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

Yuval Vinkler-Aviv, Daniel May, and Frithjof B. Anders

Phys. Rev. B 101, 165112 – Published 13 April 2020

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 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, and 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.

Received 12 December 2019
Accepted 20 March 2020

DOI:https://doi.org/10.1103/PhysRevB.101.165112

Physics Subject Headings (PhySH)

Kondo effect Spin Hall effect Topological insulators Transport phenomena

Nonequilibrium systems

Nonequilibrium Green's function Numerical Renormalization Group

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Yuval Vinkler-Aviv ¹, Daniel May², and Frithjof B. Anders²

¹University of Cologne, Institute for Theoretical Physics, 50937 Cologne, Germany
²Theoretische Physik 2, Technische Universität Dortmund, 44221 Dortmund, Germany

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Issue

Vol. 101, Iss. 16 — 15 April 2020

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Images

Figure 1
Schematic depiction of the setup considered throughout most of this paper. A 1d helical edge of a quantum spin Hall insulator consists of right-moving up-spin electrons and left-moving down-spin electrons, coupled at a single point to a generalized impurity via tunnel amplitude $t$ . This impurity encompasses a correlated spinful level interacting with an additional $S = 1 / 2$ quantum spin. Time-reversal symmetry is maintained by making the up and down level Kramers partners, and keeping a real exchange coupling elements $J$ with the impurity spin.
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Figure 2
(a) $ρ_{σ}^{r, SIAM}$ (red solid line) for finite magnetic field $B / Γ = 0.5$ compared to $ρ_{σ}^{r, helical}$ (blue dashed line) calculated for bias voltage $e V / Γ = 0.5$ and $J_{α β} = 0$ . The results for the helical model are shifted by an additional $\pm e V / 2$ . For both models $ε_{d} / Γ = - 5$ and $U / Γ = 10$ . (b) Local spin susceptibility $χ_{spin}$ for helical model in equilibrium $e V = 0$ , $ε_{d} / Γ = - 0.5, U = 0$ , and $J_{x x} = J_{y y} = Γ$ . (c) Equilibrium Kondo temperature calculated from the local spin susceptibility as a function of $J_{y y}$ for $J_{x x} = J_{z z} = Γ$ and $U = ε_{d} = 0$ .
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Figure 3
$G_{σ}^{<} (ω)$ (solid blue curves) and $2 ρ_{σ}^{r} (ω) f (ω - μ_{σ})$ (dashed green curves) for (i) the symmetrical point $J_{y y} / Γ = 1$ for spin (a) up and (b) down and (ii) for the broken $U (1)$ symmetry, $J_{y y} / Γ = 0.7$ , for spin (c) up and (d) down. $G_{σ}^{<}$ and $ρ_{σ}^{r}$ for consecutive bias voltages $e V$ are shifted by a constant offset $a = 0.25$ for better visibility. The legend applies to all subplots.
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Figure 4
(a) Linear conductivity $G = I_{B} / V$ as a function of voltage $e V / T_{K}^{eq}$ for different couplings $J_{y y}$ . $I_{B} = 0$ for the $U (1)$ -symmetric case $J_{y y} = Γ$ . The vertical black dashed line indicates $e V = T_{K}^{eq}$ . The other dashed lines represent the fits to Eq. (36). (b) Backscattering current $I_{B}$ as a function of $e V / T_{K}^{eq}$ . (c) Slopes $a_{slope}$ for the different fits in subplot (a). The error bars stem from the numerical fitting process. The values for $T_{K}^{eq}$ are shown in Fig. 2. In all cases $J_{x x} = J_{z z} = Γ$ .
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Figure 5
(a) Linear conductivity $G$ as a function of temperature $T / T_{K}^{eq}$ for a fixed $e V / Γ = 0.01$ and different couplings $J_{y y}$ . The black dashed line indicates $T_{K}^{eq}$ . Below $T / T_{K}^{eq} = 1$ the conductance $G \to 0$ when $e V < T_{K}^{eq}$ (yellow, green curves). The lines are a guide to the eye. (b) Power-law fit [Eq. (37)] to the data points (crosses) of (a) for $T / T_{K}^{eq} < 1$ . (c) Exponent $α$ of the power-law fit: $α \to 2$ (Fermi liquid) for $e V < T_{K}^{eq}$ . The error bars stem from the numerical fitting process.
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Figure 6
Low-temperature $G = I_{B} / V$ as a function of $Δ J_{y y}$ for different bias voltages $e V$ and $T \to 0$ . The upper subplot shows the corresponding equilibrium Kondo temperatures $T_{K}^{eq} (Δ J_{y y})$ .
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