We study the superfluid-insulator transition of the 2-D boson system on random surfaces. Using quantum Monte Carlo simulation and grand-canonical ensemble, we investigate the random Hubbard model and study the phase transition between the superfluid phase and the Bose glass insulator phase. Changing the chemical potential µ, we observe the transition from the superfluid phase to the insulator phase. We show the dimensional dependence of the transition by comparing the results of 2-D and 1-D boson systems. To study the localization of the boson paths, we perform the simulation on the anisotropic surfaces, and discuss the crossover from 2-D system to 1-D system.
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 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.
In this dissertation, we study the superfluid-insulator quantum phase transition in disordered boson systems. Recently, there has been considerable controversy over the validity of the scaling relations of the superfluid--Bose-glass quantum phase transition in three dimensions. Results from experimental and numerical studies on disordered quantum magnets contradict the scaling relations and the associated conventional scaling hypothesis for the singular part of the free energy. We determine various critical exponents of the superfluid--Bose-glass quantum phase transition in three-dimensional disordered Bose-Hubbard model through extensive Monte Carlo simulations. Our numerical study shows the previous studies on disordered quantum magnets were performed outside the quantum critical region, and our results for the critical exponents are in perfect agreement with previous theoretical predictions. We next move to study theoretically and numerically the superfluid-insulator quantum phase transition in one-dimensional disordered boson systems. While the superfluid-insulator transition in the weak disorder limit is well understood through the perturbative renormalization group study by Giamarchi and Schulz, transitions in the strong disorder regime are beyond the reach of this method. This problem was recently attacked by Altman et al. with the real space renormalization group method. They reached the conclusion that the superfluid-insulator transition in the strong disorder regime can be explained by the Coulomb blockade physics of weak links. However, their method is actually uncontrolled. Taking account of the crucial role of the hydrodynamic renormalization of weak links, finally, Pollet et al. put forward an asymptotically exact renormalization group theory of the superfluid-insulator transition. Based on this theory, we are able to provide an accurate description of the interplay between the well-known Giamarchi-Schulz criticality and the new weak-link criticality. A significant part of the ground-state phase diagram of one-dimensional disordered Bose-Hubbard model at unit filling is also determined numerically. In particular, we established the position of the multicritical point beyond which the new weak-link criticality takes over of the Giamarchi–Schulz criticality.
We study the zero-temperature phase transition of a two-dimensional disordered boson Hubbard model. The phase diagram is constructed in terms of the disorder strength and the chemical potential. Via Monte Carlo simulations, we find a multicritical line separating the weak-disorder regime, where the Mott-insulator-to-superfluid transition occurs, from the strong-disorder regime, where the Bose-glass-to-superfluid transition occurs. On the multicritical line, the insulator-to-superfluid transition has the dynamical critical exponent $z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.35\ifmmode\pm\else\textpm\fi{}0.05$ and the correlation length critical exponent $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.67\ifmmode\pm\else\textpm\fi{}0.03$. We suggest that the proliferation of the particle-hole pairs screens out the weak-disorder effects.
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.