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arXiv:1309.5635 · Physical Review Letters

Universal Conductivity in a Two-Dimensional Superfluid-to-Insulator Quantum Critical System

Kun Chen, Longxiang Liu, Youjin Deng, Lode Pollet, Nikolay V. Prokof'ev

Published Jan 23, 2014• 70 citations
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Abstract

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.

DOI
10.1103/physrevlett.112.030402

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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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Understanding the real time dynamics of quantum systems without quasiparticles constitutes an important yet challenging problem. We study the superfluid-insulator quantum-critical point of bosons on a two-dimensional lattice, a system whose excitations cannot be described in a quasiparticle basis. We present detailed quantum Monte Carlo results for two separate lattice realizations: their low-frequency conductivities are found to have the same universal dependence on imaginary frequency and temperature. We then use the structure of the real time dynamics of conformal field theories described by the holographic gauge/gravity duality to make progress on the dicult problem of analytically continuing the Monte Carlo data to real time. Our method yields quantitative and experimentally testable results on the frequency-dependent conductivity near the quantum critical point, and on the spectrum of quasinormal modes in the vicinity of the superfluid-insulator quantum phase transition. Extensions to other observables and universality classes are discussed.
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 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 compute the zero-temperature conductivity in the two-dimensional quantum O(N)\mathrm{O}(N)O(N) model using a nonperturbative functional renormalization-group approach. At the quantum critical point we find a universal conductivity ${\ensuremath{\sigma}}^{*}/{\ensuremath{\sigma}}_{Q}^{}$ (with ${\ensuremath{\sigma}}_{Q}^{}={q}^{2}/h$ the quantum of conductance and qqq the charge) in reasonable quantitative agreement with quantum Monte Carlo simulations and conformal bootstrap results. In the ordered phase the conductivity tensor is defined, when $N\ensuremath{\ge}3$, by two independent elements, ${\ensuremath{\sigma}}_{\mathrm{A}}(\ensuremath{\omega})$ and ${\ensuremath{\sigma}}_{\mathrm{B}}(\ensuremath{\omega})$, respectively associated with SO(N)\mathrm{SO}(N)SO(N) rotations which do and do not change the direction of the order parameter. Whereas ${\ensuremath{\sigma}}_{\mathrm{A}}(\ensuremath{\omega}\ensuremath{\rightarrow}0)$ corresponds to the response of a superfluid (or perfect inductance), the numerical solution of the flow equations shows that ${lim}_{\ensuremath{\omega}\ensuremath{\rightarrow}0}{\ensuremath{\sigma}}_{\mathrm{B}}(\ensuremath{\omega})/{\ensuremath{\sigma}}_{Q}^{}={\ensuremath{\sigma}}_{\mathrm{B}}^{*}/{\ensuremath{\sigma}}_{Q}^{}$ is a superuniversal (i.e., NNN-independent) constant. These numerical results, as well as the known exact value ${\ensuremath{\sigma}}_{\mathrm{B}}^{*}/{\ensuremath{\sigma}}_{Q}^{}=\ensuremath{\pi}/8$ in the large-NNN limit, allow us to conjecture that ${\ensuremath{\sigma}}_{\mathrm{B}}^{*}/{\ensuremath{\sigma}}_{Q}^{}=\ensuremath{\pi}/8$ holds for all values of NNN, a result that can be understood as a consequence of gauge invariance and asymptotic freedom of the Goldstone bosons in the low-energy limit.
In a clean quantum critical metal, and in the absence of umklapp, most d.c. conductivities are formally infinite due to momentum conservation. However, there is a particular combination of the charge and heat currents which has a finite, universal conductivity. In this paper, we describe the physics of this conductivity σ Q in quantum critical metals obtained by charge doping a strongly interacting conformal field theory. We show that it satisfies an Einstein relation and controls the diffusivity of a conserved charge in the metal. We compute σ Q in a class of theories with holographic gravitational duals. Finally, we show how the temperature scaling of σ Q depends on certain critical exponents characterizing the quantum critical metal. The holographic results are found to be reproduced by the scaling analysis, with the charge density operator becoming marginal in the emergent low energy quantum critical theory.