We investigate two dimensional(2D) chiral dx2−y2 ± idxy topological superconductors and three dimensional(3D) d wave topological superconductors, through concrete models. We demonstrate that these two kinds of topological superconductors are the simplest cases of more general 2D class C topological superconductors and 3D class CI topological superconductors, respectively. We then give general methods to systematically build models for all 2D class C and 3D class CI topological superconductors. Our theoretical constructions may be a critical step to experimentally realize these exotic topologically superconducting phases. The chiral edge modes or gapless surface states of our 2D or 3D models are studied in details. In all the situations, we find novel mechanisms for bulk boundary correspondence.

We construct theoretical models for two dimensional(2d) chiral d x 2 −y 2 ±id xy topological superconductors and for three dimensional(3d) d wave topological superconductors. Moreover we build models for any 2d class C and 3d class CI topological superconductors (with any even topological invariants). We also construct concrete models that can realize quantum anomalous Hall effect with arbitrary large Chern numbers. We study the chiral edge modes or gapless surface states of our 2d or 3d models in details. In all the cases, we find novel mechanisms that make the numbers of boundary states always agree with the nontrivial bulk topology, just as required by the bulk boundary correspondence.

We give a new simple derivation for the sine-Gordon description of BerezinskiiKosterlitz-Thouless(BKT) phase transition (is driven by vortices). Our derivation is simpler than traditional derivations. Besides, our derivation is a continuous field theoretic derivation by using path integration, different from the traditional derivations which are based on lattice theory or based on Coulomb gas model. Our new derivation rely on Abelian duality of two dimensional quantum field theory. By utilizing this duality in path integration, we find that the vortex configurations are naturally mapped to exponential operators in dual description, these operators are the vortex operators that can create vortices, the sine-Gordon description then naturally follows. Our method may be useful for the investigation to the BKT physics of superconductors.

We construct simple models for all topological phases of free fermions. These explicit models can realize all the nontrivial topological phases (with any possible topological invariant) of the periodic table. Many well known models for topological insulators and superconductors are special cases of our general constructions.

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