# Exact zero modes in a quantum compass chain under inhomogeneous transverse fields

## TL;DR

This paper derives an exact formula for Majorana zero modes in a quantum compass chain with inhomogeneous transverse fields, revealing how the zero modes depend on the lattice partition and field configuration.

## Contribution

It provides a rigorous derivation of the number and wavefunctions of exact Majorana zero modes in an inhomogeneous quantum compass model, including explicit solutions for finite chains.

## Key findings

- Number of EMZMs depends on the partition of sites with zero fields
- Explicit wavefunctions for EMZMs are derived
- No almost Majorana zero modes unless fields are turned off on both sublattices

## Abstract

We study the emergence of exact Majorana zero modes (EMZMs) in a one-dimensional quantum transverse compass model with both the nearest-neighbor interactions and transverse fields varying over space. By transforming the spin system into a quadratic Majorana-fermion model, we derive an exact formula for the number of the emergent EMZMs, which is found to depend on the partition nature of the lattice sites on which the magnetic fields vanish. We also derive explicit expressions for the wavefunctions of these EMZMs and show that they indeed depend on fine features of the foregoing partition of site indices. Based on the above rigorous results about the EMZMs, we provide an interpretation for the interesting dependence of the eigenstate-degeneracy on the transverse fields observed in prior literatures. As a special case, we employ a plane-wave ansatz to exactly solve an open compass chain with alternating nearest-neighbor interactions and staggered magnetic fields. Explicit forms of the canonical Majorana modes diagonalizing the model are given even for finite chains. We show that besides the possibly existing EMZMs, no almost Majorana zero modes exist unless the fields on both the two sublattices are turned off. Our results might shed light on the control of ground-state degeneracies by solely tuning the external fields in related systems.