Nonequilibrium transport of helical Luttinger liquids through a quantum dot

Abstract

We study a steady state non-equilibrium transport between two interacting helical edge states of a two dimensional topological insulator, described by helical Luttinger liquids, through a quantum dot. For non-interacting dot the current is obtained analytically by including the self-energy correction to the dot Green's function. For interacting dot we use equation of motion method to study the influence of weak on-site Coulomb interaction on the transport. We find the metal-to-insulator quantum phase transition for attractive or repulsive interactions in the leads when the magnitude of the interaction strength characterized by a charge sector Luttinger parameter K goes beyond a critical value. The critical Luttinger parameter Kcr depends on the hoping strength between dot and the leads as well as the energy level of the dot with respect to the Fermi levels of the leads, ranging from weak interaction regime for dot level off resonance to strong interaction regime for dot in resonance with the equilibrium Fermi level. Nearby the transition various singular behaviors of current noise, dot density of state, and the decoherence rate (inverse of lifetime) of the dot are briefly discussed.

Tunneling between helical Majorana modes and helical Luttinger liquids

“…Increasing the repulsive interaction strength leads to the suppression of the differential conductance on resonance and shifts the weight away from resonance. This feature is similar to the case of two Luttinger leads connected by a noninteracting quantum dot 36 with particle hole-symmetric bias voltage (µ 1 = −µ 2 = eV /2).…”

Decoherence patterns of topological qubits from Majorana modes

“…For the fermionic environment we consider the setup of a quantum wire [17,18] tunnel coupled to superconducting wires as shown in figure 1. We take the helical Luttinger wire [48], which can be realized as the edge state of some two-dimensional topological insulator [46], as a special example, but the generalization to other kinds of Luttinger liquids [47] is straightforward. For the bosonic environment we take the ring structure [19], as shown in figure 2, with external magnetic flux Φ controlling different frequency modes of bosonic couplings.…”

“…The above picture of a quantum phase transition might be understood via the renormalization group (RG) argument for the coupling constant B M . Note that the fermionic environment that we choose in this paper is the helical Luttinger liquids, which can be realized as interacting edge states of two-dimensional topological insulators [46,48]. For this kind of fermionic environment, the scaling dimension from zeroth-order renormalization group analysis [48,49] for the interaction V is κ.…”

“…a a 0 with Γ ≃ v a 0 0 as the UV cutoff for the linear spectrum of the edge state. For the electric transport discussed in [48], κ = 1 indicates the critical interaction strength for metallicinsulating behavior. That is, following equation (36) we see that for κ ⩾ ⩾ 1 12 the coupling B a increases as the cutoff decreases, while for κ > 1 the coupling B a decreases with decreasing cutoff, indicating insulating behavior for the charge transport.…”

Nonlinear spin-thermoelectric transport in two-dimensional topological insulators

“…A feasible possibility is the application of local potentials to form quantum antidots. More generally, the presence of constrictions in two-dimensional topological insulators have been proposed to give rise to coherent oscillations [18], transformations between ordinary and topological regimes [19], peaks of noise correlations [20], metal-to-insulator quantum phase transitions [21], nonequilibrium fluctuation relations [22], braiding of Majorana fermions [23], competition between Fabry-Pérot and Mach-Zehnder processes [24], control of edge magnetization [25], and detection of Kondo clouds [26]. Interestingly, König et al have experimentally demonstrated [27] the local manipulation of helical states with back-gate electrodes.…”