A new two-pole approximation, which allows to describe the transition from an insulating state to a metallic one at increase of bandwidth, and also the observable in some compounds transition from a metalic state to an insulating one with increasing temperature, is presented. The case of a paramagnetic narrow-band material at half-filling is considered. Single-particle Green function and energy spectrum of electron system are calculated. The expressions for energy gap width and the concentration of polar states (holes or doublons) are obtained. The conditions for metallic and insulating states are found.
Assuming that, in the narrow-band region, the exact Green's function of the Hubbard model essentially retains the two-peak structure proposed by Hubbard, we derive its consequences as rigorously as possible. When a quasielectron is added to the ground state of
N electrons in the narrow-band region of the Hubbard model, the whole system is shown to relax due to the strong interaction
I, thus yielding a relaxation energy $\ensuremath{\Delta}E$ between the quasielectron and the original
N electrons. This explains the conclusion in the preceding paper that a gap $\ensuremath{\Delta}E$ analogous to the energy gap in superconductivity appears between the quasielectron spectrum and the chemical potential for any occupation
n regardless of whether the lower band is exactly filled or not and that the system cannot be a normal metal. We further found that the ground state is, in fact, a bound state, and electrons involved do not obey the Landau theory of quasiparticles. However, the electron Green's function describes the motion of an electron added to this
N-electron ground state without the relaxation of the original
N electrons, and hence this added electron behaves exactly like a Landau quasiparticle and can be represented by the quasiparticle operator ${\mathrm{A}}_{k\ensuremath{\sigma}}^{\ifmmode\dagger\else\textdagger\fi{}}$ introduced by Luttinger and Nozi\`eres. Only if the original
N electrons are allowed to relax is the energy of the system reduced by $\ensuremath{\Delta}E$, yielding the energy gap $\ensuremath{\Delta}E$. When a quasielectron is removed from the system of
N+1 electrons, the same relaxation energy $\ensuremath{\Delta}E$ is removed, and again the gap $\ensuremath{\Delta}E$ is introduced between the chemical potential $\ensuremath{\mu}$ and the quasihole spectrum, thus creating a gap $2\ensuremath{\Delta}E$ between the quasielectron and quasihole spectra for any occupation
n. The Fermi level, lying at the center of the gap, never intersects the quasiparticle spectra and therefore, Luttinger's theorem that the volume within the Fermi surface is independent of interactions and equal to the value in the noninteracting limit does not apply to the Hubbard model in the narrow-band region. However, this conclusion does not immediately imply that the system is an insulator. Instead, it is shown that a current-carrying state analogous to the condensate state found by Fr\"ohlich appears under certain conditions. When the number of electrons
N is nearly or exactly equal to the number of atoms
Na, quasiparticlelike behavior disappears completely, making the system insulating. The only exception is the pathological limit $I=\ensuremath{\infty}$, where a metallic state with a Fermi volume twice that in the noninteracting limit is found.
The description of solids at a microscopic level is complex, involving the interaction of a huge number of its constituents, such as ions or electrons. It is impossible to solve the corresponding many-body problems analytically or numerically, although much insight can be gained from the analysis of simplified models. An important example is the Hubbard model, which describes interacting electrons in narrow energy bands, and which has been applied to problems as diverse as high-Tc superconductivity, band magnetism, and the metal-insulator transition. This 2005 book presents a coherent, self-contained account of the exact solution of the Hubbard model in one dimension. The early chapters will be accessible to beginning graduate students with a basic knowledge of quantum mechanics and statistical mechanics. The later chapters address more advanced topics, and are intended as a guide for researchers to some of the more topical results in the field of integrable models.
We study the transition from paramagnetic metal to paramagnetic insulator by finite temperature Quantum Monte-Carlo simulations for the 2D Hubbard model at half-filling. Working at the moderately high temperature T=0.33*t where the spin correlation length has dropped to 1.5 lattice spacings, and scanning the interaction strength U, we observe an unambiguous metal-insulator transition with onset at U_c=4t. In the metallic phase there are no indications of any `correlation narrowing' of the band width, nor for any decrease of spectral weight at the Fermi energy. The Mott-Hubbard gap opens gradually with increasing U and is accompanied by the formation of side bands near the Fermi energy. In the first stages of gap formation the gap opens at (pi,0), reminiscent of the `pseudogap' phenomenology in cuprate superconductors.