tunneling two-level system  [@zheng; @Chin]. The magnetization along the $%
z-$direction ]{}$\left\vert \left\langle \sigma _{z}\right\rangle \right\vert $[, simply the order parameter along z direction, is the renormalized factor of the tunneling amplitude $\Delta $. In this subsection, we examine the magnetization along the z direction as well as the ground-state energy by both VMPS and $3$ED, which are exhibited in Fig. \[GS\_quantity\]. ]{}

Obviously, two observables begin to deviate only after the critical point $%
\alpha _{c}$. Especially we find that the ground-state energies by the VMPS become lower than those by $3$ED after the critical points, indicating again the invalidity of ED method at strong coupling. Practically, we cannot perform ED with a very large total excitation number due to the huge Hilbert space. In this paper, our exact diagonalization is only performed up to $\ N=3$. Of course, if one can really perform true exact diagonalization without the limitation of total excitation numbers, the true ground state could be also correctly described in the ED method.

[ It is interesting to note from Figs. \[Magnetization\] and [GS\_quantity]{} that ]{}$\left\vert {\left\langle \sigma _{x}\right\rangle }%
\right\vert $[ and ]{}$\left\vert {\left\langle \sigma _{z}\right\rangle }%
\right\vert $[ exhibit different behaviors with the 