# Level crossings induced by a longitudinal coupling in the transverse field Ising chain

## TL;DR

This paper investigates how a longitudinal coupling affects the energy spectrum of the transverse field Ising chain, revealing multiple level crossings and their relation to topological phases and Majorana fermions.

## Contribution

It demonstrates that a finite longitudinal coupling induces multiple level crossings in the topological phase, supported by perturbation theory and a mapping to the open Kitaev chain.

## Key findings

- N level crossings occur as a function of the field in finite systems
- All level crossings appear simultaneously when the longitudinal coupling is introduced
- The problem is exactly solvable via Majorana fermions in the mapped Kitaev chain

## Abstract

We study the effect of antiferromagnetic longitudinal coupling on the one-dimensional transverse field Ising model with nearest-neighbour couplings. In the topological phase where, in the thermodynamic limit, the ground state is twofold degenerate, we show that, for a finite system of N sites, the longitudinal coupling induces N level crossings between the two lowest lying states as a function of the field. We also provide strong arguments suggesting that these N level crossings all appear simultaneously as soon as the longitudinal coupling is switched on. This conclusion is based on perturbation theory, and a mapping of the problem onto the open Kitaev chain, for which we write down the complete solution in terms of Majorana fermions.