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I establish the criteria and obtained analytical results for the pseudogap atthe Van Hove (antinodal) point on the Fermi surface in two dimensions. Theoriginal criterion $\xi >> \xi_{th\_db}=v_F/\pi T$ is not applicable in thiscase since Fermi velocity $v_F=0$. It turns out that the characteristic lengthfor the pseudogap crossover at the Van Hove point $\xi_{th\_vh} \propto1/T^{1/2}$, which is significantly shorter than the one at the regular Fermisurface points $\xi_{th\_db} \propto 1/T$. In particular, $\xi_{th\_vh}$ isbetween one and two lattice spacing in the intermediate interaction regime ofthe Hubbard model. I have also identified the regime where there is still asingle maximum in the spectral function, but single particle properties areabnormal. Specifically, the imaginary part of the self-energy has a minimuminstead of maximum at the Fermi level and the slope of the real part of theself-energy is positive instead of negative. The important advantages of ananalytical approach is that it provides results in both the Matsubara frequencyrepresentation and in real frequencies representation. I compare Matsubarafrequency results with the exact numerical results of the Monte Carlo methodsin the Hubbard model and then show what they correspond to in the realfrequency self-energy and spectral function.

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