Superuniversal transport near a $(2+1)$-dimensional quantum critical point

Abstract

We compute the zero-temperature conductivity in the two-dimensional quantum $\mathrm{O}(N)$ model using a nonperturbative functional renormalization-group approach. At the quantum critical point we find a universal conductivity ${\sigma }^{\ast }/{\sigma }_Q$ (with ${\sigma }_Q=q^2/h$ the quantum of conductance and $q$ 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 $N\ge 3$, by two independent elements, ${\sigma }_{\mathrm{A}}(\omega )$ and ${\sigma }_{\mathrm{B}}(\omega )$, respectively associated to $\mathrm{O}(N)$ rotations which do and do not change the direction of the order parameter. Whereas ${\sigma }_{\mathrm{A}}(\omega \to 0)$ corresponds to the response of a superfluid (or perfect inductance), the numerical solution of the flow equations shows that ${\lim }_{\omega \to 0}{\sigma }_{\mathrm{B}}(\omega )/{\sigma }_Q={\sigma }_{\mathrm{B}}^{\ast }/{\sigma }_Q$ is a superuniversal (i.e. $N$-independent) constant. These numerical results, as well as the known exact value ${\sigma }_{\mathrm{B}}^{\ast }/{\sigma }_Q=\pi /8$ in the large-$N$ limit, allow us to conjecture that ${\sigma }_{\mathrm{B}}^{\ast }/{\sigma }_Q=\pi /8$ holds for all values of $N$, a result that can be understood as a consequence of gauge invariance and asymptotic freedom of the Goldstone bosons in the low-energy limit.

6 pages, 4 figures, published version

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• Functional-renormalization-group approach to strongly coupled Bose-Fermi mixtures in two dimensions
• Operator product expansion coefficients from the nonperturbative functional renormalization group