Discrete higher Berry phases and matrix product states

Abstract

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 ${\mathrm{H}}^{3}(X;\mathbb{Z})$. In this paper, we construct two higher pumping models parametrized by $X=\mathbb{R}{P}^{2}\ifmmode\times\else\texttimes\fi{}{S}^{1}$ and $X=\mathrm{L}(3,1)\ifmmode\times\else\texttimes\fi{}{S}^{1}$ 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, 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.

Topics & Concepts