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arXiv:cond-mat/0007348 · Physical review. B, Condensed matter

Charge instabilities near a Van Hove singularity

J. González

Published Jan 1, 2001• 19 citations
arXivHTMLPDF

Abstract

The charge instabilities of electron systems in the square lattice are analyzed near the Van Hove singularity by means of a Wilsonian renormalization group approach. We show that the method preserves the spin rotational invariance at all scales, allowing a rigorous determination of spin and charge instabilities of the $t\ensuremath{-}{t}^{\ensuremath{'}}$ Hubbard model. Regarding the latter, repulsive interactions fall into two different universality classes. One of them has nonsingular response functions in the charge sector, while the other is characterized by the splitting of the Van Hove singularity. At the level of marginal perturbations, the Hubbard model turns out to be at the boundary between the two universality classes, while extended models with nearest-neighbor repulsive interactions belong to the latter class. In the case of open systems allowed to exchange particles with a reservoir, we show the existence of a range of fillings forbidden above and below the Van Hove singularity. This has the property of attracting the Fermi level in the mentioned range, as the system reaches its lowest energy when the Fermi energy is at the singularity.

DOI
10.1103/physrevb.63.045114

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The charge instabilities of electron systems in the square lattice are analyzed near the Van Hove singularity by means of a Wilsonian renormalization group approach. We show that the method preserves the spin rotational invariance at all scales, allowing a rigorous determination of spin and charge instabilities of the t\ensuremath−t\ensuremath′t\ensuremath{-}{t}^{\ensuremath{'}}t\ensuremath−t\ensuremath′ Hubbard model. Regarding the latter, repulsive interactions fall into two different universality classes. One of them has nonsingular response functions in the charge sector, while the other is characterized by the splitting of the Van Hove singularity. At the level of marginal perturbations, the Hubbard model turns out to be at the boundary between the two universality classes, while extended models with nearest-neighbor repulsive interactions belong to the latter class. In the case of open systems allowed to exchange particles with a reservoir, we show the existence of a range of fillings forbidden above and below the Van Hove singularity. This has the property of attracting the Fermi level in the mentioned range, as the system reaches its lowest energy when the Fermi energy is at the singularity.
We apply a Wilsonian renormalization group approach to the continuum limit of the attractive t − t ′ Hubbard model, taken when the Fermi level is at the Van Hove singularity of the density of states. We determine the leading instabilities by computing the different response functions up to t ′ = 0.5 t . The phase diagram shows a large boundary between superconducting and charge-density-wave phases, that merge in a triple point with a phase separation instability. The latter is realized down to very low coupling constant, as the Fermi sea degenerates towards a pair of straight lines near t ′ = 0.5 t .
Attractive non-local interactions jointly with repulsive local interaction in a microscopic modelling of electronic Fermi liquids generate a competition between an enhancement of the static charge susceptibility–ultimately signalling charge instability and phase separation–and its correlation induced suppression. We analyse this scenario through the investigation of the extended Hubbard model on a two-dimensional square lattice, using the spin rotation invariant slave-boson representation of Kotliar and Ruckenstein. The quasiparticle density of states, the renormalised effective mass and the Landau parameter F s 0 are presented, whereby the positivity of F s 0 – 1 constitutes a criterion for stability. Van Hove singularities in the density of states support possible charge instabilities. A (negative) next-nearest neighbour hopping parameter t' shifts their positions and produces a tendency towards charge instability even for low filling whereas the t' -controlled particle-hole asymmetry of the correlation driven effective mass is small. A region of instability on account of the attractive interaction V is identified, either at half filling in the absence of strong electronic correlations or, in the case of large on-site interaction U , at densities far from half filling.
We study two-dimensional fermions with a short-range interaction in the presence of a Van Hove singularity. It is shown that this system can be consistently described by an effective field theory whose Fermi surface is subdivided into regions as defined by a factorization scale, and that the theory is renormalizable in the sense that all of the counterterms are well defined in the IR limit. The theory has the unusual feature that the renormalization-group equation for the coupling has an explicit dependence on the renormalization scale, much as in theories of Wilson lines. In contrast to the case of a round Fermi surface, there are multiple marginal interactions with nontrivial RG flow. The Cooper instability remains strongest in the BCS channel. We also show that the marginal Fermi-liquid scenario for the quasiparticle width is a robust consequence of the Van Hove singularity. Our results are universal in the sense that they do not depend on the detailed properties of the Fermi surface away from the singularity.
Using the dynamical mean-field theory (DMFT) with the numerical renormalization-group impurity solver we study the paramagnetic phase of the Hubbard model with the density of states (DOS) corresponding to the three-dimensional (3D) cubic lattice and the two-dimensional (2D) square lattice, as well as a DOS with inverse square-root singularity. We show that the electron correlations rapidly smooth out the square-root van Hove singularities (kinks) in the spectral function for the 3D lattice and that the Mott metal-insulator transition (MIT) as well as the magnetic-field-induced MIT differ only little from the well-known results for the Bethe lattice. The consequences of the logarithmic singularity in the DOS for the 2D lattice are more dramatic. At half filling, the divergence pinned at the Fermi level is not washed out, only its integrated weight decreases as the interaction is increased. While the Mott transition is still of the usual kind, the magnetic-field-induced MIT falls into a different universality class as there is no field-induced localization of quasiparticles. In the case of a power-law singularity in the DOS at the Fermi level, the power-law singularity persists in the presence of interaction, albeit with a different exponent, and the effective impurity model in the DMFT turns out to be a pseudogap Anderson impurity model with a hybridization function which vanishes at the Fermi level. The system is then a generalized Fermi liquid. At finite doping, regular Fermi-liquid behavior is recovered.
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 present a general method to study weak-coupling instabilities of a large class of interacting electron models in a controlled and unbiased way. Quite generally, the electron gas is unstable towards a superconducting state even in the absence of phonons, since high-energy spin fluctuations create an effective attraction between the quasi-particles. As an example, we show the occurrence of d-wave pairing in the repulsive Hubbard model in two dimensions. In one dimension or if the Fermi surface is nested, there are several competing instabilities. The required renormalization group formalism for this case is presented to lowest (one-loop) order on a most elementary level, connecting the idea of the “parquet summation” to the more modern concept of Wilson's effective action. The validity and restrictions of the one-loop approximation are discussed in detail. As a result, three different renormalization group approaches known in the literature are shown to be equivalent within the regime of applicability. We also briefly discuss the open problem of a two-dimensional Fermi system at Van Hove filling without nesting.