The effective Hamiltonian is given by $$\begin{gathered} H_{\mathrm{eff}}=-\frac{J}{4}\sum _{\sigma ,{\sigma }^{\prime }=\pm }\sum _i\left(\begin{array}{cc}{f}_{i\sigma }^{†}& {ϵ}_{\sigma {\sigma }^{\prime }}{f}_{i{\sigma }^{\prime }}\end{array}\right)\left(\begin{array}{cc}{\chi }_{ii+1}^{†}& {\mathrm{\Delta }}_{ii+1}\\ {\mathrm{\Delta }}_{ii+1}^{†}& -{\chi }_{ii+1}\end{array}\right) \\ \times \left(\begin{array}{c}{f}_{i+1\sigma }\\ {ϵ}_{\sigma {\sigma }^{\prime }}{f}_{i+1\sigma }^{†}\end{array}\right)+\frac{J}{4}\sum _i(|{\chi }_{ii+1}{|}^2+|{\mathrm{\Delta }}_{ii+1}{|}^2), \end{gathered}$$invariant under the particle-hole SU(2) gauge transformation $$\left(\begin{array}{c}{f_{i\sigma }}^{\prime }\\ {ϵ}_{\sigma {\sigma }^{\prime }}{f}_{i\sigma }^{\prime †}\end{array}\right)={\mathit{g}}_i\left(\begin{array}{c}{f}_{i\sigma }\\ {ϵ}_{\sigma {\sigma }^{\prime }}{f}_{i\sigma }^{†}\end{array}\right)$$and $$\left(\begin{array}{cc}{\chi }_{ii+1}^{\prime †}& {{\mathrm{\Delta }}_{ii+1}}^{\prime }\\ {\mathrm{\Delta }}_{ii+1}^{\prime †}& -{{\chi }_{ii+1}}^{\prime }\end{array}\right)={\mathit{g}}_i\left(\begin{array}{cc}{\chi }_{ii+1}^{†}& {\mathrm{\Delta }}_{ii+1}\\ {\mathrm{\Delta }}_{ii+1}^{†}& -{\chi }_{ii+1}\end{array}\right){\mathit{g}}_{i+1}^{†},$$where ${\mathit{g}}_i$ belongs to the SU(2) group.

The emergence of an insulating stable fixed point seems to imply an interesting feature of the real-space renormalization group approach. The momentum-space renormalization group approach allows a metallic stable fixed point, where momentum is a good quantum number associated with emergent $U_k(1)$ symmetry at each momentum point $k$. On the other hand, the real-space renormalization group approach gives rise to the insulating stable fixed point, where position is a good quantum number related with emergent $U_x(1)$ symmetry at each position $x$. This feature looks general in our mean-field plus real-space renormalization group framework, which may be involved with the emergence of ${\mathrm{AdS}}_2$ in the IR limit.