Phase Transitions in the Hubbard Model on the Square Lattice

We study the low temperature properties of the two-dimensional weakly interacting Hubbard model on ${\text{ZZZ}}^2$ with renormalized chemical potential $\mu =2-{\mu }_0$, ${\mu }_0={10}^{-10}$ fixed, in which case the Fermi surface is close to a perfect square. Using fermionic functional integrals, cluster expansions and rigorous renormalization group analysis, we prove that the perturbation series for the two-point Schwinger function is analytic in the coupling constant $\text{ł}$ in the domain $\text{ł}\in {\text{RR}}_T=\{\text{ł}\in \text{RRR},\mathrm{\mid }\lambda {\log }^2({\mu }_{0}T\mathrm{/}{C}_1)\mathrm{\mid }\le C_2\}$ for any fixed temperature $T>0$, suggesting that there is a phase transition with critical temperature $T_c=\frac{C_1}{{\text{m}}_0}\exp (-C_2^{1\mathrm{/}2}\mathrm{\mid }\lambda {\mathrm{\mid }}^{-1\mathrm{/}2})$. Here $C_1,C_2$ are positive constants independent of $T$ and $\text{ł}$. We also prove that the second derivative of the momentum space self-energy function w.r.t. the external momentum is not uniformly bounded, suggesting that this model is {\it not} a Fermi liquid in the mathematically precise sense of Salmhofer. This result can be viewed as a first step towards rigorous study of the Fermi liquid-non Fermi liquid crossover phenomenon.