Higher Berry phase of fermionic matrix product states

Abstract

This thesis is to explore the higher Berry phase as a generalization of the normal Berry phase, for a particular class of states known as the fermionic matrix product states. The higher Berry phase has been defined through the triple inner product for bosonic matrix product states, which allows us to define a topological number in three dimensional parameter spaces, as a generalization of the Chern number in two-dimensional parameter spaces. By analyzing the algebraic structure of fermionic matrix product states, we explicitly generalize the definition and calculation methods of higher Berry phase to fermionic matrix product states. We further construct a concrete fermion model of two coupled SSH chains, and show that it has a nontrivial phase with a nonzero quantized total higher Berry phase. The concept of the triple inner product also allows the extraction of the information of on-site symmetry projective representations, which might be applicable in symmetry-protected topological states.