Deconfined criticalities and dualities between chiral spin liquid, topological superconductor and charge density wave Chern insulator

Abstract

We propose bi-critical and tri-critical theories between chiral spin liquid (CSL), topological superconductor (SC) and charge density wave (CDW) ordered Chern insulator with Chern number C=2C=2 on square, triangular and Kagome lattices. The three CDW order parameters form a manifold of S^2S2 or S^1S1 depending on whether there is easy-plane anisotropy. The skyrmion defect of the CDW order carries physical charge 2e2e and its condensation leads to a topological superconductor. The CDW-SC transitions are in the same universality classes as the celebrated deconfined quantum critical points (DQCP) between Neel order and valence bond solid order on square lattice. Both SC and CDW order can be accessed from the CSL phase through a continuous phase transition. At the CSL-SC transition, there is still CDW order fluctuations although CDW is absent in both sides. We propose three different theories for the CSL-SC transition (and CSL to easy-plane CDW transition): a U(1)U(1) theory with two bosons, a U(1)U(1) theory with two Dirac fermions, and an SU(2)SU(2) theory with two bosons. Our construction offers a derivation of the duality between these three theories as well as a promising physical realization. The SU(2)SU(2) theory offers a unified framework for a series of fixed points with explicit SO(5), O(4)SO(5),O(4) or SO(3)× O(2)SO(3)×O(2) symmetry. There is also a transparent duality transformation mapping SC order to easy-plane CDW order. The CSL-SC-CDW tri-critical points are invariant under this duality mapping and have an enlarged SO(5)SO(5) or O(4)O(4) symmetry. The DQCPs between CDW and SC inherit the enlarged symmetry, emergent anomaly, and self-duality from the tri-critical point. Our analysis unifies the well-studied DQCP between symmetry breaking phases into a larger framework where they are proximate to a topologically ordered phase. Experimentally the theory demonstrates the possibility of a rich phase diagram and criticality through closing the Mott gap of a quantum spin liquid with projective symmetry group.