Nonperturbative functional renormalization-group approach to transport in the vicinity of a $(2+1)$-dimensional O($N$)-symmetric quantum critical point

Using a nonperturbative functional renormalization-group approach to the two-dimensional quantum $\mathrm{O}(N$) 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., nondynamical) non-Abelian gauge field. While in the disordered phase the conductivity tensor $\sigma \left(\omega \right)$ is diagonal, in the ordered phase it is defined, when $N\ge 3$, by two independent elements, ${\sigma }_{\mathrm{A}}\left(\omega \right)$ and ${\sigma }_{\mathrm{B}}\left(\omega \right)$, respectively associated to $\mathrm{SO}(N$) 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 ${\sigma }_{\mathrm{A}}\left(\omega \right)$. 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 ${\sigma }^{*}/{\sigma }_q$ at the quantum critical point.

Figure 1

(a) A vertex ${\mathrm{\Gamma }}^{(n,m)}$ is represented by a black dot with $n$ solid lines and $m$ wavy lines. (b) Diagrammatic representation of $K_{\mu \nu }^{ab}$ [Eq. (33)]. Solid lines connecting vertices stand for the propagator $G={\mathrm{\Gamma }}^{(2,0)-1}$.

Figure 2

Diagrammatic representation of the RG equations (67). Signs and symmetry factors are not shown. The diagrammatic representation of the vertices is shown in Fig. 1 and the cross with $n$ wavy lines stands for ${\partial }_t{R}_k,{\partial }_t{\mathcal{R}}_k^{\left(1\right)}$, and ${\partial }_t{\mathcal{R}}_k^{\left(2\right)}$ for $n=0,1,2$, respectively.

Figure 3

Dimensionless fixed-point solutions of the flow equations at the QCP (left) and the ordered phase (right) for several values of $N$. The quantities are, respectively, expressed as a function of $\tilde{\rho }$ and $\delta \tilde{\rho }=\tilde{\rho }-{\tilde{\rho }}_{0,k}$ at the critical point and in the ordered phase. In the latter case, the fixed-point solutions ${\tilde{Z}}^{*}\left(\delta \tilde{\rho }\right)=1$ and ${\tilde{X}}_2^{*}\left(\delta \tilde{\rho }\right)=0$ are independent of $N$.

Figure 4

Flow of ${\tilde{X}}_{1,k}\left({\tilde{\rho }}_{0,k}\right)$ in the ordered phase for $N=3$ and $N=10$, and three different initial conditions, with $\alpha =2.25$. The solid lines show the system at criticality where, after a transient regime, ${\tilde{X}}_{1,k}\left({\tilde{\rho }}_{0,k}\right)$ becomes very close to its fixed point value ${\tilde{X}}_{1,k}^{*\mathrm{cr}}\left({\tilde{\rho }}_0^{*}\right)$. The dashed and dotted lines correspond to the ordered phase: ${\tilde{X}}_{1,k}\left({\tilde{\rho }}_{0,k}\right)$ is first attracted by the critical fixed point; for $k\sim {\xi }_J^{-1}$ there is a crossover to a regime dominated by the Goldstone modes and ${\tilde{X}}_{1,k}\left({\tilde{\rho }}_{0,k}\right)\simeq {\tilde{X}}_{1,k}^{*\mathrm{ord}}\left({\tilde{\rho }}_{0,k}\right)$.