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We study a two-dimensional (2D) tight-binding model of a topologicalcrystalline insulator (TCI) protected by rotation symmetry. The model is builtby stacking two Chern insulators with opposite Chern numbers which transformunder conjugate representations of the rotation group, e.g. $p_\pm$ orbitals.Despite its apparent similarity to the Kane-Mele model, it does not host stablegapless surface states. Nevertheless the model exhibits topological responsesincluding the appearance of quantized fractional charge bound to rotationaldefects (disclinations) and the pumping of angular momentum in response tothreading an elementary magnetic flux, which are described by a mutualChern-Simons coupling between the electromagnetic gauge field and an effectivegauge field corresponding to the rotation symmetry. In addition, we show thatalthough the filled bands of the model do not admit a symmetric Wannierrepresentation, this obstruction is removed upon the addition of appropriateatomic orbitals, which implies `fragile' topology. As a result, the response ofthe model can be derived by representing it as a superposition of atomicorbitals with positive and negative integer coefficients. Following theanalysis of the model, which serves as a prototypical example of 2D TCIsprotected by rotation, we show that all TCIs protected by point groupsymmetries which do not have protected surface states are either atomicinsulators or fragile phases. Remarkably, this implies that gapless surfacestates exist in free electron systems if and only if there is a stable Wannierobstruction. We then use dimensional reduction to map the problem ofclassifying 2D TCIs protected by rotation to a zero-dimensional (0D) problemwhich is then used to obtain the complete non-interacting classification ofsuch TCIs as well as the reduction of this classification in the presence ofinteractions.

Shift insulators: rotation-protected two-dimensional topological crystalline insulators