Odd-frequency pairing in a nonunitary p -wave superconductor with multiple Majorana fermions

Abstract

Odd-frequency Cooper pairs are gathering attention for the convenience of investigating the edge state of topological superconductors including Majorana fermions. Although a spinless $p$-wave superconductor has only one Majorana fermion in a topological phase, the system with magnetic fields can reach the topological phases with multiple Majorana fermions. To distinguish these multiple Majorana fermion phases, we correlate the energy spectrum with the odd-frequency pair amplitude as increasing the system size. The system size dependence tells us three pieces of information: the parity of the number of the Majorana fermions at the edge, the number of low-energy modes corresponding to the Majorana fermions with different localization lengths, and the fingerprints of the Majorana fermions. Also, we present the spatial dependence of the odd-frequency $\mathrm{f}$ vector that is created from odd-frequency pair amplitude and the spin structure of odd-frequency Cooper pairs. We find that the odd-frequency $\mathrm{f}$ vector is fixed in the same direction in any topological phase. Also, we show that the spin state of odd-frequency Cooper pairs tends to be oriented toward the direction of the magnetic fields. Our results highlight that the odd-frequency Cooper pairs can be a good indicator for the detection of the multiple Majorana fermions and the distinction among the topological phases.