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Time-reversal symmetry breaking phase in the Hubbard model: a VCA study

We study the stability of the time-reversal symmetry breaking staggered flux phase of a single band Hubbard model, within the Variational Cluster Approach (VCA). For intermediate and small values of the interaction $U$, we find metastable solutions for the staggered flux phase, with a maximum current per bond at $U\approx 3.2$. However, allowing for antiferromagnetic and superconducting long-range order it turns out that in the region at and close to half filling the antiferromagnetic phase is the most favorable energetically. The effect of nearest-neighbour interaction is also considered. Our results show that a negative nearest-neighbour interaction and finite doping favors the stability of the staggered-flux phase. We also present preliminary results for the three-band Hubbard model obtained with a restricted set of variational parameters. For this case, no spontaneous time-reversal symmetry breaking phase is found in our calculations.