Multiparticle content of Majorana zero modes in the interacting p -wave wire

Abstract

In the topological phase of $p$-wave superconductors, zero-energy Majorana quasiparticle excitations can be well defined in the presence of local density-density interactions. Here, we examine this phenomenon from the perspective of matrix representations of the commutator $\mathcal{H}=[H,\ifmmode\bullet\else\textbullet\fi{}]$, with the aim of characterizing the multiparticle content of the many-body Majorana mode. To do this we show that, for quadratic fermionic systems, $\mathcal{H}$ can always be decomposed into subblocks that act as multiparticle generalizations of the Bogoliubov--de Gennes/Majorana forms that encode single-particle excitations. In this picture, density-density-like interactions will break this exact excitation-number symmetry, coupling different subblocks and lifting degeneracies so that the eigenoperators of the commutator $\mathcal{H}$ take the form of individual eigenstate transitions $|\phantom{\rule{0.16em}{0ex}}n\ensuremath{\rangle}\ensuremath{\langle}m\phantom{\rule{0.16em}{0ex}}|$. However, the Majorana mode is special in that zero-energy transitions are not destroyed by local interactions and it becomes possible to define many-body Majoranas as the odd-parity zero-energy solutions of $\mathcal{H}$ that minimize their excitation number. This idea forms the basis for an algorithm which is used to characterize the multiparticle excitation content of the Majorana zero modes of the one-dimensional $p$-wave lattice model. We find that the multiparticle content of the Majorana zero-mode operators is significant even at modest interaction strengths. This has important consequences for the stability of Majorana-based qubits when they are coupled to a heat bath. We will also discuss how these findings differ from previous work regarding the structure of the many-body Majorana operators and point out that this should affect how certain experimental features are interpreted.