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 $\omega \to 0$ 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 $N\ge 3$, by two independent elements, ${\sigma }_{\mathrm{A}}(\omega )$ and ${\sigma }_{\mathrm{B}}(\omega )$, 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 ${\sigma }_{\mathrm{A}}(\omega )$. We show that ${\lim }_{\omega \to 0}\sigma (\omega ,\delta ){\sigma }_{\mathrm{A}}(\omega ,-\delta )/{\sigma }_q^2$ 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 ${\sigma }_{\mathrm{B}}(\omega \to 0)/{\sigma }_q$ is universal in the whole ordered phase, independent of $N$ and, when $N\to \infty$, equal to the universal conductivity ${\sigma }^{\ast }/{\sigma }_q$ at the quantum critical point.

arXiv comment25 pages, 4 figures

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