# Pretopological fractional excitations in the two-leg flux ladder

## TL;DR

This paper explores the transition from topological order to a charge density wave in a two-leg flux ladder, analyzing fractional excitations and potential realizations in cold-atom experiments, with implications for non-Abelian zero modes.

## Contribution

It provides an analytical and numerical study of the 1D-2D crossover in topological systems, introducing a method to probe fractional charges in a flux ladder and discussing experimental realizations.

## Key findings

- Charge density wave amplitude decreases exponentially with system size.
- Fractional charge excitations can be probed in a two-leg flux ladder.
- Degeneracy splitting vanishes exponentially with the number of wires.

## Abstract

Topological order, the hallmark of fractional quantum Hall states, is primarily defined in terms of ground-state degeneracy on higher-genus manifolds, e.g. the torus. We investigate analytically and numerically the smooth crossover between this topological regime and the Tao-Thouless thin torus quasi-1D limit. Using the wire-construction approach, we analyze an emergent charge density wave (CDW) signifying the break-down of topological order, and relate its phase shifts to Wilson loop operators. The CDW amplitude decreases exponentially with the torus circumference once it exceeds the transverse correlation length controllable by the inter-wire coupling. By means of numerical simulations based on the matrix product states (MPS) formalism, we explore the extreme quasi-1D limit in a two-leg flux ladder and present a simple recipe for probing fractional charge excitations in the $\nu=1/2$ Laughlin-like state of hard-core bosons. We discuss the possibility of realizing this construction in cold-atom experiments. We also address the implications of our findings to the possibility of producing non-Abelian zero modes. As known from rigorous no-go theorems, topological protection for exotic zero modes such as parafermions cannot exist in 1D fermionic systems and the associated degeneracy cannot be robust. Our theory of the 1D-2D crossover allows to calculate the splitting of the degeneracy, which vanishes exponentially with the number of wires, similarly to the CDW amplitude.