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arXiv:cond-mat/9712324 · International Journal of Modern Physics B

The Influence of Long-Range Hopping on Ferromagnetism in the Hubbard Model

Pavol Farkašovský

Published Mar 30, 1998• 4 citations
arXivHTMLPDF

Abstract

The phase diagram of the Hubbard model in an external magnetic field is examined by extrapolation of small-cluster exact-diagonalization calculations. Using a general expression for the hopping matrix elements (tij ~ q|i-j|) the influence of long-range hopping (band asymmetry) on ferromagnetism in this model is studied. It is found that the long-range hopping (nonzero q) stabilizes ferromagnetism in an external magnetic field for n > 1. In the opposite limit n≤1 the fully polarized ferromagnetic state is generally suppressed with increasing q. The critical value of magnetic field h below which the ferromagnetic state becomes unstable is calculated numerically.

DOI
10.1142/s0217979298000466

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The phase diagram of the Hubbard model in an external magnetic field is examined by extrapolation of small-cluster exact-diagonalization calculations. Using a general expression for the hopping matrix elements (tij ~ q|i-j|) the influence of long-range hopping (band asymmetry) on ferromagnetism in this model is studied. It is found that the long-range hopping (nonzero q) stabilizes ferromagnetism in an external magnetic field for n > 1. In the opposite limit n≤1 the fully polarized ferromagnetic state is generally suppressed with increasing q. The critical value of magnetic field h below which the ferromagnetic state becomes unstable is calculated numerically.
The extrapolation of small-cluster exact-diagonalization calculations is used to examine the zero-temperature phase diagram of the simplified Hubbard model in an external magnetic field. Using a general expression for hopping matrix elements ${(t}_{\mathrm{ij}}\ensuremath{\sim}{q}^{|i\ensuremath{-}j|})$ the influence of the long-range hopping (the asymmetry of the density of states) on ground-state properties, and particularly on ferromagnetism is studied. It is found that the long-range hopping (the asymmetric density of states) stabilizes the ferromagnetic state for a wide range of electron interactions UUU and electron concentrations nnn. The critical value of magnetic field hhh below which the ferromagnetic state becomes unstable is calculated numerically. It is shown that below this critical value the ferromagnetic state breaks down into a metallic state for weak coupling and an insulating state for strong coupling. The phase diagram of the model in the $h\ensuremath{-}U$ plane is presented for physically the most interesting cases.
The extrapolation of small-cluster exact-diagonalization calculations is used to examine ferromagnetism in the one-dimensional Hubbard model with a generalized type of hopping. It is found that the long-range hopping with power decaying hopping amplitudes ${(t}_{\mathrm{ij}}\ensuremath{\sim}{q}^{|i\ensuremath{-}j|})$ stabilizes the ferromagnetic state for a wide range of electron interactions U and electron concentrations n. The critical value of the interaction strength Uc(q){U}_{c}(q)Uc​(q) above which the ferromagnetic state becomes stable is calculated numerically and the ground-state phase diagram of the model (in the $U\ensuremath{-}q$ plane) is presented for physically the most interesting cases.
The extrapolation of small-cluster exact-diagonalization calculations is used to examine ferromagnetism in the one-dimensional Hubbard model with long-range and correlated hopping. It is found that the correlated hopping term stabilizes the ferromagnetic state for a wide range of electron interactions U and electron concentrations n. The critical value of the interaction strength Uc above which the ferromagnetic state becomes stable is calculated numerically and the ground-state phase diagram of the model is presented for physically the most interesting cases. 1 The Hubbard model has become, since its inception [1] in 1963, one of the most popular examples of a system of interacting electrons with short-range interactions. It has been used in the literature to study a great variety of many-body effects in metals, of which ferromagnetism, metal-insulator transitions, charge-density waves and superconductivity are the most common examples. Of all these cooperative phenomena the problem of ferromagnetism in the Hubbard model has the longest
Abstract The combination of small-cluster exact-diagonalization calculations and the quantum Monte Carlo method is used to examine ferromagnetism in the two-dimensional Hubbard model with a generalized type of hopping. It is found that the long-range hopping with exponentially decaying hopping amplitudes t ij ∼ − q Ri−Rj stabilizes the ferromagnetic state for a wide range of electron interactions U and electron concentrations n > 1. The critical value of the hopping parameter q c above which the ferromagnetic state becomes stable is calculated numerically and the ground-state phase diagram of the model is discussed for physically the most interesting cases.
Since its introduction in 1963, the Hubbard model has becomes one of the most popular models used in the literature to study cooperative phenomena in narrow-band metals (ferromagnetism, metal-insulator transitions, charge-density waves, high-Tc superconductivity). Amongst all these cooperative phenomena, the problem of itinerant ferromagnetism in the Hubbard model has the longest history. However, in spite of an impressive research activity in the past, the underlying physics (microscopic mechanisms) that leads to the stabilization of itinerant ferromagnetism in Hubbard model (narrow-band metals) is still far from being understood. In this review we present our numerical results concerning this subject, which have been reached by small cluster exact diagonalization, density matrix renormalization group and quantum Monte Carlo calculations within various extensions of the Hubbard model. Particular attention is paid to a description of crucial mechanisms (interactions) that support the stabilization of the ferromagnetic state, and namely: (i) the long-range hopping, (ii) the correlated hopping, (iii) the long-range Coulomb interaction, (iv) the flat bands and (v) the lattice structure. Most of the presented results have been obtained for the one-dimensional case, but the influence of the increasing dimension of the system on the ferromagnetic state is also intensively discussed.
Ferromagnetism in the one-dimensional Hubbard model with the next-nearest-neighbor hopping is explored by using the exact-diagonalization method in a small cluster and the equation-of-motion method in the thermodynamic limit with electron density n = 3/2. With these two complementary methods, it is found that an intermediate value of the next-nearest-neighbor hopping amplitude t1 tends to stabilize the fully polarized ferromagnetic state under the condition that the on-site coulomb interaction U is sufficiently large in our model. The ground-state phase diagram of the model is presented in the t1—U plane.