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arXiv:1511.02616 · Journal of Physics Conference Series

Charge instabilities of the two-dimensional Hubbard model with attractive nearest neighbour interaction

Raymond Frésard, Kevin Steffen, Thilo Kopp

Published Mar 1, 2016• 6 citations
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Abstract

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.

DOI
10.1088/1742-6596/702/1/012003

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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.
The paramagnetic phase of the extended attractive Hubbard model on the cubic lattice is studied within the spin rotation invariant Kotliar-Ruckenstein slave boson representation at zero temperature. It is obtained that the quasiparticle residue of the Fermi liquid phase vanishes for all densities at an interaction strength slightly smaller than [Formula: see text] that signals the Brinkman–Rice transition, and that it weakly depends on density. While for vanishing non-local interaction parameters, homogeneous static charge instabilities are found in a rather narrow window centered around quarter filling and [Formula: see text], increasing them to [Formula: see text] results into a severe narrowing of this window. On the contrary, when all interaction parameters are attractive, for example for [Formula: see text], a large parameter range in which homogeneous static charge instabilities is found. Yet, this systematically happens inside the Fermi liquid phase.
We investigate the Landau parameters for the instabilities in spin and charge channels in the nondegenerate extended Hubbard model with intersite Coulomb and exchange interactions. To this aim we use the spin rotationally invariant slave boson approach and we determine the necessary inverse propagator matrix. The analytically derived spin Landau parameter F0a{F}_{0}^{a}F0a​ for the half filled band uncovers the intrinsic instability of the nondegenerate Hubbard model towards ferromagnetism---negative intersite exchange interaction triggers a ferromagnetic instability at half filling before the metal-insulator transition, indicated by the divergence of the magnetic susceptibility at ${F}_{0}^{a}=\ensuremath{-}1$. This result is general and the instability occurs in the strongly correlated metallic regime for any lattice, in three or two dimensions. Next as an illustrative example we present numerical results obtained for the cubic lattice with nearest neighbor exchange JJJ and Coulomb VVV elements and arbitrary electron density. One finds that the range of small doping near half filling is the most unstable one towards spin polarization, but only in the case of ferromagnetic intersite exchange J<0J<0J<0. Charge Landau parameter F0s{F}_{0}^{s}F0s​ is lowered near half filling by increasing UUU when the intersite Coulomb interaction VVV is attractive, but in contrast to F0a{F}_{0}^{a}F0a​ at J<0J<0J<0 it requires an attraction beyond a critical value Vc{V}_{c}Vc​ to generate the divergence of the charge susceptibility at ${F}_{0}^{s}=\ensuremath{-}1$ in the metallic phase. This instability was found for a broad range of electronic filling away from half filling for moderate attraction.
Strong correlations effects, which are often associated to the approach to a Mott insulating state, in some cases may be observed even far from half filling. This typically happens whenever the intersite Coulomb repulsion induces a tendency toward charge ordering, an effect that confines the electrons, and in turn favors local moment formation, i.e., Mott localization. A distinct intermediate regime then emerges as a precursor of such a Wigner-Mott transition, which is characterized by both charge and spin correlations, displaying large mass enhancements and strong renormalizations of other Fermi-liquid parameters. Here we present a careful study of a quarter-filled extended Hubbard model---a simple example where such physics can be studied in detail, and discuss its relevance for the understanding of the phenomenology of low-density two-dimensional electron gases.
We study possible charge instabilities in doped Mott insulators by employing the two-dimensional t - J model with a positive value of the next nearest-neighbor hopping integral on a square lattice, which is applicable to electron-doped cuprates. Although the d -wave charge density wave (flux phase) and d -wave Pomeranchuk instability (nematic order) are dominant instabilities for a negative that corresponds to hole-doped cuprates, we find that those instabilities are strongly suppressed and become relevant only rather close to half filling. Instead, various types of bond orders with modulation vectors close to are dominant in a moderate doping region. Phase separation is also enhanced, but it can be suppressed substantially by the nearest-neighbor Coulomb repulsion without affecting the aforementioned charge instabilities.
Since the discovery of the superconducting cuprates and their hitherto unknown electronic properties there has been a continuously growing interest in a theoretical treatment of suitable models of strongly correlated fermions which at least qualitatively reflect the essential aspects of these physical systems. The Hubbard model in two dimensions plays an important role in this context. The present thesis examines instabilities and possible symmetry broken phases of the two-dimensional repulsive Hubbard model with the help of numeric and semi analytic methods. While in the first part it is pointed out how such instabilities can be treated by a mean field theory, the second part of the work supplies the theoretical bases of the numeric procedures developed in part III. For the case of a pure nearest neighbour hopping it is shown that the second order contributions of perturbation theory generate an effective attraction between the electrons. The energy scale of the superconducting instabilities is also examined; it is shown that the vanHove singularity drastically enhances this critical scale at half filling. The investigation of a possible coexistence of two different kinds of spontaneous symmetry breaking forms the emphasis of the thesis: Lately accomplished renormalization group calculations showed (for suitably chosen parameters) the occurrence of a Pomeranchuk instability of the two-dimensional Hubbard model with nonvanishing next nearest neighbour hopping, which should give rise to a ground state that spontaneously breaks the model's lattice symmetry as well as its U(1) symmetry, i.e. the ground state could be characterized by two co-existing order parameters. For treatment of the problem of co-existing instabilities the ansatz of a renormalized perturbation theory is used, which had been suggested in order to avoid the occurrence of unphysical singularities arising within the framework of naive perturbation theory; these singularities are related to interaction-induced shifts of the Fermi surface, which cannot be captured by simply shifting the chemical potential. The attractive Hubbard model is also treated within the framework of renormalized perturbation theory; the well-known reduction of the superconducting order parameter with respect to its BCS value reported in earlier investigations can be confirmed. A spontaneously broken lattice symmetrie does not arise here, in contrast to the repulsive case.
Motivated by the growing evidence of the importance of charge fluctuations in the pseudogap phase in high-temperature cuprate superconductors, we apply a large-NNN expansion formulated in a path integral representation of the two-dimensional ttt-JJJ model on a square lattice. We study all possible charge instabilities of the paramagnetic state in leading order of the 1/N1/N1/N expansion. While the ddd-wave charge density wave (flux phase) becomes the leading instability for various choices of model parameters, we find that a ddd-wave Pomeranchuk (electronic nematic phase) instability occurs as a next leading one. In particular, the nematic state has a strong tendency to become inhomogeneous. In the presence of a large second nearest-neighbor hopping integral, the flux phase is suppressed and the electronic nematic instability becomes leading in a high doping region. Besides these two major instabilities, bond-order phases occur as weaker instabilities close to half-filling. Phase separation is also detected in a finite temperature region near half-filling.