Globally symmetric topological phase: from anyonic symmetry to twist defect.

Topological phases in two dimensions support anyonic quasiparticle excitations that obey neither bosonic nor fermionic statistics. These anyon structures often carry global symmetries that relate distinct anyons with similar fusion and statistical properties. Anyonic symmetries associate topological defects or fluxes in topological phases.

Existence of Majorana-fermion bound states on disclinations and the classification of topological crystalline superconductors in two dimensions.

We prove a topological criterion for the existence of a zero-energy Majorana bound state on a disclination, a rotation symmetry breaking point defect, in fourfold symmetric topological crystalline superconductors (TCS) in two dimensions. We first establish a complete topological classification of TCS using the Chern invariant and three integral rotation invariants. By analytically and numerically studying disclinations, we algebraically deduce a Z2 index that identifies the parity of the number of Majorana zero modes at a disclination.

Disclination classes, fractional excitations, and the melting of quantum liquid crystals.

We consider how fractional excitations bound to a dislocation evolve as the dislocation is separated into a pair of disclinations. We show that some dislocation-bound excitations (such as Majorana modes and half-quantum vortices) are possible only if the elementary dislocation consists of two inequivalent disclinations, as is the case for stripes or square lattices but not for triangular lattices. The existence of multiple inequivalent disclination classes governs the two-dimensional melting of quantum liquid crystals (i.

Majorana fermions and non-Abelian statistics in three dimensions.

We show that three dimensional superconductors, described within a Bogoliubov-de Gennes framework, can have zero energy bound states associated with pointlike topological defects. The Majorana fermions associated with these modes have non-Abelian exchange statistics, despite the fact that the braid group is trivial in three dimensions. This can occur because the defects are associated with an orientation that can undergo topologically nontrivial rotations.

Geometric phase in eigenspace evolution of invariant and adiabatic action operators.

The theory of geometric phase is generalized to a cyclic evolution of the eigenspace of an invariant operator with N-fold degeneracy. The corresponding geometric phase is interpreted as a holonomy inherited from the universal Stiefel U(N) bundle over a Grassmann manifold. Most significantly, for an arbitrary initial state, this holonomy captures the inherent geometric feature of the state evolution that may not be cyclic.