Leakage of Majorana mode into correlated quantum dot nearby its singlet-doublet crossover

TL;DR

This paper investigates how Majorana modes leak into a correlated quantum dot near its singlet-doublet transition, revealing spin-dependent effects and the influence of quantum phase transitions on Majorana leakage.

Contribution

It provides a detailed analysis of Majorana mode leakage into a quantum dot with electron correlations, considering spin polarization and quantum phase transitions.

Findings

Majorana leakage depends on the quantum dot's ground state.

Spin polarization significantly affects Majorana-dot coupling.

Quantum phase transitions alter the spectral signatures of Majorana modes.

Abstract

We study quasiparticle spectrum of the correlated quantum dot deposited on superconducting substrate which is side-coupled to the Rashba nanochain, hosting Majorana end modes. Ground state of an isolated quantum dot proximitized to superconducting reservoirs is represented either by the singly occupied site or BCS-type superposition of the empty and doubly occupied configurations. Quantum phase transition between these distinct ground states is spectroscopically manifested by the in-gap Andreev states which cross each other at the Fermi level. This qualitatively affects leakage of the Majorana mode from the side-attached nanowire. We inspect the spin-selective relationship between the trivial Andreev states and the leaking Majorana mode, considering (i) perfectly polarized case, when tunneling of one spin component is completely prohibited, and (ii) another one when both spins are hybridized with the nanowire but with different couplings.

Equations

$$H_{QD}^{\mathrm{eff}}$$

$$\displaystyle\lim_{U=0}{\cal{G}}^{-1}(\omega)=\left(\begin{array}[]{cccccc}\omega-\epsilon+i\Gamma_{N}/2&0&0&\Gamma_{S}/2&-t_{m\uparrow}&-t_{m\uparrow}\\ 0&\omega+\epsilon+i\Gamma_{N}/2&-\Gamma_{S}/2&0&t_{m\uparrow}&t_{m\uparrow}\\ 0&-\Gamma_{S}/2&\omega-\epsilon+i\Gamma_{N}/2&0&-t_{m\downarrow}&-t_{m\downarrow}\\ \Gamma_{S}/2&0&0&\omega+\epsilon+i\Gamma_{N}/2&t_{m\downarrow}&t_{m\downarrow}\\ -t_{m\uparrow}&t_{m\uparrow}&-t_{m\downarrow}&t_{m\downarrow}&\omega-\epsilon_{m}&0\\ -t_{m\uparrow}&t_{m\uparrow}&-t_{m\downarrow}&t_{m\downarrow}&0&\omega+\epsilon_{m}\end{array}\right).$$

$$\displaystyle{\cal{G}}^{-1}(\omega)=\left(\begin{array}[]{cccccc}G(\omega)&0&0&F(\omega)&0&0\\ 0&-G^{*}(-\omega)&F^{*}(-\omega)&0&0&0\\ 0&F^{*}(-\omega)&G(\omega)&0&0&0\\ F(\omega)&0&0&-G^{*}(-\omega)&0&0\\ 0&0&0&0&\frac{1}{\omega-\epsilon_{m}}&0\\ 0&0&0&0&0&\frac{1}{\omega+\epsilon_{m}}\end{array}\right)^{-1}-\left(\begin{array}[]{cccccc}0&0&0&0&t_{m\uparrow}&t_{m\uparrow}\\ 0&0&0&0&-t_{m\uparrow}&-t_{m\uparrow}\\ 0&0&0&0&t_{m\downarrow}&t_{m\downarrow}\\ 0&0&0&0&-t_{m\downarrow}&-t_{m\downarrow}\\ t_{m\uparrow}&-t_{m\uparrow}&t_{m\downarrow}&-t_{m\downarrow}&0&0\\ t_{m\uparrow}&-t_{m\uparrow}&t_{m\downarrow}&-t_{m\downarrow}&0&0\end{array}\right).$$

$$G(\omega )$$

$$F(\omega )$$

$${\alpha }_s=\frac{e^{\beta U/2}+e^{-\beta E_d}}{2e^{\beta U/2}+e^{-\beta E_d}+e^{\beta E_d}}=1-{\beta }_s.$$

Full text

I Introduction

Recent development of the hybrid structures, comprising quantum dots (QDs) coupled the topologically superconducting nanowires Deng et al. (2016); Nichele et al. (2017); Deng et al. (2018); Kim et al. (2018) provide new challenges going beyond mere observation of the Majorana bound states (MBS). Since energy levels of QDs in such hybrid systems are experimentally tunable one can inspect interplay of the topological states with the correlation effects and proximity-induced electron pairing. Due to natural tendency of the Majorana modes to exist at boundaries of finite size systems, one may expect their leakage into any side-attached quantum dot (QD) Vernek et al. (2014) or more complex magnetic nanoislands Mascot et al. (2018).

It turns out, however, that efficiency of such process substantially depends on various parameters. For instance, in a weak coupling limit the Majorana modes show up either by the constructive or destructive interferometric lineshapes Górski et al. (2018). True leakage of Majorana mode into nontopological region is possible only in strongly hybridized structures Hoffman et al. (2017); Prada et al. (2017); Ptok et al. (2017a); Szumniak et al. (2017); Hoffman et al. (2017) as indeed reported experimentally Deng et al. (2016); Nichele et al. (2017); Deng et al. (2018); Kim et al. (2018). But even under such conditions, leakage of the Majorana mode into the side-coupled QD region could be affected by various effects, e.g. by correlations. We address such issue here, studying an interplay between electron pairing and Coulomb repulsion of the correlated QD proximitized to superconducting substrate where the quantum phase transition from the spinless (BCS-like) to the spinfull (singly occupied) configurations may occur. We analyze whether the Majorana mode would be able to leak into such configurations of the quantum dot.

The paper is organized as follows. We present the physical situation of our interest (Sec. II) and introduce the relevant microscopic model (Sec. III). Next, we consider leakage of the Majorana mode into the quantum dot for the fully (Sec. IV) and partly polarized (Sec. V) cases. Finally, we summarize the results (Sec. VI) and give a brief reminder about the singlet-doubled transition in absence of Majorana mode (Appendix A).

II Physical setup

We consider the strongly correlated quantum dot attached to the Rashba nanowire, both proximitized to the $s$-wave superconductor (SC). This setup (Fig. 1) could be realized in the scanning tunneling microscopy (STM) measurements, using Fe or Co nanoscopic chains on the superconducting Pb Jeon et al. (2017), Al or Re Kim et al. (2018) substrates.

In realistic situations the spin-orbit coupling along with the Zeeman effect break spin-rotational symmetry of the system. Such nanowire brought in contact with superconductor develops the intersite pairing of parallel spins. They are ‘tilted’ with respect to $z$-axis, but one can project such pairing onto $\uparrow$ and $\downarrow$ components. For each of these sectors the intersite-pairing is characterized by effective amplitudes, mainly dependent on the applied magnetic field Maśka and Domański (2017).

Upon entering the topological phase, there appear two Majorana quasiparticles simultaneously in both spin channels but with different spectral weights, what has been indeed reported in the STM measurements using a ferromagnetic tip Jeon et al. (2017). Such issue has been recently addressed by several groups Szumniak et al. (2017); Maśka and Domański (2017); Li et al. (2018). J. Klinovaja with co-workers Hoffman et al. (2017) have shown that spin up and down tunneling amplitudes between MBSs and quantum dot depend on the spin-orbit interaction length. Distance between the quantum dot and topological nanowire determines oscillations of the tunneling amplitudes. By changing such dot-nanowire distance or varying the magnetic field one can thus tune the hybrid system either to the fully or to the partly polarized tunneling regimes.

For perfect spin polarization of the QD-chain tunneling (e.g. $t_{\uparrow}\neq 0$, $t_{\downarrow}=0$) one would expect signatures of the zero-energy mode to appear only in one spin channel (for $\uparrow$ electrons). However, the proximity induced on-dot pairing between opposite spins mixes both these channels. Some aspects of this situation have been addressed in Refs Chirla and Moca (2016); Barański et al. (2017); Górski et al. (2018). Since leakage of the Majorana (zero-energy) mode is sensitive to electronic states near the Fermi level, we would like to focus on crossing of the subg-gap (Andreev) states caused by competition between on-dot pairing and Coulomb repulsion (which is spectroscopically manifested by the singlet-doublet quantum phase transition). We show that signatures of the Majorana mode look completely different for both spin channels. In our considerations we take into account the case of (i) perfectly polarized tunneling of only one spin component while the other one is completely prohibited and (ii) another case where both spin electrons can be tunnel transferred, but with different amplitudes.

III Microscopic model

To inspect the mutual relationship between the Majorana mode and in-gap features of the correlated QD we restrict our attention to the limit of large energy gap $\Delta$ of the superconducting reservoir. Under such conditions the single level quantum dot is affected by the proximity-induced pairing. The proximitized QD is described by the Hamiltonian $\sum_{\sigma}\epsilon d^{\dagger}_{\sigma}d_{\sigma}+Un_{\downarrow}n_{\uparrow}+\frac{\Gamma_{S}}{2}(d_{\uparrow}d_{\downarrow}+d^{\dagger}_{\downarrow}d^{\dagger}_{\uparrow})$, where local pairing is represented by the pair source/sink terms. We next consider coupling to the Majorana zero mode (MZM) represented by $\sum_{\sigma}\lambda\left(d^{\dagger}_{\sigma}\eta_{1}+\eta_{1}d_{\sigma}\right)+i\epsilon_{m}\eta_{1}\eta_{2}$. In our approach we recast the self-hermitian operators $\eta_{1,2}^{\dagger}=\eta_{1,2}$ by the standard fermionic ones Elliott and Franz (2015) $\eta_{1}=\frac{1}{\sqrt{2}}(f+f^{\dagger})$ and $\eta_{2}=\frac{-i}{\sqrt{2}}(f-f^{\dagger})$. Summarizing, the effective low energy Hamiltonian can be expressed as

[TABLE]

We assume that both spin components ($\sigma=\uparrow,\downarrow$) can be transferred between dot and Majorana host and $t_{m\sigma}=\lambda_{\sigma}/\sqrt{2}$ represent the hopping integrals of such processes.

Our objective is to determine the Green’s functions ${\cal{G}}(\omega)=\langle\langle\Psi;\Psi^{\dagger}\rangle\rangle$ defined in the Nambu matrix notation $\Psi^{\dagger}=(d_{\uparrow},d_{\uparrow}^{\dagger},d_{\downarrow},d_{\downarrow}^{\dagger},f,f^{\dagger})$. The equation of motion technique applied to noncorrelated problem yields

[TABLE]

To account for the correlation effects one can numerically diagonalize 8$\times$8 Hamiltonian matrix, determining the eigenenergies and transition elements between them. Another equivalent route can rely on the superconducting atomic limit solution Vecino et al. (2003), extending it to the present model

[TABLE]

where

[TABLE]

with the energy $E_{d}=\sqrt{(\epsilon+U/2)^{2}+(\Gamma_{S}/2)^{2}}$, the usual BCS-type coefficients $u_{d}^{2}=\frac{1}{2}\left[1+\frac{\epsilon+U/2}{E_{d}}\right]=1-v_{d}^{2}$ and the relative spectral weights

[TABLE]

In what follows we will quasiparticle spectrum of the correlated quantum dot, assuming that the topological nanowire is long enough so that any overlap between the end-modes is negligible ($\epsilon_{m}\simeq 0$).

IV Fully polarized case

Let us first consider the fully spin-polarized case when tunneling of one spin component, say $\downarrow$, is totally prohibited. This situation could occur for very strong magnetic fields applied along the topological nanowire. We show that combined effect of MZM leakage and proximity induced pairing give rise to zero modes apparent in energy spectrum of both spin components even if tunneling rate of one spin is turned off. We inspect the influence of these zero states on the characteristic features of a phase transition from the BCS-like singlet state ($S=0$) to the correlation-dominated doublet configuration. We show that even though zero modes appear in both spin channels, each of them have completely different character.

When a quantum dot or other nanoobject is tunnel coupled to the Rashba chain hosting Majorana particles some part of these zero-energy modes can be transferred to QD region. In consequence, the spectral function of QD reveals additional feature pinned to the Fermi level. For the perfectly polarized tunneling amplitude ($t_{m\downarrow}=0$ and $t_{m\uparrow}\neq 0$) one would expect such effect to modify only the spectral function of the directly coupled spin component Lee et al. (2013); López et al. (2014); Weymann (2017); Weymann and Wójcik (2017). However, due to the superconducting proximity effect the electrons (say $\uparrow$) leaking from nanochain into QD region are bound into local pairs with their opposite spin ($\downarrow$) partners Barański et al. (2017); Górski et al. (2018). Thereby, the Majorana mode becomes apparent in the energy spectrum of electrons for which the direct tunneling to nanochain is prohibited. This feature in $\downarrow$ spin sector, however, should be considered merely as a superconducting response for the leaking Majorana mode of the directly coupled spin $\uparrow$. Different nature of these zero-energy states is evident, both for the uncorrelated case (Fig. 2) and in a behavior of the Andreev states of the correlated quantum dot near a quantum phase transition from the BCS-type singlet to the correlation driven doublet state (Fig. 3).

IV.1 Energy spectrum of uncorrelated QD

Let us focus first on the noncorrelated case, $U=0$. By inspecting the matrix Green’s function (8) in the limit $\Gamma_{N}\rightarrow 0$ we notice that it is characterized by five poles: zero-energy state, two Andreev bound states at $\pm\sqrt{\epsilon^{2}+(\Gamma_{S}/2)^{2}}$ and another two ‘molecular’ states $\pm\sqrt{\epsilon^{2}+(\Gamma_{S}/2)^{2}+(2t_{m\uparrow})^{2}}$, resulting from hybridization of the Andreev bound states with the Rashba chain Górski et al. (2018). In Fig. 2 we visualize qualitative differences between the spectral weights of $\uparrow$ and $\downarrow$ spin sectors obtained in the strong coupling limit $t_{\uparrow}=0.6\Gamma_{S}$.

In the spectrum of $\uparrow$ electrons we observe the well pronounced zero-energy mode directly leaking from the proximitized Rashba chain in much the same fashion as initially predicted for the quantum dot attached to the Kitaev wire Vernek et al. (2014). Contrary to that, the opposite spin electrons are not directly coupled to the chain therefore zero-energy mode emerges as a consequence of the local pairing. Some tiny Majorana mode shows up in this $\downarrow$ spin sector only when the QD energy $\epsilon$ is close to the Fermi level, otherwise it quickly fades away.

Furthermore, one can notice a small dip in the zero-energy signature of spin $\uparrow$ electrons appearing near $\epsilon=0$. This comes from destructive feedback effect when the original QD level coincides with Majorana mode. Such destructive interference pattern has been described by our group Barański et al. (2017); Górski et al. (2018). We have pointed out that even when spectral weight of the zero-energy mode in spectrum of directly coupled spins is suppressed, the zero-energy mode in opposite spin channel would be enhanced. As the QD-chain tunneling amplitude of spin $\downarrow$ electrons is turned off, spin $\downarrow$ electrons do not take part in destructive interference. Consequently in the left panel of Fig. 3 we observe well shaped zero mode. It is worth noticing that spectral weights of original Andreev states ($\pm\sqrt{\epsilon^{2}+(\Gamma_{S}/2)^{2}}$) in spectrum of $\uparrow$ electrons are reduced in comparison with the molecular ones. Contrary to that, for electrons with prohibited dot-chain tunneling we observe that pure Andreev states are dominant.

IV.2 Majorana near singlet-doublet crossover

We shall now discuss the correlation effects, driven by the Coulomb repulsion $U$. In absence of the Rashba nanowire upon varying the ratio $U/\Gamma_{S}$ there appear the regular Andreev bound states at $\pm[U/2-\sqrt{(\epsilon+U/2)^{2}+(\Gamma_{S}/2)^{2}]}$ which eventually cross each other at the singlet-doublet (see Appendix A for an explanation of this quantum phase transition). Proximity induced zero-energy feature in the spin $\downarrow$ sector does not affect significantly this characteristic crossing, because it does not originate directly from the leaking Majorana mode. The strong Coulomb repulsion disfavors on-dot pairing, therefore upon increasing $U/\Gamma_{S}$ also the side effects of the local pairing are gradually suppressed. One of them is the zero-energy mode appearing in the spectrum of spin $\downarrow$ electrons, originating solely from electron pairing. For this reason, upon traversing the singlet-doublet quantum phase transition the zero-energy mode of spin $\downarrow$ electrons is gradually washed out from the strongly correlated regime (see right bottom lower panel in Fig. 3).

In the case of spin $\uparrow$ electrons the spectrum looks completely different. Upon increasing the correlations strength the ABS states are split and form the molecular branches. These new states no longer cross each other. Approaching the singlet-doublet transition we observe emergence of the zero-energy state (directly leaking from the Rashba nanochain) while the molecular Andreev states repel one another instead of the regular crossing (shown in Fig. 5). Such avoided-crossing behavior (see the upper right panel in Fig. 3) can be regarded as additional signature, of the proximity induced Majorana mode that repels the ‘trivial’ (finite-energy) Andreev states. This result generalizes the previously discussed topological/nontopological different nature of the bound states in hybrid structures comprising the uncorrelated quantum dots attached to the topological superconducting wires hosting the Majorana modes Prada et al. (2017); Ptok et al. (2017b); Szumniak et al. (2017); Hoffman et al. (2017); Deng et al. (2018). We hope that empirical detection would be feasible using the spin-polarized STM measurements and varying the level $\epsilon$ of correlated QD by the gate potential.

V Partial spin-polarization

Theoretical studies indicate that for specific conditions, it would be possible to achieve nearly perfectly spin polarized tunneling between the quantum dot and chain Hoffman et al. (2017). However in realistic experimental setups both spin electrons can be transferred, although with substantially different tunneling amplitudes. In this subsection we briefly address such partially polarized case. Motivated by the spin polarized STM measurements Jeon et al. (2017) we assume in our approach the tunneling rate of spin $\downarrow$ electrons to be $t_{m\downarrow}=0.1\Gamma_{S}$ which is three times weaker than the coupling amplitude for spin $\uparrow$ electrons ($t_{m\uparrow}=0.3\Gamma_{S}$). Fig. 4 compares the density of states of each spin electrons in the non-correlated (left panels) and strongly correlated regime (right panels), respectively.

In the strongly correlated limit we can practically observe a convolution of the features typical for each spin sectors of the fully polarized case. Yet one can clearly resolve the avoided-crossing behavior in the dominant $\uparrow$ spin coupling channel with the well pronounced Majorana mode separating them. Spectrum of the opposite spin sector, on the other hand, is predominantly reminiscent of the continuous Andreev band state branches with only very residual signatures of the zero-energy modes. Correlations could thus be very useful for distinguishing the qualitatively different character of the spin-polarized spectra of the quantum dot.

VI Summary

We have studied the energy and spin-dependent spectra of the proximitized correlated quantum dot attached to the topologically superconducting nanochain, hosting the Majorana end-modes. We have analyzed influence of the correlations (responsible for a quantum phase transition from the spinless BCS-type to the spinful configuration) on efficiency of the Majorana mode leakage. Our study predicts that the zero-energy states might appear in both spin sectors (due to the superconducting proximity effect), however their spectroscopic signatures are going to be qualitatively different. The spin channel which is directly coupled to the Majorana mode is characterized by (a) the well separated Andreev branches (of an avoided-crossing behavior) coexisting with (b) the zero-energy feature of a sizable spectral weight. The other spin sector, which is not directly coupled to the Majorana mode, in the strongly correlated limit is characterized by the continuous Andreev branches traversing the zero energy without any trace of the leaking Majorana mode. Such spin sector can eventually allow the Majorana mode to appear but only in the weak correlation regime, when the proximitized quantum dot stays in the BCS-type configuration.

Correlations can hence be very beneficial for spin-selective leakage of the Majorona modes on the quantum dots. Such effects would be easily detectable, using the polarized STM measurement analogous to those already reported in Ref. Jeon et al. (2017).

Acknowledgements.

This research has been conducted in the framework of the project “Analysis of nanoscopic systems coupled with superconductors in the context of quantum information processing” No. GB/5/2018/209/2018/DA funded in period 2018-2021 by the Ministry of National Defense Republic of Poland (J.B.,T.Z.). This project has been also supported by National Science Centre (NCN, Poland) under the grants UMO-2018/29/B/ST3/00937 (G.G.) and UMO-2017/27/B/ST3/01911 (T.D.).

Appendix A Quantum phase transition in absence of Majorana mode

For illustration of the singlet-doublet phase transition we recall briefly the results obtained for the correlated quantum dot in absence of the Rashba chain. For this purpose we analyze the subgap Andreev states in the superconducting atomic following several earlier works Martín-Rodero and Levy Yeyati (2012); Barański and Domański (2013); Bauer et al. (2007). The true eigenstates are represented by the doublet configurations $\left|\uparrow\right>$, $\left|\downarrow\right>$ (corresponding to spin $S=\frac{1}{2}$) and the BCS-type singlet states $u_{d}\left|0\right>+v_{d}\left|\downarrow\uparrow\right>$ or $-v_{d}\left|0\right>+u_{d}\left|\downarrow\uparrow\right>$ ($S=0$). It can be shown that when correlations are weak $U<\Gamma_{S}$, the ground state is represented by singlet state in full range of initial dot level ($\epsilon$). In such case, the density of states consists of two Andreev bound states branches separated by the pairing gap (Fig. 5). For stronger correlations, these subgap branches approach each other and can eventually cross at the singlet–doublet phase boundaries. The critical interaction for ground state transition at the half filling is $U=\Gamma_{S}$ (middle panel in Fig. 5). In the doublet regime ($U>\Gamma_{S}$), the subgap resonances cross each other, giving rise to a loop structure (bottom panel in Fig. 5). In section IV, we noticed that the states appearing in spin $\downarrow$ electron spectrum exhibit similar behavior (c.f. bottom panels in Fig. 3). Presence of the zero-energy state in this spin channel did not affect significantly the characteristics of the singlet-doublet transition. In contrast, for spins directly coupled to nanochain, we note the qualitatively different avoided-crossing behavior.