Discrete higher Berry phases and matrix product states

A one-parameter family of invertible states gives a topological transport phenomenon, similar to the Thouless pumping. As a natural generalization of this, we can consider a family of invertible states parametrized by some topological space X. This is called a higher pump. It is conjectured that a $(1+1)$-dimensional bosonic invertible state parametrized by X is classified by H³(X;Z). In this paper, we construct two higher pumping models parametrized by X=RP²×S¹ and X=L(3,1)×S¹ that corresponds to the torsion part of H³(X;Z). As a consequence of the nontriviality as a family, we find that a quantum mechanical system with a nontrivial discrete Berry phase is pumped to the boundary of the $(1+1)$-dimensional system. We also study higher pump phenomena by using matrix product states, and construct a higher pump invariant which takes value in a torsion part of H³(X;Z). This is a higher analog of the ordinary discrete Berry phase that takes value in the torsion part of H²(X;Z). In order to define the higher pump invariant, we utilize the smooth Deligne cohomology and its integration theory. We confirm that the higher pump invariant of the model has a nontrivial value.