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arXiv:cond-mat/9511004 · Physical review. B, Condensed matter

ε expansion of the conductivity at the superconductor-Mott-insulator transition

Rosario Fazio, Dario Zappalà

Published Apr 1, 1996• 30 citations
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Abstract

We study the critical behavior of the conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ at the zero-temperature superconductor-Mott-insulator transition in $d$ spacetime dimensions for a model of bosons with short-range interaction and no disorder. We obtain $\ensuremath{\sigma}({\ensuremath{\omega}}_{n})=(4\frac{{e}^{2}}{\ensuremath{\hbar}}){\ensuremath{\sigma}}_{\ensuremath{\epsilon}}{\ensuremath{\omega}}_{n}^{1\ensuremath{-}\ensuremath{\epsilon}}$, as predicted by the scaling theory, and the prefactor ${\ensuremath{\sigma}}_{\ensuremath{\epsilon}}$ is calculated in the $\ensuremath{\epsilon}$ expansion, to order ${\ensuremath{\epsilon}}^{2}(\ensuremath{\epsilon}=4\ensuremath{-}d)$. In two spatial dimensions ($d=3$), we find a value of the universal conductance ${\ensuremath{\sigma}}^{*}=0.315(4\frac{{e}^{2}}{h})$, in good agreement with the known Monte Carlo results.

DOI
10.1103/physrevb.53.r8883

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We study the critical behavior of the conductivity \ensuremathσ(\ensuremathω)\ensuremath{\sigma}(\ensuremath{\omega})\ensuremathσ(\ensuremathω) at the zero-temperature superconductor-Mott-insulator transition in ddd spacetime dimensions for a model of bosons with short-range interaction and no disorder. We obtain \ensuremathσ(\ensuremathωn)=(4e2\ensuremathℏ)\ensuremathσ\ensuremathϵ\ensuremathωn1\ensuremath−\ensuremathϵ\ensuremath{\sigma}({\ensuremath{\omega}}_{n})=(4\frac{{e}^{2}}{\ensuremath{\hbar}}){\ensuremath{\sigma}}_{\ensuremath{\epsilon}}{\ensuremath{\omega}}_{n}^{1\ensuremath{-}\ensuremath{\epsilon}}\ensuremathσ(\ensuremathωn​)=(4\ensuremathℏe2​)\ensuremathσ\ensuremathϵ​\ensuremathωn1\ensuremath−\ensuremathϵ​, as predicted by the scaling theory, and the prefactor \ensuremathσ\ensuremathϵ{\ensuremath{\sigma}}_{\ensuremath{\epsilon}}\ensuremathσ\ensuremathϵ​ is calculated in the \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ expansion, to order \ensuremathϵ2(\ensuremathϵ=4\ensuremath−d){\ensuremath{\epsilon}}^{2}(\ensuremath{\epsilon}=4\ensuremath{-}d)\ensuremathϵ2(\ensuremathϵ=4\ensuremath−d). In two spatial dimensions (d=3d=3d=3), we find a value of the universal conductance \ensuremathσ∗=0.315(4e2h){\ensuremath{\sigma}}^{*}=0.315(4\frac{{e}^{2}}{h})\ensuremathσ∗=0.315(4he2​), in good agreement with the known Monte Carlo results.
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.
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We investigate the superconductor-insulator phase transition in the two-dimensional boson Hubbard system with short-range interactions. Fluctuations of both the phase and the amplitude of the superfluid order parameter are included in the determination of the phase diagram at zero and finite temperatures. The mean-field phase boundary is compared to quantum Monte Carlo results. We also calculate the frequency-dependent conductivity in the vicinity of the phase transition and find it universal at the multicritical point.
I argue that the system of interacting bosons at zero temperature and in random external potential possesses a simple critical point that describes the proliferation of disorder-induced defects in the superfluid ground state, and that is located at weak disorder close to and above one dimension. This makes it possible to address the critical behavior at the superfluid-Bose glass transition in dirty boson systems by expanding around the lower critical dimension of d=1.d=1.d=1. Within the formulated renormalization procedure near d=1,d=1,d=1, the dynamical critical exponent is obtained exactly and the correlation length critical exponent is calculated as a Laurent series in the parameter $\sqrt{\ensuremath{\epsilon}},$ with $\ensuremath{\epsilon}=d\ensuremath{-}1:$ z=d,z=d,z=d, $\ensuremath{\nu}=1/\sqrt{3\ensuremath{\epsilon}}+O(\sqrt{\ensuremath{\epsilon}})$ for the short-range, and z=1,z=1,z=1, $\ensuremath{\nu}=\sqrt{2/3\ensuremath{\epsilon}}+O(\sqrt{\ensuremath{\epsilon}}),$ for the long-range Coulomb interaction between bosons. The identified critical point should be stable against the residual perturbations in the effective action for the superfluid, at least in dimensions 1< d< 2,1<~d<~2,1< d< 2, for both short-range and Coulomb interactions. For the superfluid-Mott insulator transition in the system in a periodic potential and at a commensurate density of bosons, I find $\ensuremath{\nu}=(\frac{1}{2}\sqrt{\ensuremath{\epsilon}})+\frac{1}{4}+O(\sqrt{\ensuremath{\epsilon}}),$ which yields a result in reasonable agreement with the known XY\mathrm{XY}XY critical exponent in d=2.d=2.d=2. The critical behavior of the superfluid density, phonon velocity, and the compressibility in the system with short-range interactions is discussed.
In contrast to the case of free fermions which have all states localized in two dimensions, dirty, repulsively-interacting bosons exhibit a zero-temperature localization phase transition between superconducting and insulating states as disorder is increased. The conductivity σ* = g (2 e ) 2 / h at the quantum critical point is a universal critical amplitude analogous to the universal jump in superfluid density at the classical Kosterlitz-Thouless transition. Using numerical Monte Carlo methods we compute the value of g for disorder-tuned transitions with and without long-range Coulomb interactions. We argue that this model is relevant to experiments on superconducting thin films and Josephson junction arrays.
A system of disordered spinless fermions infinitesimally above one dimension at zero temperature is studied using the perturbative renormalization-group theory to first order in disorder, and to second order in interaction. We find a superconductor-to-Anderson insulator quantum fixed point at an infinitesimal value of disorder, and calculate the correlation length and the dynamical exponents to the lowest order in \ensuremath{\epsilon} and in interaction. The scaling of conductivity with temperature and the behavior of characteristic temperature scales on both sides of the transition is determined.