Rotating Zeeman field as a tool for Majorana zero mode detection in topological superconducting wire

Technical Summary: Rotating Zeeman Field as a Tool for Majorana Zero Mode Detection in Topological Superconducting Wire

Problem Statement
The identification of true Majorana zero modes (MZMs) in solid-state systems remains a significant challenge due to the prevalence of "trivial" zero-bias peaks (ZBPs) that mimic MZM signatures. These false positives can arise from inhomogeneous electrostatic potentials, unintentionally formed quantum dots (QDs) at electrode-wire junctions, disorder, or trivial Andreev sub-bands. While recent experimental announcements (e.g., Microsoft's Majorana 1 and 2 processors) have utilized local and non-local conductance measurements to detect ZBPs and gap reopening, the distinction between topological MZMs and trivial resonances is often ambiguous. Consequently, complementary screening procedures are required to unambiguously verify the existence of MZMs.

Methodology
The authors propose a detection scheme based on the analysis of the spin polarization of a quantum dot attached to a topological superconducting wire. The core of the methodology involves manipulating the direction of the Zeeman field within the wire while maintaining a fixed Zeeman field direction in the quantum dot.

• Model System: The system consists of a quantum dot with a single active spatial level, coupled to a normal electrode and the first site of a topological wire (modeled as a Kitaev chain with Rashba spin-orbit interaction).
• Field Configuration: The Zeeman field in the wire is rotated within the $x-z$ plane (where $x$ is the wire axis and $z$ is perpendicular to the effective Rashba field). The Zeeman field in the dot remains fixed along the $z$-direction.
• Theoretical Framework: The study employs a tight-binding Hamiltonian for the wire and a Nambu basis Green's function approach for the dot. The authors calculate the retarded Green's function of the dot to determine the spectral density and spin polarization at the Fermi energy.
• Analytical Approach: A "Toy Model" is developed to provide physical insight. This model treats the wire as hosting chiral sub-bands and expresses the Majorana operators in terms of fermionic spin components, allowing for the derivation of how the coupling between the dot and the wire depends on the Zeeman field angle.

Key Contributions and Results

1. Spin Polarization as a Diagnostic Tool:
   The primary finding is that the effective spin polarization of the quantum dot at the Fermi energy exhibits a distinct dependence on the orientation of the Zeeman field in the wire only when true MZMs are present.

   • Topological State: When the wire is in the topological phase, rotating the Zeeman field from parallel ($\Theta = \pi/2$, along $x$) to perpendicular ($\Theta = 0$, along $z$) to the wire axis reveals the second spin component of the Majorana Kramers pair. Specifically, at $\Theta = \pi/2$, only one spin component (e.g., spin-down) is visible in the dot's spectral density. As the field rotates to $\Theta = 0$, the second component (spin-up) emerges. This transition causes a significant, non-linear change in the dot's spin polarization (e.g., a 52% change for weak coupling and 78% for strong coupling).
   • Trivial State: In the trivial phase, or when a trivial QD level mimics a Majorana resonance, the spin polarization remains practically constant regardless of the Zeeman field rotation.

2. Robustness of Detection:
   The proposed detection method is robust against variations in experimental parameters. The characteristic change in spin polarization is observed independently of:

   • The coupling strength between the quantum dot and the wire (tested for both weak and strong coupling regimes).
   • The specific tuning of the quantum dot's energy levels relative to the Fermi energy (detection works even when levels are detuned).

3. Identification of Topological Phase Transition:
   The paper demonstrates that the dependence of spin polarization on the magnitude of the Zeeman field exhibits a characteristic non-linearity at the critical field value ($V_{cr}$). This non-linear behavior signals the topological phase transition and is distinct from the linear dependence expected in the absence of a phase transition. This feature is observable for various field orientations and coupling strengths.

4. Physical Mechanism:
   The underlying physics is attributed to the formation of chiral sub-bands in the wire due to strong Rashba interaction. The Majorana modes inherit the chirality of these sub-bands. The rotation of the Zeeman field alters the projection of these chiral eigenstates onto the spin quantization axis of the dot, thereby modulating which spin components of the Majorana mode couple to the dot.

Significance
The paper claims that this method provides a complementary and robust procedure for screening true Majorana modes. By exploiting the spin-dependent nature of MZMs in the presence of chiral sub-bands, the technique distinguishes topological states from trivial resonances that might otherwise produce zero-bias peaks. The authors emphasize that the distinct evolution of spin polarization upon rotating the Zeeman field, combined with the non-linear response to field magnitude, offers a "twofold evidence" that can facilitate experimental verification of Majorana zero modes, addressing the current debate regarding the validity of previous observations.