Theory of Many-Body Multipole Operators in Single-Centered Electron Systems: Two-Body Toroidal Monopoles in Spinless Orbitals

TL;DR

This paper extends the multipole operator classification from one-body to many-body systems, revealing new active monopole operators in spinless interacting electron systems.

Contribution

It develops a systematic framework for classifying many-body multipole operators using spherical tensors and Grassmann algebra, including monopoles absent in one-body descriptions.

Findings

Classifies many-body monopoles including electric and magnetic toroidal monopoles.

Shows these monopoles become active in spinless interacting systems.

Provides a comprehensive irreducible decomposition of many-body operators.

Abstract

One-body multipole operators are defined as irreducible representations of rotational symmetry together with spatial-inversion and time-reversal symmetries, providing a systematic framework for classifying electronic internal degrees of freedom and for describing a wide variety of composite order parameters. While this formalism has been successfully established for the one-body operator space, a systematic classification of the many-body operator space, especially in interacting systems, remains an open challenge. In this paper, we extend the multipole formalism in the one-body operator space to the many-body operator space. By formulating fermionic creation and annihilation operators as spherical tensors and employing Clebsch-Gordan coupling combined with the exterior (Grassmann) algebra, we construct an irreducible decomposition of many-body operators that fully incorporates fermionic antisymmetrization. As a concrete application, we classify monopoles appearing in spinless many-body operators. In particular, we show that the electric toroidal monopole, a pseudoscalar breaking spatial-inversion symmetry, and the magnetic toroidal monopole, a time reversal-odd scalar, become active in spinless interacting many-body systems, although they are absent in the spinless one-body hybrid orbital space.