Exact ground states and topological order in interacting Kitaev/Majorana chains

TL;DR

This paper analytically finds exact ground states in an interacting one-dimensional Kitaev/Majorana chain, revealing topological order and a generalization of Majorana zero modes even with interactions.

Contribution

It provides an exact solution for the ground states of an interacting Kitaev chain and demonstrates the persistence of topological order and Majorana modes with interactions.

Findings

Exact ground states are analytically obtained with interactions.

Topological order persists with interactions and can be connected to non-interacting states.

Interacting Majorana-like operators are explicitly constructed.

Abstract

We study a system of interacting spinless fermions in one dimension which, in the absence of interactions, reduces to the Kitaev chain [A. Yu Kitaev, Phys.-Usp. \textbf{44}, 131 (2001)]. In the non-interacting case, a signal of topological order appears as zero-energy modes localized near the edges. We show that the exact ground states can be obtained analytically even in the presence of nearest-neighbor repulsive interactions when the on-site (chemical) potential is tuned to a particular function of the other parameters. As with the non-interacting case, the obtained ground states are two-fold degenerate and differ in fermionic parity. We prove the uniqueness of the obtained ground states and show that they can be continuously deformed to the ground states of the non-interacting Kitaev chain without gap closing. We also demonstrate explicitly that there exists a set of operators each of which maps one of the ground states to the other with opposite fermionic parity. These operators can be thought of as an interacting generalization of Majorana edge zero modes.