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arXiv:cond-mat/0306296 · Physical review. B, Condensed matter

Weak ferromagnetism and other instabilities of the two-dimensional t − t ′ Hubbard model at van Hove fillings

V. Hankevych, Bumsoo Kyung, A.–M. S. Tremblay

Published Dec 5, 2003• 30 citations
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Abstract

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.

DOI
10.1103/physrevb.68.214405

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We use a novel temperature-flow renormalization group technique to analyze magnetic and superconducting instabilities in the two-dimensional ${t\ensuremath{-}t}^{\ensuremath{'}}$ 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 ddd-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 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.
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 investigate the magnetic instabilities of the nondegenerate (s(s(s-band) and a degenerate (d(d(d-band) Hubbard model in two dimensions using many-body effects due to particle-particle diagrams and Hund's rule local correlations. The density of states and the position of the Van Hove singularity change depending on the value of next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$. The Stoner parameter is strongly reduced in the sss-band case, and ferromagnetism survives only if the electron density is small and the band has flat regions. Due to next-nearest-neighbor hopping there are flat regions in $\ensuremath{\Gamma}\ensuremath{-}X$ and $\ensuremath{\Gamma}\ensuremath{-}Y$ directions. In contrast, for the ddd-band case the reduction of the Stoner parameter which follows from particle-particle correlations is much smaller and ferromagnetism survives to a large extent. Inclusion of local spin-spin correlations has a limited destabilizing effect on the magnetic states.
Using an improved version of the projection quantum Monte Carlo technique, we study the square-lattice Hubbard model with nearest-neighbor hopping ttt and next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$ by simulation of lattices with up to $20\ifmmode\times\else\texttimes\fi{}20$ sites. For a given ${R\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2t}^{\ensuremath{'}}/t$, we consider that filling which leads to a singular density of states of the noninteracting problem. For repulsive interactions, we find an itinerant ferromagnet (antiferromagnet) for R=0.94R\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.94R=0.94 ( R=0.2R\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.2R=0.2). This is consistent with the prediction of the TTT-matrix approximation, which sums the most singular set of diagrams.