# Quaternary Jordan-Wigner mapping and topological extended-kink phase in the interacting Kitaev ring

## TL;DR

This paper introduces an exact quaternary Jordan-Wigner mapping for the interacting Kitaev ring, revealing a novel gapless topological phase with unique properties influenced by ring frustration and boundary conditions.

## Contribution

The authors develop an exact quaternary Jordan-Wigner mapping involving multiple rings, enabling the discovery of a gapless topological extended-kink phase in the interacting Kitaev ring.

## Key findings

- Discovery of a gapless topological extended-kink phase.
- Identification of a unique, symmetry-protected ground state.
- Robustness of localized kink zero modes against disorder.

## Abstract

On a ring, a single Jordan-Wigner transformation between the Kitaev model and the spin model suffers redundant degrees of freedom. However, we can establish an exact quaternary Jordan-Wigner mapping involving two Kitaev rings and two spin rings with periodic or antiperiodic boundary conditions. This mapping facilitates us to demonstrate exactly how a topological extended-kink (TEK) phase develops in the interacting Kitaev ring with odd number of lattice sites. The emergence of this new phase is attributed to the effect of geometrical ring frustration. Unlike the usual topological phases protected by energy gap in noninteracting systems, the TEK phase is gapless. And because the spectra of low energy excitations are quadratic, the specific heat per site approaches a half of Boltzmann constant near absolute zero temperature. More interestingly, the ground state is unique, immune to spontaneous symmetry breaking. It exhibits a long-range correlation function with a nonlocal factor, but no local order parameter can be defined. As a concomitant effect, a special kind of localized kink zero mode (KZM) takes place if we introduce a type of bond defect. We also show that the KZM is robust against moderate disorders.