Probing the Quantized Berry Phases in 1H-NbSe$_2$ Using Scanning Tunneling Microscopy

Topologically trivial insulators are classified into two primary categories: unobstructed and obstructed atomic insulators. While both types can be described by exponentially localized Wannier orbitals, a defining feature of obstructed atomic insulators is that the centers of charge of these orbitals are positioned at empty sites within the unit cell, rather than on atoms. Despite extensive theoretical predictions, the unambiguous and quantitative experimental identification of an obstructed atomic phase has remained elusive. In this work, we present the first direct experimental evidence of such a phase in 1H-NbSe$_2$. We develop a novel method to extract the inter-orbital correlation functions from the local spectral function probed by scanning tunneling microscopy (STM), leveraging the orbital wave functions obtained from ab initio calculations. Applying this technique to STM images, we determine the inter-orbital correlation functions for the atomic band of 1H-NbSe$_2$ that crosses the Fermi level. Our results show that this band realizes an optimally compact obstructed atomic phase, providing the first unambiguous experimental identification of such a phase. Our approach of deconvolving the STM signal using ab initio orbital wave functions is broadly applicable to other material platforms, offering a powerful tool for exploring other electronic phases.