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 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., non-dynamical) 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, σ A (ω) and σ B (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 σ A (ω). We show that limω→0 σ(ω, δ)σ A (ω, −δ)/σ 2 q 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 = q 2 /h the quantum of conductance). On the other hand we argue that the ratio σ B (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.

Criticality of the low-frequency conductivity for the bilayer quantum Heisenberg model

“…Our result supports the recent NPRG results 0.441 [3] and 0.401 [5]. Likewise, as for σ q /2πC∆ and C/Lσ 2 q , our results agree with those of the recent NPRG study [3], 1.98 and 0.2226, respectively. The latter suggests that a seemingly feasible relation C/Lσ 2 q = 1 is not validated quantitatively.…”

“…[30]. Second, for σ q /2πC∆, the NPRG-DE analysis [3] reported 1.98. Additionally, we draw reader's attention to its O(2) counterpart 1.98 as well.…”

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

“…Using Γ (1,1) i (p, φ) = φ i f (p, ρ) and Γ (0,2) (p, φ) = γ(p, ρ), where f and γ are functions of |p| and ρ, 23 we obtain…”

“…The conductivity can be calculated in a similar way. 23 However, to respect local gauge invariance, one must use the gauge-invariant regulator term ∆S k [ϕ, A] obtained from ∆S k [ϕ] by replacing the derivative ∂ µ by the covariant derivative D µ . The scale-dependent effective action is defined as a Legendre transform wrt the source J (but not A) and satisfies the flow equation…”

“…These functions are not independent but are related by Ward identities. 23 To obtain the frequency-dependent conductivity in the quantum model, one sets p = (0, 0, ω n ) and µ, ν ∈ {x, y} so that p µ = p ν = 0 and K ab µν (iω n ) = −Γ (0,2)ab µν (iω n ,φ) is fully determined by Ψ B (iω n ) and Ψ C (iω n ). In the disordered phase (ρ 0 =φ 2 /2 = 0), the conductivity tensor…”