Homeopolar excitations in a one-dimensional system of spinless fermions with nonlocal interaction

Integral equations are obtained for the ground-state energy $E_0$ and the spectrum of quasihomeopolar excitations $\varepsilon(q)$ in a one-dimensional system of spinless fermions with repulsion at neighboring sites. The fermion density $c$ and the dimensionless coupling constant $\rho=\gamma/2\beta$ vary in the ranges $0\leqslant c\leqslant 1/2$, $0<\rho<\infty$. It is found that the homeopolar excitations have an end point of their spectrum $\varepsilon(\pm2k_F)=0$ $(k_F=\pi c)$ and are symmetric about $k_F$: $\varepsilon(q)=\varepsilon(2\pi c-q)$. Asymptotic expansions for $E_0$ and $\varepsilon(q)$ as $\rho\to\infty$ are obtained. A possible connection between the zeros of $\varepsilon(q)$ and the breaking of translational symmetry of the lattice with respect to the formation of a superlattice with period $(2k_F)^{-1}$ is discussed.