# Majorana Spin Liquids, Topology and Superconductivity in Ladders

## TL;DR

This paper explores Majorana spin liquids and topological phases in ladder models related to the Kitaev honeycomb model, revealing novel phases, edge states, and superconducting transitions through theoretical analysis.

## Contribution

It introduces a generalized phase diagram for ladder systems, analyzes topological gapless modes, and links hole doping to superconducting-insulating transitions with topological features.

## Key findings

- Identification of topological gapless edge modes
- Mapping of doped holes to the SSH model
- Discovery of superconducting-insulating transition in hole pairs

## Abstract

We theoretically address spin chain analogs of the Kitaev quantum spin model on the honeycomb lattice. The emergent quantum spin liquid phases or Anderson resonating valence bond (RVB) states can be understood, as an effective model, in terms of p-wave superconductivity and Majorana fermions. We derive a generalized phase diagram for the two-leg ladder system with tunable interaction strengths between chains allowing us to vary the shape of the lattice (from square to honeycomb ribbon or brickwall ladder). We evaluate the winding number associated with possible emergent (topological) gapless modes at the edges. In the Az phase, as a result of the emergent Z2 gauge fields and pi-flux ground state, one may build spin-1/2 (loop) qubit operators by analogy to the toric code. In addition, we show how the intermediate gapless B phase evolves in the generalized ladder model. For the brickwall ladder, the $B$ phase is reduced to one line, which is analyzed through perturbation theory in a rung tensor product states representation and bosonization. Finally, we show that doping with a few holes can result in the formation of hole pairs and leads to a mapping with the Su-Schrieffer-Heeger model in polyacetylene; a superconducting-insulating quantum phase transition for these hole pairs is accessible, as well as related topological properties.