Topological properties of one-dimensional hardcore Bose?Fermi mixture

We study the topological properties of a one-dimensional (1D) hardcore Bose?Fermi mixture using the exact diagonalization method. We firstly add a hardcore boson to a fermionic system and by examining the edge states we find that the quasi-particle manifests the topological properties of the system. Then we study a mixture with 7 fermions and 1 boson. We find that the mixture also exhibits topological properties and its behaviors are similar to that of the corresponding fermionic system. We present a qualitative explanation to understand such behaviors using the mapping between a hardcore boson and a spinless fermion. These results show the existence of topological properties in a 1D hardcore Bose?Fermi mixture and may be realized using cold atoms trapped in optical lattices experimentally.