# Exact zero modes in twisted Kitaev chains

## TL;DR

This paper demonstrates the existence and properties of exact zero modes in twisted Kitaev chains with generalized boundary conditions, including their localization, conditions for occurrence, and robustness against disorder and interactions.

## Contribution

It provides a comprehensive analytical study of zero modes in twisted Kitaev chains, including conditions for their existence and their robustness, extending understanding beyond standard boundary conditions.

## Key findings

- Exact zero modes exist in large topological Kitaev chains with twisted boundaries.
- Zero modes are localized and can be rigorously characterized for specific boundary phase parameters.
- Zero modes persist even with disorder and interactions, indicating robustness.

## Abstract

We study the Kitaev chain under generalized twisted boundary conditions, for which both the amplitudes and the phases of the boundary couplings can be tuned at will. We explicitly show the presence of exact zero modes for large chains belonging to the topological phase in the most general case, in spite of the absence of "edges" in the system. For specific values of the phase parameters, we rigorously obtain the condition for the presence of the exact zero modes in finite chains, and show that the zero modes obtained are indeed localized. The full spectrum of the twisted chains with zero chemical potential is analytically presented. Finally, we demonstrate the persistence of zero modes (level crossing) even in the presence of disorder or interactions.