# Many-Body Order Parameters for Multipoles in Solids

## TL;DR

This paper introduces many-body order parameters to detect and measure multipolar charge distributions in crystalline solids, based on the topology of nested Wilson loops, and demonstrates their robustness even without symmetry constraints.

## Contribution

It proposes a novel set of many-body order parameters for bulk multipoles that are robust to symmetry breaking and validated through analytic and numerical methods.

## Key findings

- Order parameters match localized corner charges
- Order parameters measure physical multipole moments accurately
- Robustness of order parameters without symmetry constraints

## Abstract

We propose many-body order parameters for bulk multipoles in crystalline systems, which originate from the non-trivial topology of the nested Wilson loop spectrum. The many-body order parameters are designed to measure multipolar charge distribution in a crystalline unit cell, and they match the localized corner charge originating from the multipoles. We provide analytic arguments and numerical proof for the order parameters. Furthermore, we show that the many-body order parameters faithfully measure the physical multipole moments even when the symmetries quantizing multipoles are lost and thus the nested Wilson loop spectrum does not exactly reproduce the physical multipole moments.