Phase diagrams of the two-dimensional one-band $t\ensuremath{-}{t}^{\ensuremath{'}}$ Hubbard model are obtained within the two-patch and temperature-cutoff many-patch renormalization group approaches. At small ${t}^{\ensuremath{'}}$ and at van Hove band fillings antiferromagnetism dominates, while with increasing ${t}^{\ensuremath{'}}$ or changing filling antiferromagnetism is replaced by d-wave superconductivity. Near ${t}^{\ensuremath{'}}=t/2$ and close to van Hove band fillings the system is unstable towards ferromagnetism. Away from van Hove band fillings this ferromagnetic instability is replaced by a region with dominating triplet p-wave superconducting correlations. The results of the renormalization-group approach are compared with the mean-field results and the results of the T-matrix approximation.
We investigate the possibility of ferromagnetic ordering in the nondegenerate Hubbard model on the face-centered cubic lattice within the functional renormalization group technique using temperature as a scale parameter. We assume the relations between nearest, next-nearest, and next-next-nearest hopping parameters providing higher-order (giant) van Hove singularity of the density of states. The ferromagnetic instability formation with lowering temperature is described consistently in the one-loop approximation for a one-particle irreducible vertex of two-particle electron interaction. The chemical potential versus temperature phase diagrams are calculated. We find ferromagnetic order only for sufficiently strong divergence of the density of states and fillings in the vicinity of van Hove singularity. The obtained Curie temperature is more than an order of magnitude smaller than the results of the random-phase approximation. The main origin of the suppression of ferromagnetism is the screening of interaction in the particle-particle channel. We also do not find the pronounced tendency towards incommensurate order when the Fermi level is moved away from a van Hove singularity, such that the first-order quantum phase transitions from the ferro- to paramagnetic phase are obtained.
We use a novel temperature-flow renormalization group technique to analyze magnetic and superconducting instabilities in the two-dimensional ${t\ensuremath{-}t}^{\ensuremath{'}}$ Hubbard model for particle densities close to the Van Hove filling as a function of the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$. In the one-loop flow at the Van Hove filling, the characteristic temperature for the flow to strong coupling is suppressed drastically around ${t}_{c}^{\ensuremath{'}}\ensuremath{\approx}\ensuremath{-}0.33t$, suggesting a quantum critical point between
d-wave pairing at moderate ${t}^{\ensuremath{'}}>{t}_{c}^{\ensuremath{'}}$ and ferromagnetism for ${t}^{\ensuremath{'}}<{t}_{c}^{\ensuremath{'}}$. Upon increasing the particle density in the latter regime, the leading instability occurs in the triplet pairing channel.
We investigate magnetic and superconducting instabilities of the two-dimensional $t\ensuremath{-}{t}^{\ensuremath{'}}$ Hubbard model on a square lattice at van Hove densities from weak to intermediate coupling by means of the two-particle self-consistent approach. We find that as the next-nearest-neighbor hopping $|{t}^{\ensuremath{'}}|$ increases from zero, the leading instability is towards an incommensurate spin-density wave whose wave vector moves slowly away from $(\ensuremath{\pi},\ensuremath{\pi}).$ For intermediate values of $|{t}^{\ensuremath{'}}|,$ the leading instability is towards ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave superconductivity. For larger $|{t}^{\ensuremath{'}}|>0.33t,$ there are signs of a crossover to ferromagnetism at extremely low temperatures. The suppression of the crossover temperature is driven by Kanamori screening that strongly renormalizes the effective interaction and also causes the crossover temperature to depend only weakly on ${t}^{\ensuremath{'}}.$ Electronic self-energy effects for large $|{t}^{\ensuremath{'}}|$ lead to considerable reduction of the zero-energy single-particle spectral weight beginning at temperatures as high as $T\ensuremath{\lesssim}0.1t,$ an effect that may be detrimental to the existence of a ferromagnetic ground state at weak coupling.
We have investigated the magnetism and pairing correlations of the triangular lattice based on the Hubbard model using the determinant quantum Monte Carlo method and the constrained path Monte Carlo. The results show that the presence of the next-nearest-neighbor hopping integral ${t}^{\ensuremath{'}}$ introduces an additional energy scale to the system, and through ${t}^{\ensuremath{'}}$, one can regulate the shape of the density of states and thus the position of the van Hove singularity point. Increasing inverse temperature $\ensuremath{\beta}$ and on-site interaction
U favor the formation of ferromagnetic correlation in a rather large filling region, and the calculations for different lattice sizes show that the range of the ferromagnetic correlations is smaller than the smallest lattice simulated at the investigated temperatures. We study the different pairing correlations of the triangular lattice near several typical fillings and show that the
f-wave pairing dominates the system in the filling region near the van Hove singularity point with a high density of states, where the ferromagnetic correlation is also enhanced. When the filling is close to half-filling, the pairing susceptibility with
f wave is suppressed and the pairing susceptibility of
fn wave is enhanced, however, both the effective pairing interaction with
f wave and
fn wave are negative, which indicates that neither
f-wave nor
fn-wave superconductivity may exist. Finally, we find that the pairing channel of different symmetry in the system maybe closely related to the magnetic properties. Ferromagnetic fluctuation favors the formation of
f-wave pairing, while antiferromagnetic fluctuation tends to promote
fn-wave pairing.
The interplay of magnetic and superconducting fluctuations in two-dimensional systems with van Hove singularities in the electronic spectrum is considered within the functional renormalization-group (fRG) approach. While the fRG flow has to be stopped at a certain minimal temperature
TRGmin, we study temperature dependence of magnetic and superconducting susceptibilities below
TRGmin to obtain the crossover temperatures to the regime with strong magnetic and superconducting fluctuations. Near half filling we obtain the largest crossover temperature, corresponding to a regime with strong commensurate magnetic fluctuations, which is replaced by a regime with strong incommensurate fluctuations further away from half filling. With further decreasing density the system undergoes quantum phase transition from incommensurate to paramagnetic phase. Similarly to results of Hertz-Moriya-Millis approach, the temperature dependence of the inverse (incommensurate) magnetic susceptibility at the magnetic quantum-critical point is found almost linear in temperature.