Ground state phase diagram of the repulsive fermionic $t-t^{\prime}$ Hubbard model on the square lattice from weak-coupling

We obtain a complete and exact in the weak-coupling limit ($U \rightarrow 0$) ground state phase diagram of the repulsive fermionic Hubbard model on the square lattice for filling factors $0 < n < 2$ and next-nearest-neighbour hopping amplitudes $0 \le t^{\prime} \le 0.5$. Phases are distinguished by the symmetry and the number of nodes of the superfluid order parameter. The phase diagram is richer than may be expected and typically features states with a high --- higher than that of the fundamental mode of the corresponding irreducible representation --- number of nodes. The effective coupling strength in the Cooper channel $λ$, which determines the critical temperature $T_c$ of the superfluid transition, is calculated in the whole parameter space and regions with high values of $λ$ are identified. It is shown that besides the expected increase of $λ$ near the Van Hove singularity line, joining the ferromagnetic and antiferromagnetic points, another region with high values of $λ$ can be found at quarter filling and $t^{\prime}=0.5$ due to the presence of a line of nesting at $t^{\prime} \ge 0.5$. The results can serve as benchmarks for controlled non-perturbative methods and guide the ongoing search for high-$T_c$ superconductivity in the Hubbard model.

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