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Physical Review Letters

Percolation-Enhanced Localization in the Disordered Bosonic Hubbard Model

K. Sheshadri, H. R. Krishnamurthy, Rahul Anil Pandit, T. V. Ramakrishnan

Published Nov 27, 1995• 65 citations
Publisher

Abstract

We present a new (mean-field-RPA) study of the bosonic Hubbard model with chemical-potential disorder $\ensuremath{\Delta}$ at zero temperature. We get the fractional-filling phase diagram by monitoring site-dependent superfluid order parameters ${\ensuremath{\psi}}_{i}$ and fillings ${n}_{i}$. Our theory suggests a new physical picture for the superfluid-Bose glass transition, as a localization transition enhanced by the percolation of sites with negligible ${\ensuremath{\psi}}_{i}$ at a critical ${\ensuremath{\Delta}}_{c}$: for $\ensuremath{\Delta}<{\ensuremath{\Delta}}_{c}$ sites with substantial ${\ensuremath{\psi}}_{i}'\mathrm{s}$ percolate across the lattice; for $\ensuremath{\Delta}>{\ensuremath{\Delta}}_{c}$ this percolation stops. This picture, though only approximately correct especially in low dimensions, should be relevant for Monte Carlo studies of the disordered bosonic Hubbard model.

DOI
10.1103/physrevlett.75.4075

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We present a new (mean-field-RPA) study of the bosonic Hubbard model with chemical-potential disorder \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ at zero temperature. We get the fractional-filling phase diagram by monitoring site-dependent superfluid order parameters \ensuremathψi{\ensuremath{\psi}}_{i}\ensuremathψi​ and fillings ni{n}_{i}ni​. Our theory suggests a new physical picture for the superfluid-Bose glass transition, as a localization transition enhanced by the percolation of sites with negligible \ensuremathψi{\ensuremath{\psi}}_{i}\ensuremathψi​ at a critical \ensuremathΔc{\ensuremath{\Delta}}_{c}\ensuremathΔc​: for \ensuremathΔ<\ensuremathΔc\ensuremath{\Delta}<{\ensuremath{\Delta}}_{c}\ensuremathΔ<\ensuremathΔc​ sites with substantial \ensuremathψi′s{\ensuremath{\psi}}_{i}'\mathrm{s}\ensuremathψi′​s percolate across the lattice; for \ensuremathΔ>\ensuremathΔc\ensuremath{\Delta}>{\ensuremath{\Delta}}_{c}\ensuremathΔ>\ensuremathΔc​ this percolation stops. This picture, though only approximately correct especially in low dimensions, should be relevant for Monte Carlo studies of the disordered bosonic Hubbard model.
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 study the local density of the Bose-Hubbard model in the presence of on-site disorder near the Bose-glass transition using multifractal, typical medium, and percolation theories. At incommensurate filling our findings support the scenario of percolating superfluid clusters enhancing Anderson localization. Scaling analysis of the superfluid density at the incommensurate filling of $\ensuremath{\rho}=1.1$ and on-site interaction U=80tU=80tU=80t predicts a superfluid-Bose-glass transition at disorder strength of ${\mathrm{\ensuremath{\Delta}}}_{c}\ensuremath{\approx}30t$. At this filling the local-density distribution becomes more skew with increasing disorder strength. Multifractal analysis suggests a multifractal behavior resembling that of the Anderson localization. In the Bose-glass phase the mode of the local-density distribution approaches an integer value as expected from typical medium theory for the Anderson localization. Percolation analysis points to a phase transition of percolating noninteger filled sites around the same value of disorder. On the other hand, the behavior at commensurate filling is rather different. Close to the tip of the Mott lobe ($\ensuremath{\rho}=1,U=22t$) we find a Mott-insulator-Bose-glass transition at disorder strength of ${\mathrm{\ensuremath{\Delta}}}_{c}\ensuremath{\approx}16t$. An analysis of the local-density distribution shows Gaussian-like behavior for a wide range of disorders above and below the transition.
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.
The zero temperature phase diagram for ultracold bosons in a random 1D potential is obtained through a site decoupling mean-field scheme performed over a Bose-Hubbard (BH) Hamiltonian, whose hopping term is considered as a random variable. As for the model with random on-site potential, the presence of disorder leads to the appearance of a Bose glass phase. The different phases—i.e., Mott insulator, superfluid, and Bose glass—are characterized in terms of condensate fraction and superfluid fraction. Furthermore, the boundary of the Mott lobes is related to an off-diagonal Anderson model featuring the same disorder distribution as the original BH Hamiltonian.
The introduction of disorder in Bose-Hubbard model gives rise to new glassy quantum phases, namely the Bose-glass (BG) and disordered solid (DS) phases. In this work, we present the rich phase diagram of interacting bosons in disordered two-dimensional optical lattice, modelled by the disordered Bose-Hubbard model. We systematically probe the effect of long-range interaction truncated to the nearest neighbors and next-nearest neighbors on the phase diagram. We investigate the zero-temperature ground-state quantum phases using the single-site Gutzwiller mean field (SGMF) theory. We also employ strong-coupling perturbative expansion to identify the nature of ground-state solid phases analytically. At sufficiently high disorder strength, we observe a quantum phase transition between the DS and BG phases. We have investigated this transition in greater detail using cluster Gutzwiller mean field theory to study the effect of inter-site correlations which is absent in the SGMF method. We have also studied this phase transition from the perspective of percolation theory.
The introduction of disorder in the extended Bose-Hubbard model gives rise to new glassy quantum phases, namely the Bose-glass and disordered solid phases. In this work, we present a rich phase diagram of interacting bosons in a disordered two-dimensional optical lattice, modeled by the disordered Bose-Hubbard model. We systematically probe the effect of long-range interaction truncated to the nearest neighbors and next-nearest neighbors on the phase diagram. We investigate the zero-temperature ground-state quantum phases using the single-site Gutzwiller mean-field theory. We also employ strong-coupling perturbative expansion to identify the nature of ground-state solid phases analytically. At sufficiently high disorder strength, we observe a quantum phase transition between the disordered solid and Bose-glass phases. We have investigated this transition in greater detail using cluster Gutzwiller mean-field theory to study the effect of intersite correlations which is absent in the single-site Gutzwiller mean-field theory. We have also studied this phase transition from the perspective of percolation theory.