Nonzero temperature transport near fractional quantum Hall critical points

TL;DR

Nonzero temperature transport near a fractional quantum Hall critical point is analyzed using the Chen–Fisher–Wu theory of a Dirac anyon gas in a periodic potential. The authors show that the longitudinal and Hall conductivities at criticality are universal functions of $\hbar \omega / k_B T$, computed by combining a perturbative Kubo analysis with a quantum Boltzmann equation for anyons, extending prior $T=0$ results to finite $T$ and to the d.c. limit: $\sigma_{xx}(\omega)=\frac{e^2}{h} Σ_{xx}\left(\frac{\hbar \omega}{k_B T}\right)$ and $\sigma_{xy}(\omega)=\frac{e^2}{h} Σ_{xy}\left(\frac{\hbar \omega}{k_B T}\right)$. A delta-function contribution to $\widetilde{\sigma}_{xx}^{\mathrm{qp}}$ is broadened by inelastic collisions, yielding a collision-dominated scaling regime with universal functions that describe the full $\omega/T$ dependence. The work highlights the essential role of incoherent, inelastic processes at interacting quantum critical points in two dimensions and provides a calculable framework that can be extended to large-$N$ and to Coulomb-interacting variants with potential experimental relevance in microwave and transport probes.

Abstract

In an earlier work, Damle and the author (Phys. Rev. B in press; cond-mat/9705206) demonstrated the central role played by incoherent, inelastic processes in transport near two-dimensional quantum critical points. This paper extends these results to the case of a quantum transition in an anyon gas between a fractional quantized Hall state and an insulator, induced by varying the strength of an external periodic potential. We use the quantum field theory for this transition introduced by Chen, Fisher and Wu (Phys. Rev. B 48, 13749 (1993)). The longitudinal and Hall conductivities at the critical point are both $e^2/ h$ times non-trivial, fully universal functions of $\hbar ω/ k_B T$ ($ω$ is the measuring frequency). These functions are computed using a combination of perturbation theory on the Kubo formula, and the solution of a quantum Boltzmann equation for the anyonic quasiparticles and quasiholes. The results include the values of the d.c. conductivities ($\hbar ω/k_B T \to 0$); earlier work had been restricted strictly to T=0, and had therefore computed only the high frequency a.c. conductivities with $\hbar ω/ k_B T \to \infty$.

Figures (7)

• Figure 1: Real (full line) and imaginary (dashed line) parts of the perturbative result (\ref{['ecoh']}) for the conductivity $\widetilde{\sigma}_{xx}^{{\rm coh}} ( \omega )$ as a function of $\omega/T$ evaluated at $\alpha = 0.3$. The conductivity is measured in units of $q^2 e^2/h$. The singularity is at $\omega = 2 M (T)$, and is artifact of the absence of damping at this order. Note, as in (\ref{['sxxsum']}), the conductivity $\widetilde{\sigma}_{xx}$ contains another quasiparticle/quasihole contribution which is a delta function in the perturbative approach.
• Figure 2: Real and imaginary parts of the perturbative results for the Hall conductivity (\ref{['sxy1']}), (\ref{['sxy2']}) and (\ref{['sxy3']}) for $\widetilde{\sigma}_{xy} ( \omega ) = \widetilde{\sigma}_{xy}^{(1)} + \widetilde{\sigma}_{xy}^{(2)} +\widetilde{\sigma}_{xy}^{(3)}$ as a function of the $\omega/T$ evaluated at $\alpha = 0.3$. The conductivity is measured in units of $q^2 e^2/h$. Again the spurious singularities are at $\omega = 2 M(T)$. Note that unlike, $\widetilde{\sigma}_{xx}$, the Hall conductivity is adequately described by the perturbation theory.
• Figure 3: Real and imaginary parts of the solution for the universal function $k^2 G/T^2$, introduced in (\ref{['scaleg']}), as a function of $k/T$ for a few values of $\omega/ \alpha^2 T$. The combination $k^2 G/T^2$ is that appearing in the integrand of the integral (\ref{['intscale']}) for the conductivity.
• Figure 4: Real and imaginary parts of the universal function $\widetilde{\Sigma}_{xx}^{{\rm qp}}$ as a function of $\omega / \alpha^2 T$. This result is related to $\widetilde{\sigma}_{xx}^{{\rm qp}}$ by (\ref{['scalesqp']}).
• Figure 5: Real part of $1/\widetilde{\Sigma}_{xx}^{{\rm qp}}$ as a function of $\omega / \alpha^2 T$.