Manipulation of exact Majorana zero modes in spin systems by tuning external fields

Abstract

We study the emergence of \\emph{exact Majorana zero modes} (EMZMs) in one-dimensional spin systems by tuning the magnetic field acting on individual spins. By focusing on the quantum transverse compass chain with both the nearest-neighbor interactions and the transverse fields varying over space, we derive a formula for the number of the emergent EMZMs, which depends on the \\emph{partition nature} of the lattice sites on which the magnetic fields vanish. We also derive explicit expressions for the wavefunctions of these EMZMs, which are found to depend on fine features of the foregoing partition of site indices. As a specific case, the exact solution for an open compass chain with uniform nearest-neighbor interactions and an alternating magnetic field is provided. It is rigorously proved that, besides the possibly existing EMZMs, no \\emph{almost Majorana zero modes} exist unless the fields on the even and odd sublattices are both turned off. Our results provide a precise scheme to manipulate the number and spatial distributions of the EMZMs by solely tuning the external fields, and may shed light on the control of ground-state degeneracies in more general systems.