Nonperturbative functional renormalization-group approach to transport in the vicinity of a (2+1) -dimensional O( N )-symmetric quantum critical point
Résumé
Using a nonperturbative functional renormalization-group approach to the two-dimensional quantum O(N) model, we compute the low-frequency limit ω→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., nondynamical) 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≥3, by two independent elements, σA(ω) and σB(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 σA(ω). We show that limω→0σ(ω,δ)σA(ω,−δ)/σq2 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 σq=q2/h the quantum of conductance). On the other hand we argue that the ratio σB(ω→0)/σq is universal in the whole ordered phase, independent of N and, when N→∞, equal to the universal conductivity σ*/σq at the quantum critical point.