Higher Berry Phase from Projected Entangled Pair States in (2+1) dimensions

TL;DR

This paper introduces a higher Berry phase invariant for families of 2D invertible quantum states using PEPS, extending topological classification tools to higher dimensions and complex parameter spaces.

Contribution

It generalizes the concept of the higher Berry phase to two-dimensional states via a four-state inner product within the PEPS framework, enabling topological analysis of 2D invertible states.

Findings

Defined a topological invariant in H^4(X;Z) for 2D invertible states.

Constructed a non-trivial example over real projective 4-space.

Demonstrated the application of the invariant to symmetry-protected phases.

Abstract

We consider families of invertible many-body quantum states in $d$ spatial dimensions that are parameterized over some parameter space $X$. The space of such families is expected to have topologically distinct sectors classified by the cohomology group $\mathrm{H}^{d+2}(X;\mathbb{Z})$. These topological sectors are distinguished by a topological invariant built from a generalization of the Berry phase, called the higher Berry phase. In the previous work, we introduced a generalized inner product for three one-dimensional many-body quantum states, (``triple inner product''). The higher Berry phase for one-dimensional invertible states can be introduced through the triple inner product and furthermore the topological invariant, which takes its value in $\mathrm{H}^{3}(X;\mathbb{Z})$, can be extracted. In this paper, we introduce an inner product of four two-dimensional invertible quantum many-body states. We use it to measure the topological nontriviality of parameterized families of 2d invertible states. In particular, we define a topological invariant of such families that takes values in $\mathrm{H}^{4}(X;\mathbb{Z})$. Our formalism uses projected entangled pair states (PEPS). We also construct a specific example of non-trivial parameterized families of 2d invertible states parameterized over $\mathbb{R}P^4$ and demonstrate the use of our formula. Applications for symmetry-protected topological phases are also discussed.