The presence of topological edge modes at the interface two perturbed honeycomb photonic crystals with $C_6$ symmetry is often attributed to different signs Berry curvature K and K$'$ valleys. In contrast electronic counterpart, Chern number defined in valley Hall effect not a quantized quantity but can be tuned finite values including zero simply by changing geometrical perturbations. Here, we...

Geometric phases in quantum mechanics play an extraordinary role in broadening our understanding of fundamental significance of geometry in nature. One of the best known examples is the Berry phase [M.V. Berry (1984), Proc. Royal. Soc. London A, 392:45], which naturally emerges in quantum adiabatic evolution. So far the applicability and measurements of the Berry phase were mostly limited to sy...

We derive an effective Hamiltonian for a quantum system constrained to a submanifold ~the constraint manifold! of configuration space ~the ambient space! by an infinite restoring force. We pay special attention to how this Hamiltonian depends on quantities which are external to the constraint manifold, such as the extrinsic curvature of the constraint manifold, the curvature of the ambient spac...

Topological aspects of electron wavefunction play a crucial role in determining the physical properties materials. Berry curvature and Chern number are used to define topological structure electronic bands. While its effects materials have been studied, detecting changes invariant, number, is challenging. In this regard, twisted double bilayer graphene (TDBG) has emerged as promising platform g...