Superuniversal transport near a (2+1) -dimensional quantum critical point

Abstract

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 * /σ Q (with σ Q = q 2 /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→0 σB(ω)/σ Q = σ * B /σ Q is a superuniversal (i.e. N -independent) constant. These numerical results, as well as the known exact value σ * B /σ Q = π/8 in the large-N limit, allow us to conjecture that σ * B /σ Q = 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.

Nonperturbative renormalization-group approach preserving the momentum dependence of correlation functions

“…In the disordered phase the solution of the RG flow for the conductivity differs from the exact solution. 18 In the ordered phase, to leading order in 1/N , σ A (ω) is determined by ρ 0,k and Z k (p), which reproduce the exact solution in the large-N limit. No simple analytic form has been found for the next-to-leading order contribution to σ A (ω) which depends on X 1,k and X 2,k .…”

“…An important result obtained by the LPA is the superuniversality of one of the elements of the conductivity tensor, σ * B /σ Q = π/8, in the ordered phase. 18 Finally we would like to point out that the LPA might offer the possibility to avoid the analytic continuation of numerical data using Padé approximants (or alternative methods). Indeed, by approximating the propagators in the internal loops of the flow equations by their LPA expressions, it becomes possible to perform exactly both Matsubara-frequency sums and analytic continuation to real frequencies, [59][60][61][62][63][64][65] which would allow to obtain the frequency dependence of correlation functions in the hydrodynamic regime |ω| T .…”

“…Flow equations for the vertices are obtained by taking functional derivatives of Eqs. (11), (15) and (18). Replacing the vertices by their LPA expressions, we derive equations for the various functions of interest: W k (ρ) = U k (ρ), Z k (p), Y k (p), f k (p), γ k (p), X 1,k (p) and X 2,k (p).…”

“…17 It was used in Ref. 18 for the calculation of the conductivity of the two-dimensional quantum O(N ) model in order to circumvent the failure of BMW and DE approximation schemes. 19 Our aim is to benchmark the LPA considering as a test-bed the two-dimensional quantum O(N ≥ 2) model at zero temperature.…”

Operator product expansion coefficients from the nonperturbative functional renormalization group

“…(n,m) k can be obtained from (30) by taking functional derivatives wrt φ and h. The presence of the source h in addition to the field φ allows one to follow the flow of composite fields, an approach which proved to be useful in tackling a wide range of issues [51,[62][63][64][65][66][67][68][69][70].…”