Nonperturbative functional renormalization-group approach to transport in the vicinity of a $(2+1)$-dimensional O($N$)-symmetric quantum critical point

Abstract

Using a nonperturbative functional renormalization-group approach to the two-dimensional quantum O($N$) model, we compute the low-frequency limit $ω\to 0$ of the zero-temperature conductivity in the vicinity of the quantum critical point. Our results are obtained from a derivative expansion to second order of a scale-dependent effective action in the presence of an external (i.e., non-dynamical) non-Abelian gauge field. While in the disordered phase the conductivity tensor $σ(ω)$ is diagonal, in the ordered phase it is defined, when $N\geq 3$, by two independent elements, $σ_{\rm A}(ω)$ and $σ_{\rm B}(ω)$, respectively associated to SO($N$) rotations which do and do not change the direction of the order parameter. For $N=2$, the conductivity in the ordered phase reduces to a single component $σ_{\rm A}(ω)$. We show that $\lim_{ω\to 0}σ(ω,δ)σ_{\rm A}(ω,-δ)/σ_q^2$ is a universal number which we compute as a function of $N$ ($δ$ measures the distance to the quantum critical point, $q$ is the charge and $σ_q=q^2/h$ the quantum of conductance). On the other hand we argue that the ratio $σ_{\rm B}(ω\to 0)/σ_q$ is universal in the whole ordered phase, independent of $N$ and, when $N\to\infty$, equal to the universal conductivity $σ^*/σ_q$ at the quantum critical point.