Bond excitations in the pseudogap phase of the Hubbard Model
  Using the dynamical cluster approximation, we calculate the correlation functions associated with the nearest neighbor bond operator which measure the z component of the spin exchange in the two-dimensional Hubbard model with $U$ equal to the bandwidth. We find that in the pseudogap region, the local bond susceptibility diverges at T=0. This shows the existence of degenerate bond spin excitation and implies quantum criticality and bond order formation when long range correlations are considered. The strong correlation between excitations on parallel neighboring bonds suggests bond singlet dimerization. The suppression of divergence for $n< \approx 0.78$ implies that tor these model parameters this is quantum critical point which separates the unconventional pseudogap region characterized by bond order from a conventional Fermi liquid. 
Alexandru Macridin, Mark Jarrell