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arXiv:2603.10620 · Journal of the Physical Society of Japan

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

Shingo Kuniyoshi, Rikuto Oiwa, Satoru Hayami

Published Jun 30, 2026• 1 citation
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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.

DOI
10.7566/jpsj.95.073704

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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.
The multipole decomposition of the single-site density matrix provides a symmetry-adapted representation of local electronic degrees of freedom. Conventional, so-called fixed-shell, formulations do not span the full local single-particle operator space, as only operators mapping within the same orbital lll are resolved. Here we construct a complete orthogonal basis of real Hermitian multipole operators for the local density matrix by extending the existing formulation to inter-shell operators. We revisit the multipole decomposition as a decomposition of the operator space by SO(3)\mathrm{SO}(3)SO(3) by first decomposing the orbital and spin operator spaces. Then by coupling them we arrive at the spin-12\frac{1}{2}21​ local single-particle operator space, staying consistent with the existing fixed-shell formulations. We then classify the multipoles by parity and time-reversal symmetry, which allows for unique identification of multipole moments of the density matrix that contribute to expectation values of fully symmetry-resolved observables. As an application, we analyze the two enantiomers of chiral trigonal tellurium by computing the electric toroidal monopole moment selected by symmetry.
The conventional multipole framework provides a standard symmetry-based language for one-body electronic degrees of freedom, but fails to distinguish physically distinct sectors in many-body operator space. We show that canonical symmetries compatible with rotations provide this structure through two additional 2 labels associated with the particle-number gauge transformation Gπ/2 and the particle-hole transformation CA. These labels separate operators with different body numbers and particle-number changes, leading to a sixteenfold classification that organizes selection rules for nonzero expectation values, induced multipoles, and symmetry-allowed couplings in many-body multipole space.
A geometric theory of the irreducible tensor operators of quantum spin systems. It is based upon the Maxwell-Sylvester geometric representation of the multipolar electrostatic potential. In the latter, an order-ℓ\ellℓ multipolar potential is represented by a collection of ℓ\ellℓ equal length vectors, i.e. by ℓ\ellℓ points on a sphere, instead of by its components on some fixed (but arbitrary) basis. The geometric representation offers a much more appropriate tool for getting physical insight on specific characteristics of a multipole, such as its symmetries, or its departure from ideal symmetry. We derive explicit expressions enabling to perform any calculations we may need to perform on multipoles. All relevant quantities are eventually expressed in terms of scalar products of pairs of vectors (i.e., in terms of geometric quantities such as lengths and angles). The whole formalism is entirely independent of any particular choice of coordinate, and needs no use of the somehow abstract formalism traditionally used when dealing with angular momenta. The formalism is then applied to treat the problem of the irreducible tensor operators of quantum spin systems. It enables to completely dispense with the calculation and use of the Stevens operators, which can be quite complicated even for moderate values of ℓ\ellℓ. Explicit expressions for the calculation of expectations values of physical observables are derived. They essentially consist in combinations of scalar products of vector pairs. Together with the coherent state representation of the quantum states of spin systems, this provides a complete geometric, coordinate-free, description of the states, dynamics and physical properties of these systems.
We provide a systematic treatment of the tenfold way of classifying fermionic systems that naturally allows for the study of those with arbitrary N-body interactions. We identify four types of symmetries such systems can possess, which consist of one ordinary type (usual unitary symmetries), and three non-ordinary symmetries (such as time reversal, charge conjugation and sublattice). Focusing on systems that possess no non-trivial ordinary symmetries, we demonstrate that the non-ordinary symmetries are strongly constrained. This approach not only leads very naturally to the tenfold classes, but also obtains the canonical representations of these symmetries in each of the ten classes. We also provide a group cohomological perspective of our results in terms of projective representations. We then use the canonical representations of the symmetries to obtain the structure of Hamiltonians with arbitrary N-body interactions in each of the ten classes. We show that the space of N-body Hamiltonians has an affine subspace (of a vector space) structure in classes which have either or both charge conjugation and sublattice symmetries. Our results can help address many open questions including the topological classification of interacting fermionic systems. (C) 2017 Elsevier Inc. All rights reserved.
Symmetry plays a central role in modern physics, from classifying quantum states to characterizing phases of matter through spontaneous symmetry breaking. In interacting fermionic systems with multiple internal degrees of freedom, however, determining the full continuous symmetry group and classifying possible order parameters remain challenging. In this work, we present a systematic framework for analyzing continuous symmetries and identifying candidate order parameters in such systems. By mapping the Hamiltonian to a Majorana representation, we obtain the generators of continuous symmetries from the Lie algebra of operators that commute with the Hamiltonian. We then identify the structure of this Lie algebra using the theory of semisimple Lie algebras. Building on representation theory, we further develop a systematic method for exhaustively enumerating candidate order parameters. By decomposing the exterior-power representations induced by the symmetry algebra on the Majorana space and incorporating discrete lattice symmetries, we classify these order parameters according to the symmetries they break. (Abridged. Please see the PDF manuscript for the complete abstract and specific model applications.)