Probing topological degeneracy on a torus using superconducting altermagnets

The notion of topological order (TO) can be defined through the characteristic ground state degeneracy of a system placed on a manifold with nonzero genus $g$, such as a torus. This ground state degeneracy has served as a key tool for identifying TOs in theoretical calculations but, to our knowledge, 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 $\mathrm{TO}\ensuremath{\bigotimes}\overline{\mathrm{TO}}$ 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 ${p}_{x}\ifmmode\pm\else\textpm\fi{}i{p}_{y}$ chiral superconductor which is closely related to the Moore-Read Pfaffian fractional quantum Hall state. Specifically, we introduce a simple model with an altermagnetic normal state which, in the presence of an attractive interaction, hosts a helical ${({p}_{x}\ensuremath{-}i{p}_{y})}^{\ensuremath{\uparrow}}\ensuremath{\bigotimes}{({p}_{x}+i{p}_{y})}^{\ensuremath{\downarrow}}$ 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.