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Philosophical Magazine B

The Hubbard model, spin degeneracy and Ising spins

K. Ziegler

Published May 1, 2002• 2 citations
Publisher

Abstract

The Hubbard model is used as a starting point for a study of an electronic system with spin ½. Using a functional integral representation at half-filling it is demonstrated that the spin degeneracy is equivalent to dynamic Ising spins coupled to the fermions. A magnetic phase transition of the model is related to a transition of the Ising spins from a paramagnetic to an antiferromagnetic phase. A metal-insulator transition in the paramagnetic phase can be described within this approach by the Green's function of non-interacting fermions, coupled to Ising spins. This picture is compared with the earlier Ising spin representation of the Hubbard model by Hirsch and discussed in terms of a small hopping amplitude.

DOI
10.1080/13642810208224373

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arXiv:cond-mat/9612142•Journal of the Physical Society of Japan•May 15, 1997•38 citations
The Hubbard model is used as a starting point for a study of an electronic system with spin ½. Using a functional integral representation at half-filling it is demonstrated that the spin degeneracy is equivalent to dynamic Ising spins coupled to the fermions. A magnetic phase transition of the model is related to a transition of the Ising spins from a paramagnetic to an antiferromagnetic phase. A metal-insulator transition in the paramagnetic phase can be described within this approach by the Green's function of non-interacting fermions, coupled to Ising spins. This picture is compared with the earlier Ising spin representation of the Hubbard model by Hirsch and discussed in terms of a small hopping amplitude.
The Hubbard model is used to study an electronic system at half filling. Starting from a functional integral representation the spin-up Grassmann field is integrated out. It is shown that the resulting spinless fermion theory has an instantaneous cluster interaction, and that the spinless fermions are coupled to thermally fluctuating Ising spins. The coupling parameter of the spinless fermion interaction is a product of the Hubbard interaction and the hopping rate. As an example the strongly metallic as well as the strongly insulating regime are investigated in terms of the effective Ising statistics.
Antiferromagnetic and metal to insulator transitions in the ground state of two-dimensional half-filled and nearly half-filled Hubbard models with the nearest neighbor and the second nearest neighbor transfers (- t and t ′ , respectively) are studied by using the Green's function approach with the self-energy up to the second order in the coupling constant U . For a choice of the parameter value t ′ / t =0.2 the half-filled model undergoes the first order phase transition between paramagnetic metallic (PM) phase and antiferromagnetic insulator (AFI) phase at U / t =4.0 while the first order transition between PM and antiferromagnetic metallic (AFM) phases takes place for nearly half-filled band. The one-electron spectral density is calculated for the PM, AFM, and AFI phases near the phase boundaries. In the PM phase the quasi-particle dispersion indicates the band narrowing from the Hartree-Fock results and the flat band behavior as was indicated by previous stadies and was observed in high- T c cuprates. The quasi-particle dispersion in the AFM phase is seen to be close to the Hartree-Fock results.
We present a slave-boson formulation for the Hubbard model that preserves the spin-rotation and particle-hole symmetries of the paramagnetic phase at half filling. Fermion and boson fields are treated on an equal footing in a functional integral formulation. We show that, while in two and three dimensions there is a metal-insulator transition as the interaction U increases, in one dimension our approach always gives an insulating state. Short-range spin-spin correlations are included at the saddle-point level for all values of U, giving a good description of the insulating state. We show that the energy gap in the one-particle spectrum increases from zero as (U-Uc{\mathit{U}}_{\mathit{c}}Uc​${)}^{\mathrm{\ensuremath{\alpha}}}$, where Uc{\mathit{U}}_{\mathit{c}}Uc​ is the critical value of the on-site interaction and the exponent \ensuremath{\alpha} is 1 and 1/2 for 2 and 3 dimensions, respectively. \textcopyright{} 1996 The American Physical Society.
We investigate the phase diagram of the square lattice bilayer Hubbard model at half-filling with the variational Monte Carlo method for both the magnetic and the paramagnetic case as a function of the interlayer hopping and on-site Coulomb repulsion U . With this study we resolve some discrepancies in previous calculations based on the dynamical mean-field theory, and we are able to determine the nature of the phase transitions between metal, Mott insulator and band insulator. In the magnetic case we find only two phases: an antiferromagnetic Mott insulator at small for any value of U and a band insulator at large . At large U values we approach the Heisenberg limit. The paramagnetic phase diagram shows at small a metal to Mott insulator transition at moderate U values and a Mott to band insulator transition at larger U values. We also observe a re-entrant Mott insulator to metal transition and metal to band insulator transition for increasing in the range of . Finally, we discuss the phase diagrams obtained in relation to findings from previous studies based on different many-body approaches.
Abstract. We investigate the phase diagram of the square lattice bilayer Hubbard model at half filling with the variational Monte Carlo method for both the magnetic and the paramagnetic case as a function of the interlayer hopping t ⊥ and on-site Coulomb repulsion U. With this study we resolve some discrepancies in previous calculations based on the dynamical mean field theory, and we are able to determine the nature of the phase transitions between metal, Mott insulator and band insulator. In the magnetic case we find only two phases: An antiferromagnetic Mott insulator at small t ⊥ for any value of U and a band insulator at large t⊥. At large U values we approach the Heisenberg limit. The paramagnetic phase diagram shows at small t ⊥ a metal to Mott insulator transition at moderate U values and a Mott to band insulator transition at larger U values. We also observe a re-entrant Mott insulator to metal transition and metal to band insulator transition for increasing t ⊥ in the range of 5.5t < U < 7.5t. Finally, we discuss the obtained phase diagrams in relation to previous studies based on different many-body approaches. PACS numbers: 71.10.Fd, 71.27.+a, 71.30.+h 1.
A slave-boson functional-integral method has been developed for the Hubbard model with arbitrary, orbital degeneracy DDD. Its saddle-point mean-field theory is equivalent to the Gutzwiller approximation, as in the case of single-band Hubbard model. Our theory is applied to the doubly degenerate (D=2D = 2D=2) model, and numerical calculations have been performed for this model in the paramagnetic states. The effect of the exchange interaction on the metal-insulator (MI) transition is discussed. The critical interaction for the MI transition is analytically calculated as functions of orbital degeneracy and electron occupancy.