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arXiv:cond-mat/9704134 · Physical Review Letters

Dual Superfluid-Bose-Glass Critical Point in Two Dimensions and the Universal Conductivity

Igor F. Herbut

Published Nov 3, 1997• 40 citations
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Abstract

The continuum version of the dual theory for a system of two-dimensional quantum disordered bosons with statistical particle-hole symmetry and at $T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$ admits a new disordered critical point within the renormalization group calculation at fixed dimension. We obtain the universal conductivity and the critical exponents at the superfluid-Bose glass transition in the system: ${\ensuremath{\sigma}}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.25(2e{)}^{2}/h$, $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.38$, and $z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.93$, to the lowest order in fixed-point values of the dual coupling constants.

DOI
10.1103/physrevlett.79.3502

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The continuum version of the dual theory for a system of two-dimensional quantum disordered bosons with statistical particle-hole symmetry and at T=0T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0T=0 admits a new disordered critical point within the renormalization group calculation at fixed dimension. We obtain the universal conductivity and the critical exponents at the superfluid-Bose glass transition in the system: \ensuremathσc=0.25(2e)2/h{\ensuremath{\sigma}}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.25(2e{)}^{2}/h\ensuremathσc​=0.25(2e)2/h, \ensuremathν=1.38\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.38\ensuremathν=1.38, and z=1.93z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.93z=1.93, to the lowest order in fixed-point values of the dual coupling constants.
I study the zero-temperature phase transition between the superfluid and the insulating ground states of the Bose-Hubbard model in a random chemical potential and at large integer average number of particles per site. Duality transformation maps the pure Bose-Hubbard model onto the sine-Gordon theory in one dimension (1D), and onto the three dimensional Higgs electrodynamics in two dimensions (2D). In 1D the random chemical potential in the dual theory couples to the space derivative of the dual field, and appears as a random magnetic field along the imaginary time direction in 2D. I show that the transition from the superfluid state both in 1D and 2D is always controlled by the random critical point. This arises due to a coupling constant in the dual theory with replicas that becomes generated at large distances by the random chemical potential, and represents a relevant perturbation at the pure superfluid-Mott insulator fixed point. At large distances the dual theory in 1D becomes equivalent to Haldane's macroscopic representation of a disordered quantum fluid, where the generated term is identified with the random backscattering. In 2D the generated coupling corresponds to the random mass of the complex field that represents vortex loops. I calculate the critical exponents at the superfluid-Bose-glass fixed point in 2D to be $\ensuremath{\nu}=1.38$ and z=1.93z=1.93z=1.93, and the universal conductivity at the transition ${\ensuremath{\sigma}}_{c}{=0.25e}_{*}^{2}/h$, using the one-loop field-theoretic renormalization group in fixed dimension.
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 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 ${\mathrm{\ensuremath{\sigma}}}_{\mathit{Q}}$=e∗2{\mathit{e}}_{\mathrm{*}}^{2}e∗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.
The ground-state of hard-core bosons in a random one-body potential is studied numerically to observe the transition from superfluid to insulating ``Bose glass'' phase. Finite-size scaling analyses are performed to estimate the transition point and critical exponents. The dynamical critical exponent is z=1.95\ifmmode\pm\else\textpm\fi{}0.25 and is consistent with the prediction of Fisher et al. [Phys. Rev. B 40, 546 (1989)]. Values for other exponents and preliminary results regarding the universal conductivity are presented.
The generic Hamiltonian describing the zero temperature transition between the insulating Bose glass phase and the superfluid phase lacks particle-hole symmetry, but a statistical version of this symmetry is believed to be restored at the critical point. We show that the renormalization group relevance of particle-hole asymmetry may be explored in a controlled fashion only for small time dimensions, ${\ensuremath{\epsilon}}_{\ensuremath{\tau}}\ensuremath{\ll}1$, where we find a stable particle-hole asymmetric and an unstable particle-hole symmetric fixed point, but we provide evidence that the two merge for some finite ${\ensuremath{\epsilon}}_{\ensuremath{\tau}}\ensuremath{\approx}\frac{2}{3}$, which tends to confirm symmetry restoration at the physical ${\ensuremath{\epsilon}}_{\ensuremath{\tau}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$.