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arXiv:cond-mat/0109392 · Physical Review Letters

Magnetic and Superconducting Instabilities of the Hubbard Model at the Van Hove Filling

Carsten Honerkamp A, Manfred Salmhofer B

Published Oct 16, 2001• 138 citations
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Abstract

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.

DOI
10.1103/physrevlett.87.187004

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We use a novel temperature-flow renormalization group technique to analyze magnetic and superconducting instabilities in the two-dimensional t\ensuremath−t\ensuremath′{t\ensuremath{-}t}^{\ensuremath{'}}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′{t}^{\ensuremath{'}}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 tc\ensuremath′\ensuremath≈\ensuremath−0.33t{t}_{c}^{\ensuremath{'}}\ensuremath{\approx}\ensuremath{-}0.33ttc\ensuremath′​\ensuremath≈\ensuremath−0.33t, suggesting a quantum critical point between ddd-wave pairing at moderate t\ensuremath′>tc\ensuremath′{t}^{\ensuremath{'}}>{t}_{c}^{\ensuremath{'}}t\ensuremath′>tc\ensuremath′​ and ferromagnetism for t\ensuremath′<tc\ensuremath′{t}^{\ensuremath{'}}<{t}_{c}^{\ensuremath{'}}t\ensuremath′<tc\ensuremath′​. Upon increasing the particle density in the latter regime, the leading instability occurs in the triplet pairing channel.
We derive a differential equation for the one-particle-irreducible vertex functions of interacting fermions as a function of the temperature. Formally, these equations correspond to a Wilsonian renormalization group (RG) scheme that uses the temperature as an explicit scale parameter. Our method allows us to analyze the competition between superconducting and various magnetic Fermi surface instabilities in the one-loop approximation. In particular, this includes ferromagnetic fluctuations, which are difficult to treat on an equal footing in conventional Wilsonian momentum space techniques. Applying the scheme to the two-dimensional $t\ensuremath{-}{t}^{\ensuremath{'}}$ Hubbard model we investigate the RG flow of the interactions at the van Hove filling with varying next-nearest-neighbor hopping ${t}^{\ensuremath{'}}.$ Starting at ${t}^{\ensuremath{'}}=0$ we describe the evolution of the flow to strong coupling from an antiferromagnetic nesting regime over a d-wave regime at moderate ${t}^{\ensuremath{'}}$ to a ferromagnetic region at larger absolute values of ${t}^{\ensuremath{'}}.$ Upon increasing the particle density in the latter regime the ferromagnetic tendencies are cut off and the leading instability occurs in the triplet superconducting 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 present a stability analysis of the two-dimensional $t\ensuremath{-}{t}^{\ensuremath{'}}$ Hubbard model for various values of the next-nearest-neighbor hopping ${t}^{\ensuremath{'}},$ and electron concentrations close to the Van Hove filling by means of the flow equation method. For ${t}^{\ensuremath{'}}>~\ensuremath{-}t/3$ a ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave Pomeranchuk instability dominates (apart from antiferromagnetism at small ${t}^{\ensuremath{'}}).$ At ${t}^{\ensuremath{'}}<\ensuremath{-}t/3,$ the leading instabilities are a g-wave Pomeranchuk instability and p-wave particle-hole instability in the triplet channel at temperatures T<0.15t,T<0.15t,T<0.15t, and an s∗{s}^{*}s∗-magnetic phase for T>0.15t;T>0.15t;T>0.15t; upon increasing the electron concentration, the triplet analog of the flux phase occurs at low temperatures. Other weaker instabilities are also found.
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 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 UUU 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 fff-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 fff wave is suppressed and the pairing susceptibility of fn{f}_{n}fn​ wave is enhanced, however, both the effective pairing interaction with fff wave and fn{f}_{n}fn​ wave are negative, which indicates that neither fff-wave nor fn{f}_{n}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 fff-wave pairing, while antiferromagnetic fluctuation tends to promote fn{f}_{n}fn​-wave pairing.
We study a ferromagnetic tendency in the two-dimensional Hubbard model near van Hove filling by using a functional renormalization-group method. We compute temperature dependences of magnetic susceptibilities including incommensurate magnetism. The ferromagnetic tendency is found to occur in a dome-shaped region around van Hove filling with an asymmetric property: incommensurate magnetism is favored near the edge of the dome above van Hove filling whereas a first-order-like transition to the ferromagnetic ground state is expected below van Hove filling. The dome-shaped phase diagram is well captured in the Stoner theory by invoking a smaller Coulomb interaction. Triplet p-wave superconductivity tends to develop at low temperatures inside the dome and extends more than the ferromagnetic region above van Hove filling.