Superuniversal transport near a $(2 + 1)$-dimensional quantum critical point
Résumé
We compute the zero-temperature conductivity in the two-dimensional quantum O(N) model using a nonperturbative functional renormalization-group approach. At the quantum critical point we find a universal conductivity σ*/σQ (with σQ=q2/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≥3, by two independent elements, σA(ω) and σB(ω), respectively associated with SO(N) rotations which do and do not change the direction of the order parameter. Whereas σA(ω→0) corresponds to the response of a superfluid (or perfect inductance), the numerical solution of the flow equations shows that limω→0σB(ω)/σQ=σB*/σQ is a superuniversal (i.e., N-independent) constant. These numerical results, as well as the known exact value σB*/σQ=π/8 in the large-N limit, allow us to conjecture that σB*/σQ=π/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.