regime belong to the energy levels in different $\left\langle \hat{N}\right\rangle $ subspaces, leading to level crossing at some coupling strengths. Thus the first derivative of the ground-state energy with respect to the coupling strength must be discontinuous at these coupling strengths, so instability of $\left\langle \hat{N}\right\rangle $ before $\alpha _{c}$ just corresponds to the first-order phase transitions. In addition, the jump of $%
\left\langle \hat{N}\right\rangle $ can also account for the back and forth of the parity between $1 $ and $-1$.

One can also note that the total excitation number$\ \left\langle \hat{N}%
\right\rangle $ by VMPS increases abruptly at the critical coupling $\alpha
_{c}$. The $N$ED approach can only describe the phase with excitation number less than or equal to $N$. The total excitation number is not limited in VMPS, so it in principle can describe all phases. The total excitation numbers are not conserved in the ground state above $\alpha _{c}$, because of the symmetry breaking. At the critical points, $\left\langle \hat{N}%
\right\rangle $ does not jump to a plateau with the finite larger integer, different from the first-order phase transitions.

The instability of total excitation number $\left\langle \hat{N}%
\right\rangle $ at the phase transitions can account for 