# Non-local String Order Parameter in the $S = 1/2$ Kitaev-Heisenberg Ladder

## TL;DR

This paper demonstrates the existence of a non-local string order parameter in the spin-1/2 Kitaev-Heisenberg ladder, revealing hidden order in a phase previously considered disordered, and maps it to an XY chain with Ising coupling.

## Contribution

It introduces a non-local string order parameter for the Kitaev phase in the ladder model, persisting with Heisenberg interaction, and relates it to a solvable XY chain model.

## Key findings

- Existence of a non-local string order parameter in the Kitaev phase.
- Identification of two phases with non-zero SOP for different Kitaev interactions.
- Phase diagram similar to the 2D honeycomb Kitaev-Heisenberg model.

## Abstract

We study the spin-$\frac{1}{2}$ Kitaev-Heisenberg (KJ) model in a two-leg ladder. Without a Heisenberg interaction, the Kitaev phase in the ladder model has Majorana fermions with local Z$_2$ gauge fields, and is usually described as a disordered phase without any order parameter. Here we prove the existence of a non-local string order parameter (SOP) in the Kitaev phase which survives with a finite Heisenberg interaction. The SOP is obtained by relating the Kitaev ladder, through a non-local unitary transformation, to a one-dimensional $XY$ chain with an Ising coupling to a dangling spin at every site. This differentiates the Kitaev phases from other nearby phases including a rung singlet. Two phases with non-zero SOP corresponding to ferromagnetic and antiferromagnetic Kitaev interactions are identified. The full phase diagram of the KJ ladder is determined using exact diagonalization and density matrix renormalization group methods, which shows a striking similarity to the KJ model on a two-dimensional honeycomb lattice.