Discrete Higher Berry Phases and Matrix Product States

A $1$-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 $(1+1)$-dimensional bosonic invertible state parametrized by $X$ is classified by $\mathrm{H}^{3}(X;\mathbb{Z})$. In this paper, we construct two higher pumping models parametrized by $X=\mathbb{R}P^{2}\times S^1$ and $X=\mathrm{L}(3,1)\times S^1$ that corresponds to the torsion part of $\mathrm{H}^{3}(X;\mathbb{Z})$. 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 (MPS), and construct a higher pump invariant which takes value in a torsion part of $\mathrm{H}^{3}(X;\mathbb{Z})$. This is a higher analog of the ordinary discrete Berry phase that takes value in the torsion part of $\mathrm{H}^{2}(X;\mathbb{Z})$. 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.