Criteria and analytical results for the Pseudogap at the Van Hove point in two dimensions

Abstract: I establish the criteria and obtained analytical results for the pseudogap at the Van Hove (antinodal) point on the Fermi surface in two dimensions. The original criterion $\xi >> \xi_{th_db}=v_F/\pi T$ is not applicable in this case since Fermi velocity $v_F=0$. It turns out that the characteristic length for the pseudogap crossover at the Van Hove point $\xi_{th_vh} \propto 1/T^{1/2}$, which is significantly shorter than the one at the regular Fermi surface points $\xi_{th_db} \propto 1/T$. In particular, $\xi_{th_vh}$ is between one and two lattice spacing in the intermediate interaction regime of the Hubbard model. I have also identified the regime where there is still a single maximum in the spectral function, but single particle properties are abnormal. Specifically, the imaginary part of the self-energy has a minimum instead of maximum at the Fermi level and the slope of the real part of the self-energy is positive instead of negative. The important advantages of an analytical approach is that it provides results in both the Matsubara frequency representation and in real frequencies representation. I compare Matsubara frequency results with the exact numerical results of the Monte Carlo methods in the Hubbard model and then show what they correspond to in the real frequency self-energy and spectral function.