Probing topological degeneracy on a torus using superconducting altermagnets

Abstract

The notion of topological order (TO) can be defined through the characteristic ground state degeneracy of a system placed on a manifold with non-zero genus $g$, such as a torus. This ground state degeneracy has served as a key tool for identifying TOs in theoretical calculations but it has never been possible to probe experimentally because fabricating a device in the requisite toroidal geometry is generally not feasible. Here we discuss a practical method that can be used to overcome this difficulty in a class of topologically ordered systems that consist of a TO and its time reversal conjugate $\overline{\mathrm{TO}}$. The key insight is that a system possessing such $\mathrm{TO}\otimes \overline{\mathrm{TO}}$ order fabricated on an annulus behaves effectively as TO on a torus, provided that one supplies a symmetry-breaking perturbation that gaps out the edge modes. We illustrate this general principle using a specific example of a spin-polarized $p_x\pm i{p}_y$ chiral superconductor which is closely related to the Moore-Read Pfaffian fractional quantum Hall state. Specifically, we introduce a simple model with altermagnetic normal state which, in the presence of an attractive interaction, hosts a helical $(p_x-i{p}_y{)}^{\uparrow }\otimes (p_x+i{p}_y{)}^{\downarrow }$ superconducting ground state. We demonstrate that when placed on an annulus with the appropriate symmetry-breaking edge perturbation this planar two-dimensional system, remarkably, exhibits the same pattern of ground state degeneracy as a $p_x+i{p}_y$ superconductor on a torus. We discuss broader implications of this behavior and ways it can be tested experimentally.

15 pages, 10 figures