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Journal of Physical Studies

Charge and magnetic states for the Hubbard model on a three-site cluster

O. P. Matveev, Andrij M. Shvaika

Published Jan 1, 2006• 1 citation
Publisher

Abstract

The Hubbard model for small size clusters is investigated.Using direct diagonalization of the small cluster Hamiltonian, electronic, magnetic and spin susceptibilities are calculated and their behavior at low temperature is investigated.The temperature behavior of susceptibilities for twoand three-site clusters is investigated for both center and edge of the Brillouin zone.If at the given concentration the ground state is polarized, the susceptibilities diverge following the Curie law and the system is transformed into the magnetically ordered state at T = 0.

DOI
10.30970/jps.10.208

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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 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.
We investigate the spin and charge susceptibilities of the two-dimensional Hubbard model based upon the perturbative calculation in the strength of correlation U. For U comparable to a bare bandwidth, the charge susceptibility decreases near half filling as hole doping approaches zero. This behavior suggesting the precursor of the Mott-Hubbard gap formation cannot be obtained without the vertex corrections beyond the random-phase approximation. In the low-temperature region, the spin susceptibility deviates from the Curie-Weiss-like law and finally turns to decrease with the decrease of temperature. This spin-gap-like behavior originates from the van Hove singularity in the density of states. \textcopyright{} 1996 The American Physical Society.