Many-Body Order Parameters for Multipoles in Solids

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.Comment: Added discussions on Bosonic systems and C4-symmetric insulators; Main Text: 6 pages, 3 figures. Supplemental Material: 15 pages, 10 figure