The $\beta$-expansion of the $D=1$ fermionic spinless Hubbard model off the half-filling regime

We found that when the spinless model is off the half-filling regime ($\mu \neq V$),theHelmholtzfreeenergy(HFE)canbewrittenastwo$\beta$-expansions: one expansion comes from the half-filling configuration and another one that depends on the parameter $x=\mu -V$. We show numerically that the chemical potential as a function of temperature satisfies a relation similar to the one derived from the particle-hole symmetry of the fermionic spinless model. We extend the $\beta$-expansion of the HFE of the one-dimensional fermionic spinless Hubbard model up to order ${\beta }^8$.