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arXiv:1503.05544 · Physical Review B

Fermi-liquid Landau parameters for a nondegenerate band: Spin and charge instabilities in the extended Hubbard model

Grégoire Lhoutellier, Raymond Frésard, Andrzej M. Oleś

Published Jun 8, 2015• 18 citations
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Abstract

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 ${F}_{0}^{a}$ 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 $J$ and Coulomb $V$ 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 $J<0$. Charge Landau parameter ${F}_{0}^{s}$ is lowered near half filling by increasing $U$ when the intersite Coulomb interaction $V$ is attractive, but in contrast to ${F}_{0}^{a}$ at $J<0$ it requires an attraction beyond a critical value ${V}_{c}$ 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.

DOI
10.1103/physrevb.91.224410

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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 F0a=\ensuremath−1{F}_{0}^{a}=\ensuremath{-}1F0a​=\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 F0s=\ensuremath−1{F}_{0}^{s}=\ensuremath{-}1F0s​=\ensuremath−1 in the metallic phase. This instability was found for a broad range of electronic filling away from half filling for moderate attraction.
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.
Magnetic susceptibilities of interacting (d) electrons in the Hubbard model and the three-band CuO2{\mathrm{CuO}}_{2}CuO2​ model are worked out within the Gaussian fluctuations of the functional integral in the slave boson representation. The normalized Landau Fermi-liquid parameter |F0a{\mathit{F}}_{0}^{\mathit{a}}F0a​|/\ensuremath{\alpha} (\ensuremath{\alpha}\ensuremath{\equiv}W0{\mathit{W}}_{0}W0​NF0{\mathit{N}}_{\mathit{F}}^{0}NF0​, NF0{\mathit{N}}_{\mathit{F}}^{0}NF0​ the density of states at the Fermi level, and W0{\mathit{W}}_{0}W0​ the bare bandwidth) defined from the susceptibility expression is investigated for the two models as functions of Coulomb repulsion U, charge-transfer energy \ensuremath{\Delta}, and doping concentration \ensuremath{\delta}. It is shown that the Landau parameter for the three-band model is larger and exhibits different renormalization behaviors than that for the Hubbard model in both the weak- and strong-coupling regimes except in the Mott-Hubbard weak-coupling regime, where it takes the same random-phase approximation form U/W0{\mathit{W}}_{0}W0​ as the latter. The reasons for these results are analyzed. The slave-boson Landau parameter for the Hubbard model exactly reproduces the T-matrix expression |F0a{\mathit{F}}_{0}^{\mathit{a}}F0a​|/\ensuremath{\alpha}=U/W0{\mathit{W}}_{0}W0​/(1+U/W0{\mathit{W}}_{0}W0​) in the low-density limit, whose renormalization has the different origin from that in the dilutely doped regime. \textcopyright{} 1996 The American Physical Society.
We study the two-dimensional one-band Hubbard model on the square lattice through the spin rotation invariant Kotliar-Ruckenstein slave-boson method, with an emphasis on the electron-doped regime. The phase diagram determined within mean field approximation reveals three different magnetically ordered phases separated by first-order transitions, and a wide instability region as indicated by a negative compressibility. We show that this conundrum of the unstable mean field domain is resolved by spatially mixed phases of neighboring stable states. We also find charge density wave type instabilities in the antiferromagnetically ordered (AFM) state at wave vectors commensurate with the doping. The corresponding charge and spin density response functions are calculated from a Gaussian fluctuations formalism about a magnetic saddle point, for which we provide a detailed derivation. Further, we present density and spin excitation spectra in the AFM phase, allowing for a quantitative calculation of the spin wave parameters, the gap in the longitudinal spin excitation spectrum, and the density excitations in the upper Hubbard band.
We study the phase diagram of a one-dimensional extended Hubbard model with antiferromagnetic exchange interaction analytically and numerically. The bosonization and transfer-matrix renormalization group methods are used in the corresponding coupling regimes. At half-filling, the system is a Mott insulator with a finite spin excitation gap if the on-site Coulomb repulsion is fairly smaller than the antiferromagnetic exchange JJJ. This Mott insulator is characterized by the bond-charge-density-wave order or spontaneously dimerization. In the weak-coupling regime where the spin-charge separation holds approximately, the critical point separating the gapless and gapped spin liquid phases is ${U}_{c}\ensuremath{\sim}J∕2$. However, as JJJ increases, the spin-charge couplings become important and the critical point Uc{U}_{c}Uc​ is significantly suppressed and eventually tends to zero as $J\ensuremath{\rightarrow}\ensuremath{\infty}$. Away from half-filling, the charge gap completely collapses but the spin gap persists.
We examine the two existing slave-boson methods for the finite-U Hubbard model by Kotliar and Ruckenstein (KR) and by Newns et al. and Zou and Anderson (NZA) in the light of the Fermi-liquid quasiparticle picture. It is shown that to treat doubly occupied states the KR slave-boson scheme is more consistent with this picture and that the well-known erroneous gauge-symmetry breaking encountered in the mean-field treatment of the NZA double-occupancy boson is due to the presence of the hidden double-fermion operators. Then, the effective intersite antiferromagnetic interaction between quasiparticles is derived in the KR slave-boson approach by taking into account empty and double-occupancy slave-boson fluctuations to O(1/N2{\mathit{N}}^{2}N2) in the 1/N expansion of the functional-integral formalism. The obtained effective exchange coupling constant Jeff{\mathit{J}}_{\mathrm{eff}}Jeff​(\ensuremath{\delta})=J(2p02{\mathit{p}}_{0}^{2}p02​-\ensuremath{\delta}/2) with J\ensuremath{\equiv}tij2{\mathit{t}}_{\mathit{i}\mathit{j}}^{2}tij2​/U is strongly dependent upon doping \ensuremath{\delta} and is scaled with the reduced Coulomb repulsion u\ensuremath{\equiv}U/Uc{\mathit{U}}_{\mathit{c}}Uc​ in the two-dimensional (2D) and 3D Hubbard models, where Uc{\mathit{U}}_{\mathit{c}}Uc​ is the critical value for the metal-to-insulator transition. The roles of this intersite correlation and the spin-dependent band-renormalization correlation included in the mean-field theory for the paramagnetic-to-antiferromagnetic instability are compared in the two Hubbard models. The intersite contribution is more significant in the 2D case than in the 3D case, but is generally small compared with the band effect except for small dopings \ensuremath{\delta} and smaller U values. The maximum doping ${\mathrm{\ensuremath{\delta}}}_{\mathrm{cr}}$ for the paramagnetic-to-ferromagnetic instability in the strong-coupling limit (U\ensuremath{\rightarrow}\ensuremath{\infty}) of the 2D and 3D Hubbard models is found to be ${\mathrm{\ensuremath{\delta}}}_{\mathrm{cr}}$\ensuremath{\sim}0.28 and ${\mathrm{\ensuremath{\delta}}}_{\mathrm{cr}}$\ensuremath{\sim}0.38, respectively.