Vortex-core Majorana coupling to a chiral edge in a $p_x+ip_y$ superconductor: Nonmonotonic spectral reorganization and coherent fermion-parity dynamics

Abstract

We study how vortex--edge coupling reorganizes the low-energy sector of a finite two-dimensional \(p_x+ip_y\) superconducting disk as a function of the vortex--boundary separation \(d\) and examine what this reorganization implies for the parity memory associated with a prescribed vortex-core Majorana wave packet, a resource relevant to Majorana-based quantum operations. Bogoliubov--de Gennes calculations reveal nonmonotonic core--edge reorganization of the lowest positive-energy finite-disk eigenstate, with particularly rapid variation near \(d\simeq7\xi\), where \(\xi\) is the coherence length. To separate this eigenstate reorganization from the spectral representation of a prescribed state, we rigidly translate a centered-vortex core-reference packet to each fixed vortex position, restrict it to the target disk, and project it, without intermediate normalization, onto the particle-hole-complete low-energy subspace. For \(\Delta_0/E_F=0.36\) and disk radius \(R=30\xi\), the resulting retained norm exceeds \(0.98\) at all six sampled separations, \(4.25\leq d/\xi\leq8.25\), while, depending on \(d\), the spectral measure is concentrated near zero energy, fragmented over several low-energy levels, or dominated by finite-energy weight. Correspondingly, the signed parity correlator displays slow temporal variation, rapid coherent dephasing, or sign-changing oscillations, with possible finite-size recurrences at later times. Thus a large retained norm does not by itself imply spectral concentration or persistent parity memory.