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arXiv:cond-mat/0611710 · Europhysics Letters (EPL)

Superfluid–Bose-glass transition in two dimensions at T = 0: Analytic solution of a mode-coupling toy model

E. V Zenkov

Published Jan 20, 2005• 2 citations
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Abstract

Analytic expression for the memory function and the optical conductivity of the two-dimensional Bose gas with logarithmic interaction at T = 0 in the presence of point-like impurities is obtained within the mode-coupling approximation. Depending on the value of a dimensionless combination of the model parameters proportional to the strength of the impurity potential, two different phases are distinguished, viz the disordered superfluid and insulator (Bose glass), separated by an intermediate (quasi)metal state.

DOI
10.1209/epl/i2004-10348-5

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Analytic expression for the memory function and the optical conductivity of the two-dimensional Bose gas with logarithmic interaction at T = 0 in the presence of point-like impurities is obtained within the mode-coupling approximation. Depending on the value of a dimensionless combination of the model parameters proportional to the strength of the impurity potential, two different phases are distinguished, viz the disordered superfluid and insulator (Bose glass), separated by an intermediate (quasi)metal state.
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.
We investigate the superfluid-insulator transition in the disordered two-dimensional Bose-Hubbard model through quantum Monte Carlo simulations. The Bose-Hubbard model is studied in the presence of site disorder, and the quantum critical point between the Bose glass and superfluid is determined in the grand canonical ensemble at $\ensuremath{\mu}/U=0$ (close to $\ensuremath{\rho}=0.5$), $\ensuremath{\mu}/U=0.375$ (close to $\ensuremath{\rho}=1$), and $\ensuremath{\mu}/U=1$ as well as in the canonical ensemble at $\ensuremath{\rho}=0.5$ and 1. Particular attention is paid to disorder averaging, and it is shown that a large number of disorder realizations are needed in order to obtain reliable results. Typically, more than 100000100\phantom{\rule{0.16em}{0ex}}000100000 disorder realizations were used. In the grand canonical ensemble, we find Ztc/U=0.112(1)Z{t}_{c}/U=0.112(1)Ztc​/U=0.112(1) with $\ensuremath{\mu}/U=0.375$, significantly different from previous studies. When compared to the critical point in the absence of disorder (Ztc/U=0.2385Z{t}_{c}/U=0.2385Ztc​/U=0.2385), this result confirms previous findings showing that disorder enlarges the superfluid region. At the critical point, we then study the dynamic conductivity.
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{d}_{c}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^{4}\mathrm{He}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 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.
We consider the quantum critical properties of a disordered two dimensional Bose gas based on the renormalization group approach. To account for disorder, we consider that the Bose gas interacts with an external random field. The system’s behavior in the critical region of the quantum phase transition is obtained in the universality class z =2 and includes contributions beyond the Gaussian approximation by considering interactions between bosonic degrees of freedom. The evaluation of the system’s conductivity close to the transition point proves the occurrence of a nonzero conductivity (metallic-like) state in the system in the ω =0 and T →0 K limit.
We consider a one-dimensional system of interacting bosons in a random potential. At zero temperature, it can be either in the superfluid or in the insulating phase. We study the transition at weak disorder and moderate interaction. Using a systematic approach, we derive the renormalization group equations at two-loop order and discuss the phase diagram. We find the universal form of the correlation functions at the transitions and compute the logarithmic corrections to the main universal power-law behavior. In order to mimic large density fluctuations on a single site, we study a simplified model of disordered two-leg bosonic ladders with correlated disorder across the rung. Contrarily to the single-chain case, the latter system exhibits a transition between a superfluid and a localized phase where the exponents of the correlation functions at the transition do not take universal values.