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\begin{document}
\onecolumngrid
\section{Explicit proof of constraints and effective theories presented in Table I}

In this appendix, we give a general derivation the constraints and effective theories shown in Table I. On a $C_m$ invariant line in a $C_n$ invariant system ($m$ being a factor of $n$), each band corresponds to a 1D representation of cyclic group $C_m$ (spinless), or double cyclic group $C^D_m$ (spinful). In either case, an eigenvalue of $C_m$ is in the form:\bea\alpha_p=\exp(i2\pi p/m+iF\pi/m),\eea where $p=0,1,...m-1$. Now suppose the $C_m$ eigenvalues of the conduction and valence bands are\bea u_c=\alpha_p,\\
\nonumber u_v=\alpha_q.\eea We have if $p=q$, then the two bands will \emph{not} cross on this $C_m$-invariant line because of the presence of a symmetry allowed a constant off-diagonal term $\delta|\psi_u\rangle\langle\psi_v|+h.c.$ which can always open a gap. If $p\neq q$, then the matrix representation of $C_m$, $\mathcal{C}_m$ is given by\bea\mathcal{C}_m=\exp(i\pi\frac{F+p+q}{m})\exp(i\pi\frac{p-q}{m}\sigma_z).\eea The transform of $H_{eff}(\bq)$ under $C_m$ is given by\bea\label{eq:temp1}\mathcal{C}_m{}H_{eff}(\bq)\mathcal{C}^{-1}_m&=&g(\bq)\sigma_z+f(\bq)\exp(i\pi\frac{p-q}{m}\sigma_z)\sigma_+\exp(-i\pi\frac{p-q}{m}\sigma_z)+h.c.\\
\nonumber&=&g(\bq)\sigma_z+f(\bq)e^{-i2\pi(p-q)/m}\sigma_++h.c.\eea
In the basis of $q_\pm$, the $R_m\bq$ is given by\bea R_m(q_+,q_-)=(q_+e^{i2\pi/m},q_-e^{-i2\pi/m}).\label{eq:temp2}\eea Substituting Eq.(\ref{eq:temp1},\ref{eq:temp2}) into Eq.(1) in main text, we obtain\bea e^{-i2\pi(p-q)/m}f(q_+,q_-)=f(q_+e^{i2\pi/m},q_-e^{-i2\pi/m})\label{eq:temp3},\\
\nonumber g(q_+,q_-)=g(q_+e^{i2\pi/m},q_-e^{-i2\pi/m}).\eea This is the general constraint on $f,g$ by $C_m$ symmetry for a given pair of $(u_c,u_v)$. From Eq.(\ref{eq:temp3}) we learn that the forms of $f$ and $g$ only depend on $p-q$, or $u_c/u_v$. These results comprise the central column of Table I.

To generate the last column from the general constraints Eq.(\ref{eq:temp3}), we start from an expansion of \bea f(q_+,q_-)=\sum_{n_1n_2}A_{n_1n_2}q_+^{n_1}q_-^{n_2},\eea where $A_{n_1n_2}$ is an arbitrary complex coefficient. Eq.(\ref{eq:temp3}) gives $A_{n_1n_2}=0$ if $n_2-n_1\neq p-q\;\textrm{mod}\;m$. Then we pick up $(n_1,n_2)$ with smallest $n_1+n_2$ and nonzero $A_{n_1n_2}$ to obtain the last column of Table I.

Physically, one can easily understand Table I as the consequence of the `semi-conservation' of total angular momentum $J_z$. Here `semi-conservation' means that $J_z$ is only conserved up to a multiple of $m$. This is because in a lattice the continuous rotation symmetry downgrades to discrete rotation symmetry of order $m$. If $u_c/u_v=e^{i2\pi(p-q)/m}$, if means the conduction and valence bands differ by $\delta J_z=p-q$. Therefore, the off-diagonal term of the effective Hamiltonian, $|\psi_c\rangle\langle\psi_v|$, must couple a $k_-^{p-q}$ or $k_+^{m-p+q}$ to conserve the total angular momentum up to a multiple of $m$.

\end{document}