Analytical and Numerical study of the out-of-equilibrium current through a helical edge coupled to a magnetic impurity

TL;DR

This paper investigates the conductance of a helical edge coupled to a magnetic impurity, analyzing how anisotropic exchange interactions and finite bias influence the transition from perfect conductance to backscattering regimes using analytical and numerical methods.

Contribution

It provides a detailed analysis of out-of-equilibrium conductance in helical edges with magnetic impurities, highlighting the role of anisotropic exchange and the application of TD-NRG for strongly correlated systems.

Findings

Conductance follows a power-law decay $G \\sim (T/T_K)^2$ at low temperatures.

Anisotropic exchange coupling can break the $U(1)$ symmetry, affecting conductance.

Finite bias voltage cuts off the RG flow, influencing the crossover to the strong-coupling regime.

Abstract

We study the conductance of a time-reversal symmetric helical electronic edge coupled antiferromagnetically to a magnetic impurity, employing analytical and numerical approaches. The impurity can reduce the perfect conductance $G_0$ of a noninteracting helical edge by generating a backscattered current. The backscattered steady-state current tends to vanish below the Kondo temperature $T_K$ for time-reversal symmetric setups. We show that the central role in maintaining the perfect conductance is played by a global $U(1)$ symmetry. This symmetry can be broken by an anisotropic exchange coupling of the helical modes to the local impurity. Such anisotropy, in general, dynamically vanishes during the renormalization group (RG) flow to the strong coupling limit at low-temperatures. The role of the anisotropic exchange coupling is further studied using the time-dependent Numerical Renormalization Group (TD-NRG) method, uniquely suitable for calculating out-of-equilibrium observables of strongly correlated setups. We investigate the role of finite bias voltage and temperature in cutting the RG flow before the isotropic strong-coupling fixed point is reached, extract the relevant energy scales and the manner in which the crossover from the weakly interacting regime to the strong-coupling backscattering-free screened regime is manifested. Most notably, we find that at low temperatures the conductance of the backscattering current follows a power-law behavior $G\sim (T/T_K)^2$, which we understand as a strong nonlinear effect due to time-reversal symmetry breaking by the finite-bias.