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physica status solidi (b)

New Green's function approach describing the ferromagnetic state in the Hubbard model with correlated hopping

Grzegorz Górski, Jerzy Mizia, Krzysztof Kucab

Published Jul 11, 2014• 4 citations
Publisher

Abstract

We consider a Hubbard model (HM) with occupation dependent hopping integrals. Using the Hartree–Fock (H–F) approximation and the new Green's function approach with inter‐site kinetic averages included, we analyze the influence of the correlated hopping on ferromagnetic ordering. The influence of correlated hopping on the nonlocal quasiparticle energies and corresponding k‐dependent spectral weights is included. In addition, we obtain the shift of the spin bands, which is a major factor in creating spontaneous ferromagnetism. At some parameters of the model, the correlated hopping effect is strong enough to achieve saturated ferromagnetism. This state may be obtained at the asymmetric density of states (DOS) and the Fermi energy located in the region of large spectral weight near the band edge. The results are compared with DMFT‐Quantum Monte Carlo calculations and with the Hubbard III approximation, which includes the correlated hopping effect.

DOI
10.1002/pssb.201350313

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We consider a Hubbard model (HM) with occupation dependent hopping integrals. Using the Hartree–Fock (H–F) approximation and the new Green's function approach with inter‐site kinetic averages included, we analyze the influence of the correlated hopping on ferromagnetic ordering. The influence of correlated hopping on the nonlocal quasiparticle energies and corresponding k‐dependent spectral weights is included. In addition, we obtain the shift of the spin bands, which is a major factor in creating spontaneous ferromagnetism. At some parameters of the model, the correlated hopping effect is strong enough to achieve saturated ferromagnetism. This state may be obtained at the asymmetric density of states (DOS) and the Fermi energy located in the region of large spectral weight near the band edge. The results are compared with DMFT‐Quantum Monte Carlo calculations and with the Hubbard III approximation, which includes the correlated hopping effect.
We consider a Hubbard model with occupation dependent hopping integrals. Using the Hartree-Fock (H-F) approximation and the new Green function approach with inter-site kinetic averages included, we analyze the influence of the correlated hopping on ferromagnetic ordering. The influence of correlated hopping on the nonlocal quasiparticle energies and corresponding k-dependent spectral weights is included. In addition we obtain the shift of the spin bands which is a major factor in creating spontaneous ferromagnetism. At some parameters of the model the correlated hopping effect is strong enough to achieve saturated ferromagnetism. This state may be obtained at the asymmetric density of states (DOS) and the Fermi energy located in the region of large spectral weight near the band edge. The results are compared with DMFT-Quantum Monte Carlo calculations and with the Hubbard III approximation which includes the correlated hopping effect.
Using equations of motion of connected Green functions the Hubbard III approximation is reexamined. It is shown that two important higher-order correlation effects are neglected in the Hubbard III treatment. One is bandwidth correction, the other is band-shift correction. These two corrections make important effects to the metal-insulator transition and spontaneous ferromagnetism in the Hubbard model. Due to the bandwidth correction the critical value of Uc{U}_{c}Uc​ in the metal-insulator transition decreases from 3/2W(W\sqrt{3}/2W(W3​/2W(W represents Bloch bandwidth) in the Hubbard III approximation to 0.67W,0.67W,0.67W, which approaches significantly the critical value of 0.5W0.5W0.5W by the quantum Monte Carlo simulation, while the band-shift correction leads to a ferromagnetic solution missed in the Hubbard III approximation in the limit of $\stackrel{\ensuremath{\rightarrow}}{U}\ensuremath{\infty},$ which is in qualitative agreement with the conclusion that the Hubbard model can describe the ferromagnetism resulted from the spectral density method and the quantum Monte Carlo simulation. Our studies show that the higher-order correlation effects play an important role in describing the properties of the strongly correlated systems.
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.
Using exact diagonalization techniques, the Hubbard model with hopping between nearest (t) and next-nearest neighbors ${(t}^{\ensuremath{'}})$ is studied in a two-dimensional cluster with $4\ifmmode\times\else\texttimes\fi{}4$ sites. Within a range of values of ${t}^{\ensuremath{'}}<0$ and low densities, states with different total spin are degenerate in the noninteracting system. At a finite value of the Coulomb repulsion U, states with high spin are energetically favored and the ground state can be fully polarized. The behavior of the density of states differs from that obtained within the Hartree-Fock approximation. The low energy region of the density of states (DOS) for the majority spins remains weakly modified by U. Our results bring further support to the idea that itinerant ferromagnetism occurs in strongly correlated systems with high noninteracting DOS at the bottom of the band.
The equation for the electron Green's function of the fermionic Hubbard model, derived using the strong coupling diagram technique, is solved self-consistently for the near-neighbor form of the kinetic energy and for half-filling. In this case the Mott transition occurs at the Hubbard repulsion Uc ≈ 6.96t, where t is the hopping constant. The calculated spectral functions, density of states (DOS) and momentum distribution are compared with results of Monte Carlo simulations. A satisfactory agreement was found for U > Uc and for temperatures, at which magnetic ordering and spin correlations are suppressed. For U < Uc and lower temperatures the theory describes qualitatively correctly the positions and widths of spectral continua, variations of spectral shapes and occupation numbers with changing wave vector and repulsion. The locations of spectral maxima turn out to be close to the positions of δ-function peaks in the Hubbard-I approximation.
We study the possibility and stability of band ferromagnetism in the single-band Hubbard model for the simple cubic (sc) lattice. A nonlocal self-energy is derived within a modified perturbation theory. Results for the spectral density and quasiparticle density of states are shown with special attention to the effects of k\mathbf{k}k dependence. The importance of nonlocal correlations for the fulfillment of the Mermin-Wagner theorem is our main result. A phase diagram showing regions of ferromagnetic order is calculated for the three-dimensional lattice. Besides, we show results for the optical conductivity and prove that the renormalized one-loop contribution to the conductivity already cancels the Drude peak exactly in case of a local self-energy which is not true anymore for a nonlocal self-energy.