Can non-local conductance spectra conclusively signal Majorana zero modes? -- Insights from von Neumann entropy

Abstract

The topological origin of the zero bias conductance signatures obtained via conductance spectroscopy in topological superconductor hybrid systems is a much contended issue. Recently, nonlocal conductance signatures that exploit the non-locality of the zero modes in three terminal hybrid setups have been proposed as means to ascertain the definitive presence of Majorana modes. The topological entanglement entropy, which is based on the von Neumann entropy, is yet another way to gauge the non-locality in connection with the bulk-boundary correspondence of a topological phase. We show that while both the entanglement entropy and the non-local conductance exhibit a clear topological phase transition signature for long enough pristine nanowires, non-local conductance fails to signal a topological phase transition for shorter disordered wires. While recent experiments have indeed shown premature gap-closure signatures in the non-local conductance spectra, we believe that the entanglement entropy can indeed signal a genuine transition, regardless of the constituent non-idealities in an experimental situation. Our results thereby point toward furthering the development of experimental techniques beyond conductance measurements to achieve a conclusive detection of Majorana zero modes.

Conductance Spectroscopy of Majorana Zero Modes in Superconductor-Magnetic Insulating Nanowire Hybrid Systems

“…These observations are due to the hybridization of the MZMs when they overlap in finite nanowires, resulting in a splitting of the zero mode [53]. The hybridization of the MZMs through the nanowire is suppressed with increasing length [54][55][56], which is consistent with our observations. In Fig.…”

“…where η is an infinitesimal positive damping parameter introduced for numerical stability, and I is the identity matrix of the dimension of the Hamiltonian matrix in Nambu space. In the wide-band approximation [54][55][56], the self energies for the metallic contacts, Σ L(R) , are written in their eigenbasis and are hence diagonal, as detailed in the supplementary material. We use the Usadel equation, which is derived from a quasi-classical approximation to the Gorkov equations, to find the Green's function, and hence, the self-energy, Σ for the SC-MI bilayer [48].…”

“…We use the retarded Green's function defined above to calculate the conductance matrix for this setup [54][55][56]69]. As shown in Fig.…”

“…1, we apply voltages V L(R) to the left and right contacts and measure terminal currents I L(R) . We use the Keldysh non-equilibrium Green's function formalism to evaluate the terminal currents [54][55][56]69]. The terminal electronic current at the left contact [73] can be derived in the Landauer Büttiker form as: CAR (E) represent the energy resolved transmission probabilities for the direct, Andreev and crossed-Andreev processes involving the left and right contacts for the electronic sector of the Nambu space and I' is the extra current due to the SC-MI bilayer acting as an effective contact, derived in the supplementary material.…”

“…They are characterised by a parameter γ L/R , which represents the escape rate into the contacts. The current operator for an N-S-N device, derived from first principles in various references [55,56], to discern the various components of the current:…”