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arXiv:cond-mat/9302037 · Physical review. B, Condensed matter

Universal conductivity in the two-dimensional boson Hubbard model

G. G. Batrouni, B. E. Larson, Richard T. Scalettar, Jan Tobochnik, J. Wang

Published Oct 1, 1993• 51 citations
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Abstract

We use quantum Monte Carlo methods to evaluate the conductivity \ensuremath{\sigma} of the two-dimensional disordered boson Hubbard model at the superfluid--Bose-glass phase boundary. At the critical point for particle density \ensuremath{\rho}=0.5, we find ${\mathrm{\ensuremath{\sigma}}}_{\mathit{c}}$=(0.45\ifmmode\pm\else\textpm\fi{}0.07)${\mathrm{\ensuremath{\sigma}}}_{\mathit{Q}}$, where ${\mathrm{\ensuremath{\sigma}}}_{\mathit{Q}}$=${\mathit{e}}_{\mathrm{*}}^{2}$/h from a finite-size scaling analysis of the superfluid density. We obtain ${\mathrm{\ensuremath{\sigma}}}_{\mathit{c}}$=(0.47\ifmmode\pm\else\textpm\fi{}0.08)${\mathrm{\ensuremath{\sigma}}}_{\mathit{Q}}$ from a direct calculation of the current-current correlation function. Simulations at the critical points for other particle densities, \ensuremath{\rho}=0.75 and 1.0, give similar values for \ensuremath{\sigma}. We discuss possible origins of the difference in this value from that recently obtained by other numerical approaches.

DOI
10.1103/physrevb.48.9628

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We use quantum Monte Carlo methods to evaluate the conductivity \ensuremath{\sigma} of the two-dimensional disordered boson Hubbard model at the superfluid--Bose-glass phase boundary. At the critical point for particle density \ensuremath{\rho}=0.5, we find \ensuremathσc{\mathrm{\ensuremath{\sigma}}}_{\mathit{c}}\ensuremathσc​=(0.45\ifmmode\pm\else\textpm\fi{}0.07)\ensuremathσQ{\mathrm{\ensuremath{\sigma}}}_{\mathit{Q}}\ensuremathσQ​, where \ensuremathσQ{\mathrm{\ensuremath{\sigma}}}_{\mathit{Q}}\ensuremathσQ​=e∗2{\mathit{e}}_{\mathrm{*}}^{2}e∗2​/h from a finite-size scaling analysis of the superfluid density. We obtain \ensuremathσc{\mathrm{\ensuremath{\sigma}}}_{\mathit{c}}\ensuremathσc​=(0.47\ifmmode\pm\else\textpm\fi{}0.08)\ensuremathσQ{\mathrm{\ensuremath{\sigma}}}_{\mathit{Q}}\ensuremathσQ​ from a direct calculation of the current-current correlation function. Simulations at the critical points for other particle densities, \ensuremath{\rho}=0.75 and 1.0, give similar values for \ensuremath{\sigma}. We discuss possible origins of the difference in this value from that recently obtained by other numerical approaches.
We compute the universal conductivity of the (2+1)-dimensional XY universality class, which is realized for a superfluid-to-Mott insulator quantum phase transition at constant density. Based on large-scale Monte Carlo simulations of the classical (2+1)-dimensional J-current model and the two-dimensional Bose-Hubbard model, we can precisely determine the conductivity on the quantum critical plateau, σ(∞) = 0.359(4)σQ with σQ the conductivity quantum. The universal conductivity curve is the standard example with the lowest number of components where the bottoms-up AdS/CFT correspondence from string theory can be tested and made to use [R. C. Myers, S. Sachdev, and A. Singh, Phys. Rev. D 83, 066017 (2011)]. For the first time, the shape of the σ(iω(n)) - σ(∞) function in the Matsubara representation is accurate enough for a conclusive comparison and establishes the particlelike nature of charge transport. We find that the holographic gauge-gravity duality theory for transport properties can be made compatible with the data if temperature of the horizon of the black brane is different from the temperature of the conformal field theory. The requirements for measuring the universal conductivity in a cold gas experiment are also determined by our calculation.
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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 zero-temperature universal conductivity of two-dimensional (2D) films at the supeconductor-insulator transition is studied. The existence of a finite conductivity at T=0 and the universality class for this transition is discussed. Neglecting disorder as a first approximation, so the transition is from a commensurate Mott-Hubbard insulator to a superconductor, we calculate analytically the universal conductivity for the 2D pure boson Hubbard model up to the first order in a large-N expansion and numerically by Monte Carlo simulation of the (2+1)-D XY model. From the Monte Carlo results we find the universal conductivity to be ${\mathrm{\ensuremath{\sigma}}}^{\mathrm{*}}$=(0.285\ifmmode\pm\else\textpm\fi{}0.02)${\mathrm{\ensuremath{\sigma}}}_{\mathit{Q}}$, where ${\mathrm{\ensuremath{\sigma}}}_{\mathit{Q}}^{\mathrm{\ensuremath{-}}1}$==RQ{\mathit{R}}_{\mathit{Q}}RQ​==h/(2e)2{)}^{2})2\ensuremath{\approxeq}6.45 k\ensuremath{\Omega}. An analysis in 1D suggests that in the presence of disorder, the universal conductivity in films might be somewhat smaller than this value. The possible existence of universal dissipation in 4He^{4}\mathrm{He}4He films is also discussed briefly.
We investigate the superfluid (SF) to Bose-glass (BG) quantum phase transition using extensive quantum Monte Carlo simulations of two-dimensional hard-core bosons in a random box potential. T=0 critical properties are studied by thorough finite-size scaling of condensate and SF densities, both vanishing at the same critical disorder Wc=4.80(5). Our results give the following estimates for the critical exponents: z=1.85(15), ν=1.20(12), η=-0.40(15). Furthermore, the probability distribution of the SF response P(lnρSF) displays striking differences across the transition: while it narrows with increasing system sizes L in the SF phase, it broadens in the BG regime, indicating an absence of self-averaging, and at the critical point P(lnρSF+zlnL) is scale invariant. Finally, high-precision measurements of the local density rule out a percolation picture for the SF-BG transition.
We study the phase diagram of two-dimensional, interacting bosons in the presence of a correlated disorder in continuous space, by using large-scale quantum Monte Carlo simulations at finite temperature. We show that the superfluid transition is strongly protected against disorder. It remains of the Berezinskii-Kosterlitz-Thouless type up to disorder strengths comparable to the chemical potential. Moreover, we study the transport properties in the strong disorder regime where a zero-temperature Bose-glass phase is expected. We show that the conductance exhibits a thermally activated behavior vanishing only at zero temperature. Our results point towards the existence of a Bose bad-metal phase as a precursor of the Bose-glass phase.