We present exact wave functions for the ground state of a lattice gas of spinless one-dimensional fermions, at filling factors of
1/M. The one-body hopping operator decreases exponentially. The two-body interaction is Coulomb-like: A neutralizing background is required for the system to be extensive. There is a parameter in the problem, which can be taken to be the strength of the charge. All correlation functions are evaluated, and the system is shown to have long-ranged crystalline order. For filling factor
1/2, the momentum distribution and pair correlation function are explicitly evaluated. There is no Fermi surface. The crystalline order parameter is evaluated.
The overlap between the ground-state wave functions of a one-dimensional electron gas with the Hamiltonians Ĥ 0 and
H^0 , where
H^0+V^ is the impurity potential, is calculated. It is shown that in the limit of an infinite potential the overlap vanishes as M −1/8 as M→∞, where M is the number of filled levels, while in the case of a weak potential this overlap differs little from 1. A relation is found between the magnitude of the overlap and the behavior of the density of states near the Fermi energy (statistics of the levels). The possibility of linearization of the spectrum and the possibility of performing a bosonization procedure are discussed in light of the results obtained.
Using eigen-functional bosonization method, we study quantum many-particle systems, and show that the quantum many-particle problems end in to solve the differential equation of the phase fields which represent the particle correlation strength. Thus, the physical properties of these systems are completely determined by the differential equation of the phase fields. We mainly focus on the study of D-dimensional electron gas with/without transverse gauge fields, two-dimensional electron gas under an external magnetic field, D-dimensional boson systems, a D-dimensional Heisenberg model and a one-band Hubbard model on a square lattice, and give their exact (accurate for Heisenberg model) functional expressions of the ground state energy and action, and the eigen-functional wave functions of the fermions/bosons. With them, we can calculate a variety of correlation functions of the systems, such as single particle Green's functions and their ground state wave functions. In present theoretical framework, we can unifiably represent the Landau Fermi liquid, non-Fermi liquid (
D≥2) and Tomonaga-Luttinger liquid.