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arXiv:1512.04271 · Physical review. B./Physical review. B

Ground-state phase diagram of the repulsive fermionic t−t′ Hubbard model on the square lattice from weak coupling

Fedor Šimkovic, Xuan-Wen Liu, Youjin Deng, Evgeny Kozik

Published Aug 5, 2016• 22 citations
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Abstract

We obtain a complete and numerically exact in the weak-coupling limit $(U\ensuremath{\rightarrow}0)$ ground-state phase diagram of the repulsive fermionic Hubbard model on the square lattice for filling factors $0<n<2$ and next-nearest-neighbor hopping amplitudes $0\ensuremath{\le}{t}^{\ensuremath{'}}\ensuremath{\le}0.5$. 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 $\ensuremath{\lambda}$, which determines the critical temperature ${T}_{c}$ of the superfluid transition, is calculated in the whole parameter space and regions with high values of $\ensuremath{\lambda}$ are identified. It is shown that besides the expected increase of $\ensuremath{\lambda}$ near the Van Hove singularity line, joining the ferromagnetic and antiferromagnetic points, another region with high values of $\ensuremath{\lambda}$ can be found at quarter filling and ${t}^{\ensuremath{'}}=0.5$ due to the presence of a line of nesting at ${t}^{\ensuremath{'}}\ensuremath{\ge}0.5$. The results can serve as benchmarks for controlled nonperturbative methods and guide the ongoing search for high-${T}_{c}$ superconductivity in the Hubbard model.

DOI
10.1103/physrevb.94.085106

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We determine the symmetry of Cooper pairs, on the basis of the perturbation theory in terms of the Coulomb interaction U , for the two-dimensional Hubbard model on the square lattice. The phase diagram is investigated in detail. The Hubbard model for small U is mapped on to an effective Hamiltonian with the attractive interaction using the canonical transformation: H eff = e S H e − S . The gap equation of the weak coupling formulation is solved without numerical ambiguity to determine the symmetry of Cooper pairs. The superconducting gap crucially depends on the position of the van Hove singularity. We show the phase diagram in the plane of the electron filling n e and the next nearest-neighbor transfer t '. The d-wave pairing is dominant for the square lattice in a wide range of n e and t '. The d-wave pairing is also stable for the square lattice with anisotropic t '. The three-band d–p model is also investigated, for which the d-wave pairing is stable in a wide range of n e and t pp (the transfer between neighboring oxygen atoms). In the weak coupling analysis, the second-neighbor transfer parameter - t ' could not be so large so that the optimum doping rate is in the range of 0.8 < n e < 0.85.
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