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Modern Physics Letters B

GROUND STATE PROPERTIES OF THE SIMPLIFIED HUBBARD MODEL

Pavol Farkašovský

Published Aug 30, 1992• 3 citations
Publisher

Abstract

We present the exact solution of the simplified Hubbard model in which only one kind of electrons can hop and this quantum mechanical hopping of electrons is assumed to be unconstrained. It is shown that the model still behaves non-trivially, although it no longer depends on the lattice structure and the dimensionality of the system. For this case we find: (i) a gap in the ground state energy always exists at the half-filled band point (n = 1), (ii) a preferred magnetic state at n = 1 and large U is a total spin singlet, (iii) U-dependence of the ground state energy has qualitatively the same form as one of the conventional Hubbard model with the (t2/U)-behavior at large U. A phase diagram of the model is discussed.

DOI
10.1142/s0217984992000934

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We present the exact solution of the simplified Hubbard model in which only one kind of electrons can hop and this quantum mechanical hopping of electrons is assumed to be unconstrained. It is shown that the model still behaves non-trivially, although it no longer depends on the lattice structure and the dimensionality of the system. For this case we find: (i) a gap in the ground state energy always exists at the half-filled band point (n = 1), (ii) a preferred magnetic state at n = 1 and large U is a total spin singlet, (iii) U-dependence of the ground state energy has qualitatively the same form as one of the conventional Hubbard model with the (t2/U)-behavior at large U. A phase diagram of the model is discussed.
The ground-state wave function and energy of the Hubbard model with unlimited hopping can be obtained for arbitrary fillings using the transformation recently introduced by Brandt and Giesekus. A simple linear behavior is obtained for the ground-state energy dependence on the number of particles: EGS{\mathit{E}}_{\mathrm{GS}}EGS​=-2t(L-N), where -t is the hopping energy (t>0), L is the number of sites, and N is the number of electrons. The ground state shows a very large total spin degeneracy. The system is shown to behave as a Fermi liquid with a spectral weight Z=1-N/L for all fillings less than 1 (N\ensuremath{\le}L).
The exact phase diagrams of the one-dimensional Hubbard model, both attractive and repulsive, in the presence of an arbitrary magnetic field h for various electron concentrations n and on-site interaction strengths U <0 or U >0 are calculated and investigated. The exact ground-state properties, namely, the ground-state energy, the average spin (magnetization), the concentration of the doubly occupied sites, the kinetic energy, the chemical potential, the spin (magnetic) susceptibility and the charge compressibility, are calculated and examined over a wide range of interaction strengths U for various h and n . It is found that the spin susceptibility at half-filling is non-analytic and changes discontinuously as U →0. The exact theory shows the absence of a charge energy gap in the U ⩽0 region for all n and provides, for the chemical potential, the rigorous upper and lower bounds for half-filled and empty bands respectively. The analytical results derived for the weak-coupling limit and asymptotic expansions for strong coupling are in full agreement with the numerical calculations.
The recently introduced concept of exact mean-field theories is applied to the extended simplified Hubbard model, which is a one-band extended Hubbard model (including nearest-neighbor interaction) where only the \ensuremath{\downarrow} electrons can hop. An exact mean-field Hamiltonian is derived for the model on a Bethe lattice in the limit of a large coordination number (Z\ensuremath{\rightarrow}\ensuremath{\infty}). The mean-field Hamiltonian is used to calculate the Green functions and the free energy at half-filling. This exact solution is analyzed analytically at small and large values of the on-site (U) and nearest-neighbor (V) interaction strengths. One finds that a phase transition occurs at sufficiently low temperatures for all U,V>0. The low-temperature phase is a charge-density wave for V>1/2U and a spin-density wave for V1/2U. Special attention is paid to (i) the critical temperature as a function of U and V, (ii) the order parameter as a function of temperature, and (iii) the density of states. The density of states shows that the low-temperature phase is an insulator for all U,V provided that V/U\ensuremath{\ne}1/2, while the high-temperature phase is a paramagnetic metal for all V if U2, and a paramagnetic insulator if U>2. Many detailed results are given.
The exact phase diagram and the ground-state properties of the one-dimensional Hubbard model with arbitrary on-site interaction of electrons are calculated over a wide range of magnetic field and electron concentrations by means of the Bethe-ansatz formalism. The ground-state properties, including the total energy, the average spin (magnetization) and spin (magnetic) susceptibility are investigated for both signs of the interaction strength U/t. The critical behavior near the onset of magnetization and magnetic saturation are also analyzed. At the onset of magnetization and near the magnetic saturation the spin susceptibility χ diverges at all U/t for half-filling case n=1, whereas for n≠1 it is always finite. The reverse susceptibility χ-1(U) exhibits anomalous hump, which increases with h or n, and shows discontinuity as U/t→±0 at infinitesimal h→0. The analytical results for the ground-state properties in strong and weak interaction limits are in full agreement with our numerical calculations.
The Hubbard Hamiltonian with infinite on-site repulsion and negative transfer is diagonalized numerically on 3×3 frustrated lattices, such as the square lattice with next-nearest-neighbor transfer and the triangular lattice. The number of electrons is chosen to be less by just one electron than that in the exactly half-filled band. When the strength of the frustration is sufficiently large, a singlet state appears as the ground state. Then, the total spin of the lowest excited state is unity. There is an excitation gap for the square lattice but not for the triangular lattice.