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

TL;DR

This paper employs a nonperturbative functional renormalization-group method to analyze the low-frequency conductivity near a (2+1)-dimensional O(N) quantum critical point, revealing universal ratios and phase-dependent conductivity behaviors.

Contribution

It introduces a derivative expansion approach within the functional renormalization-group framework to compute universal conductivity ratios near quantum criticality in the O(N) model.

Findings

Universal number for the product of conductivities in the disordered phase.

Universal ratio of conductivity components in the ordered phase.

Connection between ordered phase conductivity and quantum critical point conductivity at large N.

Abstract

Using a nonperturbative functional renormalization-group approach to the two-dimensional quantum O($N$) model, we compute the low-frequency limit $\omega\to 0$ 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 $\sigma(\omega)$ is diagonal, in the ordered phase it is defined, when $N\geq 3$, by two independent elements, $\sigma_{\rm A}(\omega)$ and $\sigma_{\rm B}(\omega)$, 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 $\sigma_{\rm A}(\omega)$. We show that $\lim_{\omega\to 0}\sigma(\omega,\delta)\sigma_{\rm A}(\omega,-\delta)/\sigma_q^2$ is a universal number which we compute as a function of $N$ ($\delta$ measures the distance to the quantum critical point, $q$ is the charge and $\sigma_q=q^2/h$ the quantum of conductance). On the other hand we argue that the ratio $\sigma_{\rm B}(\omega\to 0)/\sigma_q$ is universal in the whole ordered phase, independent of $N$ and, when $N\to\infty$, equal to the universal conductivity $\sigma^*/\sigma_q$ at the quantum critical point.