  Z2 spin liquids have topological order. One manifestation of this is that a
Z2 spin liquid on a torus exhibits a four-fold degeneracy. Recent numerical
evidence has argued for the existence of a spin liquid ground state in the
Hubbard model on a honeycomb lattice near U approximately 4. The evidence for
this claim involves the presence of a gapped state that lacks any identifiable
order. This argument relies on being able to distinguish small order from no
order which is notoriously difficult. In this paper we demonstrate an approach
which uses quantum Monte Carlo to search for one of the key features that
positively identify the topological spin liquid: the topological degeneracy.
For any finite system, this topological degeneracy is split where the splitting
decays exponentially with system size. We search for low lying states in the
energy spectrum that could be identified as these topologically degenerate
states. We show that, for system sizes up to N=162 sites, there is no evidence
for these states being significantly below the first excited state giving
evidence against the existence of a topological phase. We discuss the possible
options for the Hubbard model on the honeycomb lattice in the absence of such
degeneracy.
