Existence of topologically ordered Euler phases

Determine whether intrinsically topologically ordered Euler phases exist in two dimensions, i.e., whether there are interacting phases characterized by a nontrivial Euler class that also exhibit intrinsic topological order (such as nonzero topological entanglement entropy and ground-state degeneracy on a torus), rather than being merely symmetry-protected without intrinsic topological order.

Background

The paper constructs a spinful interacting projected entangled pair state (PEPS) by applying a Gutzwiller projection to two copies of a non-interacting Euler insulator PEPS. Using tensor network methods, the authors analyze entanglement entropy, entanglement spectrum, transfer operator gap, and static structure factor.

Their results indicate no topological correction to the area law (suggesting absence of intrinsic topological order), a cusp in the entanglement spectrum reminiscent of non-interacting Euler insulators, evidence consistent with a finite gap, and the presence of Bragg peaks indicating local order. These findings suggest a symmetry-protected, non-intrinsically ordered Euler phase in their construction.

Given these observations, the authors pose the broader unresolved question of whether any topologically ordered Euler phases exist at all, leaving this as future work.