Parameterized families of 2+1d $G$-cluster states

We construct a $G$-cluster Hamiltonian in 2+1 dimensions and analyze its properties. This model exhibits a $G\times2\mathrm{Rep}(G)$ symmetry, where the $2\mathrm{Rep}(G)$ sector realizes a non-invertible symmetry obtained by condensing appropriate algebra objects in $\mathrm{Rep}(G)$. Using the symmetry interpolation method, we construct $S^1$- and $S^2$-parameterized families of short-range-entangled (SRE) states by interpolating an either invertible $0$-form or $1$-form symmetry contained in $G\times2\mathrm{Rep}(G)$. Applying an adiabatic evolution argument to this family, we analyze the pumped interface mode generated by this adiabatic process. We then explicitly construct the symmetry operator acting on the interface and show that the interface mode carries a nontrivial charge under this symmetry, thereby demonstrating the nontriviality of the parameterized family.