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Edge states and topological orders in the spin liquid phases of star lattice

A group of novel materials can be mapped to the star lattice, which exhibits some novel physical properties. We give the bulk-edge correspondence theory of the star lattice and study the edge states and their topological orders in different spin liquid phases. The bulk and edge-state energy structures and Chern number depend on the spin liquid phases and hopping parameters because the local spontaneous magnetic flux in the spin liquid phase breaks the time reversal and space inversion symmetries. We give the characteristics of bulk and edge energy structures and their corresponding Chern numbers in the uniform, nematic and chiral spin liquids. In particular, we obtain analytically the phase diagram of the topological orders for the chiral spin liquid states SL[ϕ,ϕ,-2ϕ], where ϕis the magnetic flux in two triangles and a dodecagon in the unit cell. Moreover, we find the topological invariance for the spin liquid phases, SL[ϕ_{1},ϕ_{2},-(ϕ_{1}+ϕ_{2})] and SL[ϕ_{2},ϕ_{1},-(ϕ_{1}+ϕ_{2})]. The results reveal the relationship between the energy-band and edge-state structures and their topological orders of the star lattice.