Revisiting the Poor Man's Majoranas: The Spin-Exchange Induced Spillover Effect

TL;DR

This work reexamines the two-site Kitaev chain realizing Poor Man's Majoranas in a QD-based platform. It uses a Green's function framework to show that exchanging a PMM-hosting QD with a proximal spin $S$ via exchange coupling $J$ yields a spin-statistics dependent subgap multiplet: $2S+1$ satellites for fermionic spins and $2S+2$ satellites for bosonic spins. It also identifies an environmentally induced protection from a multi-terminal environment, characterized by $\Gamma$ and $\eta$, that partially localizes PMMs and suppresses spillover for moderate $J$ fluctuations. These results propose a spectroscopic protocol for quantum spin characterization and introduce a dissipative stabilization concept for PMMs, with potential implications for PMM-based qubits and parity readout.

Abstract

We give a review on Poor Man's Majorana (PMM) modes, which are theoretically established in the minimal Kitaev chain implementation consisting of two grounded, spinless quantum dots (QDs) operating at the sweet spot condition, where electron cotunneling and crossed Andreev reflection amplitudes achieve precise balance. Particularly, we systematically review, within the Green's functions theoretical framework, the PMM hybridization dynamics under spin-exchange perturbations proposed by some of us in J. Phys.: Condens. Matter 37, 205601 (2025), which demonstrates a characteristic spatial delocalization when subjected to an exchange coupling $J$ mediated by a quantum spin $S$. This spin-exchange induced PMM spillover effect provides a spectroscopic protocol for determining the quantum statistics of $S$ through the emergent multi-level structure in the proximal QD's density of states. Our principal theoretical result establishes that the exchange interaction generates $2S+2$ ($2S+1$) satellite states symmetrically distributed about the zero-bias anomaly, serving as a definitive signature of bosonic (fermionic) spin statistics. As novelty, we demonstrate that multi-terminal environmental coupling induces significant suppression of the spin-exchange spillover mechanism. Under constrained variations of $J$, this effectively localizes the perturbed PMM within its host QD, preventing spatial hybridization with adjacent site. The absence of topological protection in this minimal Kitaev realization is strategically leveraged to: (i) Develop a novel spectroscopic technique for quantum spin characterization through PMM hybridization signatures; (ii) Propose the "environmentally induced protection", an engineered dissipative spectral stabilization for PMMs against exchange fluctuations in multi-terminal architectures.

Figures (6)

• Figure 1: The minimal Kitaev chain made by two quantum dots (QDs) exhibit electron cotunneling ($t$) and crossed Andreev reflection ($\Delta$), where at the condition $t=\Delta$ and chemical potentials $\mu_{L}=\mu_{R}=0$ (inset), the system functions as a spin-statistics probe for the spin $S$ quasiparticles via the “ Poor Man's Majorana” (PMM) spillover effect. This phenomenon arises from exchange coupling ($J$) fluctuations, mediating a projection of the left PMM onto the right QD. The spin-statistics signature manifests as a multi-level spectrum with multiplicity $2S+2$ ($2S+1$) for bosonic (fermionic) spin $S$ states, localized predominantly on the left dot. Given the hybridization of both dots with distinct fermionic reservoirs (gray metallic leads), the asymmetric coupling $\Gamma_{\alpha}=\delta_{\alpha L}\Gamma+\eta$ ($\alpha=L(R)$ for left (right) dot), which is considered to mimic the experimental conditions of Ref.Kouwenhoven2023, emerges as critical in stabilizing the PMMs $\gamma_{L1}$ and $\gamma_{R2}$ under constrained $J$ variations. Consequently, $\gamma_{L1},\gamma_{L2},$ and $\gamma_{R1}$ form an unconventional trimer configuration, where the zero-energy mode is exclusively hosted by $\gamma_{L1}$ and absent in $\gamma_{R1}$. This dichotomy stems from the interplay between $\Gamma$ and $\eta$, which isolates $\gamma_{L1}$ and $\gamma_{R2}.$
• Figure 2: Chemical Potential Induced Spillover: The Majorana ($\pi\Gamma{\cal {A}}_{\gamma_{\alpha j}}$) and electronic ($\pi\Gamma{\cal {A}}_{d_{L\uparrow}d_{L\uparrow}^{\dagger}}$) spectral functions are analyzed. (a)-(c) At the sweet spot ($t=\Delta$ with vanishing local potentials $\mu_{L}=\mu_{R}=J=0$), a zero-energy mode emerges in $\pi\Gamma{\cal {A}}_{\gamma_{L1}}$ [panel (a), where $\pi\Gamma{\cal {A}}_{\gamma_{L1}}(0)=1$] and in $\pi\Gamma{\cal {A}}_{\gamma_{R2}}$ [panel (b), unnormalized]. These zero-modes correspond to the spatially isolated PMM states $\gamma_{L1}$ and $\gamma_{R2}$, localized at the left and right QDs, respectively. The satellite peaks originate from the bonding ($-2\Delta\approx-24\text{$\mu$eV}$) and antibonding ($+2\Delta\approx+24\text{$\mu$eV}$) hybridized states of the $\gamma_{L2}-\gamma_{R1}$ dimer formed. Consequently, $\pi\Gamma{\cal {A}}_{\gamma_{L2}}(0)=\pi\Gamma{\cal {A}}_{\gamma_{R1}}(0)=0$. Panel (c) exhibits the density of states (DOS) $\pi\Gamma{\cal {A}}_{d_{L\uparrow}d_{L\uparrow}^{\dagger}}(0)=\frac{1}{2}$, characteristic of the half-electron state associated with $\gamma_{L1}$. (d)-(f) Introducing a finite potential asymmetry ($\mu_{L}\neq0$) induces PMM spillover, reducing $\pi\Gamma{\cal {A}}_{\gamma_{L1}}(0)<1$ while activating $\pi\Gamma{\cal {A}}_{\gamma_{R1}}(0)\neq0,$ a direct consequence of the $\gamma_{L1}-\gamma_{L2}-\gamma_{R1}$ trimer formation. The persistent zero-mode in $\pi\Gamma{\cal {A}}_{\gamma_{L1}}$ and $\pi\Gamma{\cal {A}}_{\gamma_{R1}}$ reflects a nonbonding state. Panel (f) shows the PMM spillover via $\pi\Gamma{\cal {A}}_{d_{L\uparrow}d_{L\uparrow}^{\dagger}}(0)<\frac{1}{2}$, accompanied by a peak at $\mu_{L}=10$.
• Figure 3: Spin-Exchange Induced Spillover as a Spin Statistics Probe ($t=\Delta$, $\mu_{L}=\mu_{R}=0$, and $J\neq0$). The system is governed by the trimer configuration $\gamma_{L1}-\gamma_{L2}-\gamma_{R1}$, mediated by the exchange coupling $J$. (a)-(c) The PMM spillover persists, evidenced by $\pi\Gamma{\cal {A}}_{\gamma_{L1}}(0)<1$ [panel (a)], $\pi\Gamma{\cal {A}}_{\gamma_{R1}}(0)\neq0$ [panel (b)], and $\pi\Gamma{\cal {A}}_{d_{L\uparrow}d_{L\uparrow}^{\dagger}}(0)<\frac{1}{2}$ [panel (c)]. For half-integer (fermionic) spin $S$, a distinct multi-level structure with multiplicity $2S+1$ emerges around the zero-mode, prominently visible in $\pi\Gamma{\cal {A}}_{\gamma_{L1}}$, $\pi\Gamma{\cal {A}}_{\gamma_{L2}}$, and $\pi\Gamma{\cal {A}}_{d_{L\uparrow}d_{L\uparrow}^{\dagger}}$ due to strong spin-proximity effects. In contrast, the right QD exhibits weaker signatures, with $\pi\Gamma{\cal {A}}_{\gamma_{R1}}$ showing only marginal spectral weight. (d)-(f) For integer (bosonic) $S$, the spectral structure transitions to a $2S+2$ multiplicity, particularly pronounced in $\pi\Gamma{\cal {A}}_{\gamma_{L2}}$ and $\pi\Gamma{\cal {A}}_{d_{L\uparrow}d_{L\uparrow}^{\dagger}}$, while $\pi\Gamma{\cal {A}}_{\gamma_{L1}}$ displays only $2S$ side peaks. This distinction arises because the bosonic case preserves the original bonding ($-2\Delta\approx-24\text{$\mu$eV}$) and antibonding ($+2\Delta\approx+24\text{$\mu$eV}$) states of the $\gamma_{L2}-\gamma_{R1}$ dimer ($J=0$), superimposing them onto the $2S$-fold multiplicity. Notably, $\pi\Gamma{\cal {A}}_{\gamma_{R2}}$ remains devoid of multi-level structure, reflecting the persistent isolation of $\gamma_{R2}$ [panels (b), (e)].
• Figure 4: PMM dissipative spectral stabilization (“ environmentally induced protection” ): (a)-(b) Spectral functions ${\cal {A}}_{\gamma_{L1}}(0)$ and ${\cal {A}}_{\gamma_{R1}}(0)$ as functions of $\mu_{L}$ and $J$. For $J$-dependent coupling at $\mu_{L}=0$ (red curves), the presence of hybridized electronic environments (characterized by couplings $\Gamma$ and $\eta$) enables $\gamma_{L1}$ to maintain partial Majorana dissipative spectral stabilization. This manifests as plateaus in ${\cal {A}}_{\gamma_{L1}}(0)$ (finite) and ${\cal {A}}_{\gamma_{R1}}(0)$ (null) within the perturbation regime $-1\lesssim J\lesssim+1$ (gray shaded region) (or $-8{\mu eV}\lesssim J\lesssim+8{\mu eV}$ in realistic units). The observed spectral stability indicates complete suppression of PMM spillover from the left to right QD throughout this parameter window. Remarkably, both fermionic and bosonic spin statistics yield qualitatively similar spectral responses [panels (c)-(d)]. In contrast, for $\mu_{L}$-dependent tuning at $J=0$ (blue curves), environmental coupling fails to prevent PMM delocalization: finite spectral weight ${\cal {A}}_{\gamma_{R1}}(0)$ emerges upon deviation from $\mu_{L}=0$, as evidenced [panels (a)-(d)].
• Figure 5: Spectral functions ${\cal {A}}_{\gamma_{L1}}(0)$ and ${\cal {A}}_{\gamma_{R1}}(0)$ in terms of of $\mu_{L}$ and $J:$ regardless the spin $S$ and perturbation, if the coupling $\Gamma$ is turned-off, the dissipative spectral stabilization PMM picture breaks down. Solely at $\mu_{L}=J=0,$ the spectral weight ${\cal {A}}_{\gamma_{R1}}(0)=0.$