Higher Berry Curvature from the Wave function II: Locally Parameterized States Beyond One Dimension

TL;DR

This paper introduces a systematic wave function-based method to construct higher-dimensional topological invariants, like higher Berry curvature, for lattice systems using local parameter spaces, applicable to tensor network ground states.

Contribution

It generalizes the concept of Berry curvature to higher forms in multiple dimensions and provides a construction for topological invariants in short-range entangled lattice systems.

Findings

Constructed $(d+2)$-form higher Berry curvature for $d$-dimensional systems.

Demonstrated the approach with exactly solvable 2D lattice models.

Showed the quantization of topological invariants for short-range entangled states.

Abstract

We propose a systematic wave function based approach to construct topological invariants for families of lattice systems that are short-range entangled using local parameter spaces. This construction is particularly suitable when given a family of tensor networks that can be viewed as the ground states of $d$ dimensional lattice systems, for which we construct the closed $(d+2)$-form higher Berry curvature, which is a generalization of the well known 2-form Berry curvature. Such $(d+2)$-form higher Berry curvature characterizes a flow of $(d+1)$-form higher Berry curvature in the system. Our construction is equally suitable for constructing other higher pumps, such as the (higher) Thouless pump in the presence of a global on-site $U(1)$ symmetry, which corresponds to a closed $d$-form. The cohomology classes of such higher differential forms are topological invariants and are expected to be quantized for short-range entangled states. We illustrate our construction with exactly solvable lattice models that are in nontrivial higher Berry classes in $d=2$.