Higher Berry Curvature from the Wave Function I: Schmidt Decomposition and Matrix Product States

TL;DR

This paper introduces a new method to compute higher Berry curvature in extended quantum systems using wave functions and matrix product states, revealing quantized invariants and demonstrating applicability through models.

Contribution

It provides a simple formula for higher Berry curvature calculation in 1D systems via Schmidt decomposition and MPS, including quantization results for translationally invariant states.

Findings

Derived a formula for HBC using wave functions and Schmidt decomposition.

Established a quantized invariant for translationally invariant MPS.

Validated the approach with exactly solvable and numerical models.

Abstract

Higher Berry curvature (HBC) is the proposed generalization of Berry curvature to infinitely extended systems. Heuristically HBC captures the flow of local Berry curvature in a system. Here we provide a simple formula for computing the HBC for extended $d = 1$ systems at the level of wave functions using the Schmidt decomposition. We also find a corresponding formula for matrix product states (MPS), and show that for translationally invariant MPS this gives rise to a quantized invariant. We demonstrate our approach with an exactly solvable model and numerical calculations for generic models using iDMRG