Detecting Majorana fermions in quasi-one-dimensional topological phases using nonlocal order parameters

TL;DR

This paper introduces nonlocal order parameters to identify Majorana-hosting topological phases in quasi-1D and 2D systems, enabling bulk detection of symmetry-protected topological states in cold atom experiments.

Contribution

It proposes simple nonlocal order parameters for interacting 1D topological superconductors and classifies 2D topological phases protected by translation symmetry.

Findings

Nonlocal order parameters distinguish topological phases in 1D and 2D.

Classification of 2D topological superconductors with translation symmetry.

Physical interpretation of phases as defect condensates.

Abstract

Topological phases which host Majorana fermions can not be identified via local order parameters. We give simple nonlocal order parameters to distinguish quasi-one-dimensional (1D) topological superconductors of spinless fermions, for any interacting model in the absence of time reversal symmetry. These string or "brane" order parameters are natural for measurements in cold atom systems using quantum gas microscopy. We propose them as a way to identify symmetry-protected topological phases of Majorana fermions in cold atom experiments via bulk rather than edge degrees of freedom. Subsequently, we study two-dimensional (2D) topological superconductors via the quasi-1D limit of coupling $N$ identical chains on the cylinder. We classify the symmetric, interacting topological phases protected by the additional $\mathbb{Z}_N$ translation symmetry. The phases include quasi-1D analogs of (i) the $p+ip$ chiral topological superconductor, which can be distinguished up to the 2D Chern number mod 2, and (ii) the 2D weak topological superconductor. We devise general rules for constructing nonlocal order parameters which distinguish the phases. These rules encode the signature of the fermionic topological phase in the symmetry properties of the terminating operators of the nonlocal string or brane. The nonlocal order parameters for some of these phases simply involve a product of the string order parameters for the individual chains. Finally, we give a physical picture of one of the topological phases as a condensate of certain defects, which motivates the form of the nonlocal order parameter and is reminiscent of higher dimensional constructions of topological phases.