be also called the first-order phase transition, because the first derivative of the ground-state energy is discontinuous. Due to the possible sequence of instabilities with the coupling strength, we can truncate the Hilbert space up to a finite $N$ excitation number. To this end, we first separate the Hilbert space into several subspaces with different excitation numbers $%
l=0,1,2...N$. The wave function in $l$-subspace $\left\vert \psi
_{l}\right\rangle $ can be written explicitly with $l$ excitations. E. g. the wavefunctions for $l=0,1,2$ and $3$-subspace are listed in the following $$\begin{aligned}
\left\vert \psi _{0}\right\rangle &=&\left\vert 0\right\rangle \left\vert
\downarrow \right\rangle , \\
\left\vert \psi _{1}\right\rangle &=&c\left\vert 0\right\rangle \left\vert
\uparrow \right\rangle +\sum_{k}d_{k}{a_{k}}^{\dag }\left\vert
0\right\rangle \left\vert \downarrow \right\rangle , \\
\left\vert \psi _{2}\right\rangle &=&\sum_{k}e_{k}{a_{k}}^{\dag }\left\vert
0\right\rangle \left\vert \uparrow \right\rangle +\sum_{kk^{^{\prime
}}}f_{kk^{^{\prime }}}{a_{k}}^{\dag }{a_{k^{\prime }}}^{\dag }\left\vert
0\right\rangle \left\vert \downarrow \right\rangle , \\
\left\vert \psi _{3}\right\rangle &=&\sum_{kk^{^{\prime }}}p_{kk^{^{\prime
}}}{a_{k}}^{\dag }{a_{k^{\prime }}}^{\dag }\left\vert 0\right\rangle
\left\vert \uparrow \right\rangle \\
&&+\sum_{kk^{^{\prime }}k^{^{\prime \prime }}}q_{kk^{^{\prime }}k^{^{\prime
\prime }}}{a_{k}}^{\dag }{a_{k^{\prime }}}^{\dag }{a_{k^{\prime \prime }}}%
^{\dag }\left\vert 0\right\rangle \left\vert \downarrow \right\rangle .\end{aligned}$$Then the wavefunction in the truncated Hilbert space up to $N$ excitations can be expressed as$$\left\vert \psi \right\rangle ^{\leq N}=\sum_{l=0}^{N}\left\vert \psi
_{l}\right\rangle .  \label{ED}$$For $N=0$, $\left\vert \psi \right\rangle ^{\leqslant 0}=\left\vert
0\right\rangle \left\vert 