Interaction effects on a Majorana zero mode leaking into a quantum dot

We have recently shown [Phys. Rev. B 89, 165314 (2014)] that a noninteracting quantum dot coupled to a one-dimensional topological superconductor and to normal leads can sustain a Majorana mode even when the dot is expected to be empty, i.e., when the dot energy level is far above the Fermi level of the leads. This is due to the Majorana bound state of the wire leaking into the quantum dot. Here we extend this previous work by investigating the low-temperature quantum transport through an interacting quantum dot connected to source and drain leads and side coupled to a topological wire. We explore the signatures of a Majorana zero mode leaking into the quantum dot for a wide range of dot parameters, using a recursive Green's function approach. We then study the Kondo regime using numerical renormalization group calculations. We observe the interplay between the Majorana mode and the Kondo effect for different dot-wire coupling strengths, gate voltages, and Zeeman fields. Our results show that a “0.5” conductance signature appears in the dot despite the interplay between the leaked Majorana mode and the Kondo effect. This robust feature persists for a wide range of dot parameters, even when the Kondo correlations are suppressed by Zeeman fields and/or gate voltages. The Kondo effect, on the other hand, is suppressed by both Zeeman fields and gate voltages. We show that the zero-bias conductance as a function of the magnetic field follows a well-known universality curve. This can be measured experimentally, and we propose that the universal conductance drop followed by a persistent conductance of $0.5e^2/h$ is evidence of the presence of Majorana-Kondo physics. These results confirm that this “0.5” Majorana signature in the dot remains even in the presence of the Kondo effect.

Figure 1

(a) Schematic representation of a QD coupled to one end of a topological quantum wire. The quantum wire is described by a tight-binding chain with hopping parameter $t$, Rashba spin-orbit interaction $\alpha$, and induced superconducting pairing $\Delta$. The QD is modeled as a single orbital of energy ${\varepsilon }_{\text{dot}}$ with local Coulomb interaction $U$, coupled to metallic leads with a coupling $V_{\mathbf{k}}$. The energy shift in the QD level from the applied local gate voltage is given by $V_g$. An applied magnetic field induces a Zeeman splitting $V_Z$ in the wire and $V_Z^{\left(\text{dot}\right)}$ in the QD, where $V_Z\ne V_Z^{\left(\text{dot}\right)}$ because of different $g$ factors, $g_{\text{wire}}\ne g_{\text{dot}}$. (b) Single-particle energy level structure of the QD as a function of $V_Z^{\left(\text{dot}\right)}$ for a fixed gate voltage. (c) A spinless regime can be accessed through the application of a large magnetic field in the QD. For comparison with the spinful case $(V_Z^{\left(\text{dot}\right)}=0)$, the spin-down QD level ${\varepsilon }_{0,\downarrow }$ can be fixed at the energy ${\varepsilon }_{\text{dot}}$ by simultaneously applying a gate voltage $V_g=V_Z^{\left(\text{dot}\right)}$ for positive $V_Z^{\left(\text{dot}\right)}$.

Figure 2

Density of states at the QD site as a function of the Zeeman splitting in the wire, calculated using the Hubbard I method. The top panels [(a) and (c)] are for the QD $g$ factor $g_{\text{dot}}=0$, whereas the bottom panels [(b) and (d)] correspond to $g_{\text{dot}}=0.1g_{\text{wire}}$. Results for the noninteracting case $\left(U=0\right)$ are presented in panels (a) and (b); results for the interacting case are shown in panels (c) and (d). These calculations are carried out with the QD spin-down level fixed at ${\varepsilon }_{0,\downarrow }={\varepsilon }_{\text{dot}}$. For this purpose, a compensating gate voltage $V_g=V_Z^{\left(\text{dot}\right)}$ is applied (see the discussion in Sec. 3c). Parameters: $t=10\mathrm{meV},E_{\mathrm{SO}}=50\mu \mathrm{eV},\Delta =250\mu \mathrm{eV},\tilde{\mu }=-0.01t$; $\Gamma =1\mu \mathrm{eV},{\varepsilon }_{\text{dot}}=-6.25\Gamma$, and $t_0=40\Gamma$.

Figure 3

Spin-up local density of states of the QD for the wire in the topological phase, with $t=10\mathrm{meV},E_{\mathrm{SO}}=50\mu \mathrm{eV},V_Z=500\mu \mathrm{eV}$, and $\tilde{\mu }=-0.01t$. QD parameters are $\Gamma =1\mu \mathrm{eV}$ and $t_0=40\Gamma$.

Figure 4

Spin-down local density of states of the QD for the wire in the topological phase for the same parameters of Fig. 3. Note the reduced amplitude and the shift toward negative energies of the central peak in panel (d) for finite $U$ and ${\varepsilon }_{0,s}<0$.

Figure 5

Spin-down local density of states at the QD site for the microscopic model Eq. (2) (solid line) and for the effective model Eq. (16) (squares). The parameters are the same as in Fig. 4 but with a finite Zeeman energy $V_Z^{\left(\mathrm{dot}\right)}=0.1V_Z=50\mu \mathrm{eV}$. A gate voltage $V_g=V_Z^{\left(\text{dot}\right)}$ is also applied in order to fix ${\varepsilon }_{0,\downarrow }={\varepsilon }_{\text{dot}}$ for comparison with Fig. 4.

Figure 6

NRG calculations of the zero-temperature spin-up local density of states at the QD site, in the absence of a magnetic field $[V_Z^{\left(\text{dot}\right)}=0]$. The interacting (noninteracting) case is presented in the right (left) panels, where the Coulomb interaction is $U=12.5\Gamma$ $\left(U=0\right)$. The QD level position is indicated in the panels, and the Majorana-QD coupling is $\lambda =0.707\Gamma$.

Figure 7

NRG calculations of the zero-temperature spin-down local density of states at the QD site, in the absence of a magnetic field $[V_Z^{\left(\text{dot}\right)}=0]$. The interacting (noninteracting) case is presented in the right (left) panels, where the Coulomb interaction is $U=12.5\Gamma$ $\left(U=0\right)$. The QD level position is indicated in the panels, and the Majorana-QD coupling is $\lambda =0.707\Gamma$.

Figure 8

NRG calculations of the zero-temperature spin-down local density of states at the QD site in the presence of a strong Zeeman field $V_Z^{\left(\text{dot}\right)}$ within the QD. Parameters: $U=12.5\Gamma ,\lambda =\Gamma$.

Figure 9

Extracting the Majorana energy scale $T_M$ from the spin-down density of states, for three values of $\lambda$. The spin-down DOS is shown close to the Fermi level in panel (a). In panel (b) it is shown with the energy axis in logarithmic scale. For $T_M\gg T_K$ (triangles), $T_M$ is given by the width at half-maximum of the “0.5” peak. In that case, $T_M\sim {10}^{-1}\Gamma$. When $T_M\ll T_K$ (squares), $T_M$ is given by the width of the dip that indicates the onset of the Majorana-dominated regime, in this case giving $T_M\sim {10}^{-7}\Gamma$. In the Kondo-Majorana crossover (circles) $T_M$ cannot be clearly distinguished. (c) $T_K$ and $T_M$ in units of $\Gamma$, as functions of $\lambda$. The Kondo temperature was extracted from the spin-up density of states as the width at half-maximum of the Kondo peak.

Figure 10

Conductance effects of the detuning from particle-hole symmetry produced by a gate voltage $V_g$. For $|V_Z^{\left(M\right)}|\lesssim T_K$, the spin-up DOS at the Fermi level drops as the Kondo peak is shifted to positive (negative) values for $V_g>0$ $\left(V_g<0\right)$, due to the Majorana-induced Zeeman splitting $V_Z^{\left(M\right)}$. This is shown in panels (a) and (b), respectively. The Kondo effect finally disappears for $|V_g|\gtrsim U/2$ due to the charge fluctuations of the mixed-valence regime. The conductance effects of the Kondo quench are shown in panel (c). The enhanced conductance at zero detuning $\left(V_g=0\right)$ comes from both the “0.5” and the Kondo peaks, whereas after the suppression of the Kondo effect the persistent $0.5G_0$ conductance comes only from the “0.5” peak. Parameters: ${\varepsilon }_{\text{dot}}=-U/2=-6.25\Gamma$; $V_Z^{\left(\text{dot}\right)}=0$.

Figure 11

Low-temperature zero-bias conductance of the QD as a function of the Zeeman splitting $V_Z^{\left(\text{dot}\right)}$. Inset: Universality curve resulting from the rescaling $V_Z^{\left(\text{dot}\right)}/T_K\left(\lambda \right)$. $T_K\left(\lambda \right)$ was obtained from Fig. 9. The dashed curve corresponds to the spin-up conductance for $\lambda =0$ and is given as a reference. Parameters: ${\varepsilon }_{\text{dot}}=-U/2=-6.25\Gamma$.

Figure 12

Quantum dot occupancy (charge) and spin polarization for the parameters of Fig. 10.