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arXiv:cond-mat/0611238 · Physical Review B

Ferromagnetism in the t−t′ Hubbard model: Interplay of lattice, band dispersion, and interaction effects studied within a Goldstone-mode preserving scheme

Sudhakar Pandey, Avinash Singh

Published Feb 9, 2007• 24 citations
arXivHTMLPDF

Abstract

Ferromagnetism in the Hubbard model is investigated on sc, bcc, and fcc lattices using a systematic inverse-degeneracy $(1∕\mathcal{N})$ expansion which incorporates self-energy and vertex corrections such that spin-rotation symmetry and the Goldstone mode are explicitly preserved. First-order quantum corrections to magnon energies are evaluated for several cases, providing a comprehensive picture of the interplay of lattice, band dispersion, and interaction effects on the stability of the ferromagnetic state with respect to both long- and short-wavelength fluctuations. Our results support the belief that ferromagnetism is a generic feature of the Hubbard model at intermediate and strong coupling provided the density of states (DOS) is sufficiently asymmetric and strongly peaked near the band edge, as for the fcc lattice with finite ${t}^{\ensuremath{'}}$. For short-wavelength modes, the behavior of a characteristic energy scale ${\ensuremath{\omega}}^{*}\ensuremath{\sim}{T}_{c}$ (magnon DOS peak energy) is in excellent agreement with the ${T}_{c}$ vs $n$ behavior within dynamical mean field theory, with respect to both the stable range of densities $(0.20<n<0.85)$ as well as the optimal density $n=0.65$. However, our finding of vanishing spin stiffness near optimal density highlights the role of long-wavelength fluctuations in further reducing the stable range of densities.

DOI
10.1103/physrevb.75.064412

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Ferromagnetism in the Hubbard model is investigated on sc, bcc, and fcc lattices using a systematic inverse-degeneracy (1∕N)(1∕\mathcal{N})(1∕N) expansion which incorporates self-energy and vertex corrections such that spin-rotation symmetry and the Goldstone mode are explicitly preserved. First-order quantum corrections to magnon energies are evaluated for several cases, providing a comprehensive picture of the interplay of lattice, band dispersion, and interaction effects on the stability of the ferromagnetic state with respect to both long- and short-wavelength fluctuations. Our results support the belief that ferromagnetism is a generic feature of the Hubbard model at intermediate and strong coupling provided the density of states (DOS) is sufficiently asymmetric and strongly peaked near the band edge, as for the fcc lattice with finite t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′. For short-wavelength modes, the behavior of a characteristic energy scale \ensuremathω∗\ensuremath∼Tc{\ensuremath{\omega}}^{*}\ensuremath{\sim}{T}_{c}\ensuremathω∗\ensuremath∼Tc​ (magnon DOS peak energy) is in excellent agreement with the Tc{T}_{c}Tc​ vs nnn behavior within dynamical mean field theory, with respect to both the stable range of densities (0.20<n<0.85)(0.20<n<0.85)(0.20<n<0.85) as well as the optimal density n=0.65n=0.65n=0.65. However, our finding of vanishing spin stiffness near optimal density highlights the role of long-wavelength fluctuations in further reducing the stable range of densities.
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Recent ultracold-atom experiments have observed ferromagnetic correlations in a geometrically tunable Fermi--Hubbard lattice at intermediate densities. Motivated by subsequent theoretical work that connected the low-density limit of the same model to the Mueller--Hartmann mechanism for itinerant ferromagnetism, we investigate the stability of ferromagnetism across a broad density range using the T-matrix approximation, a controlled approach in the dilute limit. We reproduce the ferromagnetic regime observed experimentally at intermediate densities and systematically compare finite-size and thermodynamic-limit calculations in the dilute regime. We find that the low-density ferromagnetic phase reported for the one-diagonal hopping lattice is strongly suppressed with increasing system size and disappears in the thermodynamic limit, indicating that it originates from finite-size effects. By contrast, low-density ferromagnetism remains robust in the square-lattice Hubbard model with hopping along both diagonal directions. We show that this qualitative difference cannot be explained by the degeneracy of the band minima alone, which occurs in both models. Instead, it is controlled by the singular behavior of the density of states at the band bottom: a weak divergence is insufficient to stabilize ferromagnetism, whereas a much stronger quasi-one-dimensional singularity supports a fully polarized ground state even in the dilute limit. Our results demonstrate that band-minimum degeneracy alone is not sufficient for low-density ferromagnetism and that the nature of the band-bottom density-of-states singularity ultimately controls its stability.
Single-particle dynamics in the 120\ifmmode^\circ\else\textdegree\fi{} ordered antiferromagnetic state of the triangular-lattice Hubbard model is studied using a physically transparent fluctuation approach in terms of multiple magnon emission and absorption processes within the noncrossing approximation. Hole and electron spectral features are evaluated at intermediate UUU, and analyzed in terms of a competition between the frustration-induced direct hopping and the virtual hopping. Finite UUU-induced competing interactions and frustration effects contributing through the magnon dispersion are also discussed. Finite contribution to self-energy correction from long-wavelength (Goldstone) modes, together with the high density of electron scattering states in the narrow, sharp peak in the upper band, result in strong fermion-magnon scattering leading to pronounced incoherent behavior in the electron dynamics. The fluctuation-induced first-order metal-insulator transition due to vanishing band gap is also discussed.
On the basis of a regular perturbation theory in the form of diagrammatic technique with X-operators, a ferromagnetic state in the Hubbard model with infinite on-site Coulomb interaction U is studied for a wide range of electron concentration n. Treating the kinetic energy of electrons as perturbation, we get the exact graphic representations for three Green’s functions [Formula: see text] and [Formula: see text] for electrons with spin ↑ and ↓ (spontaneous magnetic moment is in ↑ direction) and [Formula: see text] for spin waves. All of them are expressed through the unique system of four-point and three-point vertex parts for effective electron-magnon and electron-electron interactions. These vertex parts are calculated in two approximations: a low-density approximation for n≪1 or nh=1–n≫1, and the generalized random-phase approximation (GRPA), which was suggested earlier by us for the description of paramagnetic phase in this model. An important result in both cases is the understanding of essential difference of electron states with spin ↑ and ↓. The state for ↑ has coherent character in all region of electron concentration (n>nc) where a ferromagnetic state exists, while the state for spin ↓ has mixed characters including both coherent and incoherent contribution. For the saturated ferromagnetism (n>ns) when nh≪1, [Formula: see text] is an entirely incoherent (nonquasiparticle) state. Appearance of incoherent state is probably a general property of strongly correlated systems distinguishing its behavior from the Fermi liquid. We show also that the Hubbard-I approximation has no region of applicability for the electron with spin ↓ in a ferromagnet and we found a new factor describing the correlation-narrowing of ↓ spin electron band. Green’s function [Formula: see text] calculated in two different limits n≪1 and nh≪1, are jointed together to allow us to calculate ns which lies in the intermediate concentration regime. For simple cubic lattice we found ns=0.87. In the limit nh≪1 our results reduce to the earlier known ones, including Nagaoka’s result for the spin wave energy. We point out the way to construct a rigorous theory for ferromagnetic state in the intermediate range of electron concentration. In the conclusion several important problems are discussed in connection to the continuation of the present work.