Modulating Hamiltonian approach to quantum many-body systems and crystalline topological phases protected by generalized magnetic translations

We discuss the topology of the parameter space of invertible phases with an onsite symmetry $G$, i.e., quantum many-body ground states that have neither fractionalization nor spontaneous breaking of the symmetry. The classification of invertible phases is known to be obtained by counting the connected components in the parameter space of the invertible phases. We consider its generalization --- the deformation classes of the mappings from $n$-dimensional spheres ${S}^{n}$ to this parameter space for arbitrary integer $n$. We argue a direct one-to-one correspondence in the framework of lattice models between the noncontractibility of ${S}^{n}$ and (i) the classification of invertible phases in $d$ dimensions when $d\ensuremath{\ge}n$; or (ii) zero-dimensional invertible Hamiltonians parametrized by ${S}^{n\ensuremath{-}d}$ when $d<n$, using an isotropic modulating Hamiltonian approach. Explicitly, we construct the noncontractible spheres of two-dimensional invertible phases, i.e., $n=2$ and $d=2$. We also propose a large class of crystalline topological phases protected by a generalized magnetic translations.