry. We follow the approach of a recent series of papers by Altman, Kafri, Polkovnikov, and Refael, in which the strong disorder renormalization group is used to study disordered bosons in one dimension. Adapting this method to two dimensions, we study several different species of disorder and uncover universal features of the superfluid-insulator transition. In particular, we locate an unstable finite disorder fixed point that governs the transition between the superfluid and a gapless, glassy insulator. We present numerical evidence that this glassy phase is the incompressible Mott glass and that the transition from this phase to the superfluid is driven by a percolation-type process. Finally, we provide estimates of the critical exponents governing this transition.
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.
We study the superfluid-insulator transition in a one-dimensional system of interacting bosons, modeled as a disordered Josephson array, using a strong-randomness real-space renormalization-group technique. Unlike perturbative methods, this approach does not suffer from run-away flows and allows us to study the complete phase diagram. We show that the superfluid-insulator transition is always Kosterlitz-Thouless like in the way that length and time scales diverge at the critical point. Interestingly however, we find that the transition at strong disorder occurs at a nonuniversal value of the Luttinger parameter, which depends on the disorder strength. This result places the transition in a universality class different from the weak disorder transition first analyzed by Giamarchi and Schulz [Europhys. Lett. 3, 1287 (1987)]. While the details of the disorder potential are unimportant at the critical point, the type of disorder does influence the properties of the insulating phases. We find three classes of insulators which arise for different classes of disorder potential. For disorder only in the charging energies and Josephson coupling constants, at integer filling we find an incompressible but gapless Mott-glass phase. If both integer and half-integer filling factors are allowed then the corresponding phase is a random-singlet insulator, which has a divergent compressibility. Finally in a generic disorder potential the insulator is a Bose glass with a finite compressibility.
We present a nonperturbative renormalization-group approach to the Bose-Hubbard model. By taking as initial condition of the renormalization-group flow the (local) limit of decoupled sites, we take into account both local and long-distance fluctuations in a nontrivial way. This approach yields a phase diagram in very good quantitative agreement with quantum Monte Carlo simulations, and reproduces the two universality classes of the superfluid-Mott-insulator transition. The critical behavior near the multicritical points, where the transition takes place at constant density, agrees with the original predictions of Fisher et al. [Phys. Rev. B 40, 546 (1989)] based on simple scaling arguments. At a generic transition point, the critical behavior is mean-field like with logarithmic corrections in two dimensions. In the weakly correlated superfluid phase (far away from the Mott insulating phase), the renormalization-group flow is controlled by the Bogoliubov fixed point down to a characteristic (Ginzburg) momentum scale kG, which is much smaller than the inverse healing length kh. In the vicinity of the multicritical points, when the density is commensurate, we identify a sharp crossover from a weakly to a strongly correlated superfluid phase where the condensate density and the superfluid stiffness are strongly suppressed and both kG and kh are of the order of the inverse lattice spacing.