The poor man's Majorana tetron

Abstract

The Majorana tetron is a prototypical topological qubit stemming from the ground state degeneracy of a superconducting island hosting four Majorana modes. This degeneracy manifests as an effective non-local spin degree of freedom, whose most paradigmatic signature is the topological Kondo effect. Degeneracies of states with different fermionic parities characterize also minimal Kitaev chains which have lately emerged as a platform to realize and study unprotected versions of Majorana modes, dubbed poor man's Majorana modes. Here, we introduce the ``poor man's Majorana tetron'', comprising four quantum dots coupled via a floating superconducting island. Its charging energy yields non-trivial correlations among the dots, although, unlike a standard tetron, it is not directly determined by the fermionic parity of the Majorana modes. The poor man's tetron displays parameter regions with a two-fold degenerate ground state with odd fermionic parity, that gives rise to an effective Anderson impurity model when coupled to external leads. We show that this system can approach a regime featuring the topological Kondo effect under a suitable tuning of experimental parameters. Therefore, the poor man's tetron is a promising device to observe the non-locality of Majorana modes and their related fractional conductance.

Figures (9)

• Figure 1: (a) Sketch of the poor man's tetron: two semiconducting nanowires (blue) are connected by a shared SC floating island (silver); a set of electrostatic gates is used to define a quantum dot $D_i$ on each side of each nanowire (dashed circles). The superconducting segments of the nanowires are treated as proximitized dots $PD_{{{\sf u}},{{\sf d}}}$ (orange ellipses). Gate voltages are used to control the external dot energy levels $\mu_D$ ($V_i$ voltages), and the induced charge $n_g$ ($V_{g\tau}$ voltages). The voltage $V_{\rm bg}$ of a backgate is used to tune the potential $\mu_{\rm SC}$ in the proximitized dots. The device is then connected with four external leads (lateral gold contacts) to explore the emerging Kondo physics. (b) Schematic representation of the Hamiltonian terms of the poor man's tetron. The arrows indicate the sign of the spin-orbit coupling in the tunneling term.
• Figure 2: Energy of the states of the superconducting island for $N=0,1,2$ excess electrons as a function of $n_g$. The parabolas describe the charging energy in Eq. \ref{['charging']}. For odd particle numbers the energy is lifted by the SC gap $\tilde{\Delta}$. The energy difference between $N=2$ and $N=0$ states must be counterbalanced by the dot potential for optimal CAR processes (purple dashed line).
• Figure 3: Alternation of the parity sectors that express the global ground state of the poor man's tetron as a function of $n_g$ and $\mu$. The parameters are: $\Delta=4t$, $\mu_{SC}=-5t$, $E_C=0.15t$, $\alpha=\pi/6$. The dashed line corresponds to the constraint \ref{['tuning']} and the energies in Fig. \ref{['fig:spectrum']}. The labels indicate the particle number $N_t$ and the parity $P_{{\sf d}}$.
• Figure 4: Energies of the ground states of the parity sectors of the poor man's tetron with $N_t=1,\ldots,5$ in the perturbative regime as a function of the induced charge $n_g$. The labels indicate the particle number $N_t$ and the parity $P_{{\sf d}}$. The chemical potential $\mu(n_g)$ is chosen to fulfill Eq. \ref{['tuning']}. The other parameters are $\Delta=4t$, $\mu_{SC}=-5t$, $E_C=0.15t$, $\alpha=\pi/6$. For this choice of parameters $\Delta_{\rm CAR}\approx 0.087t$, $t_{\rm COT}(0) \approx -0.056t$ and $t_{\rm COT}(2) \approx -0.063t$.
• Figure 5: Relative error $\varepsilon/A_+$ (blue) and energy gap $\delta E$ (orange) between the even excited states and the odd ground states as a function of $\mu_{\rm SC}$. The data are calculated by considering the optimal induced charge $n_g$ fulfilling the constraint \ref{['tuningTK']}. The other system parameters are the same as Fig. \ref{['fig:spectrum']}.