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arXiv:2607.13053

On multipoles, their decomposition by time-reversal symmetry, and the electric toroidal monopole

Vinzenz A. Müller, Nora Taufertshöfer, Nicola A. Spaldin

quant-phcond-mat.mtrl-scimath-phSubmitted Jul 3, 2026• Announced Jul 6, 2026• 0 citations
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Abstract

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 $l$ 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 $\mathrm{SO}(3)$ by first decomposing the orbital and spin operator spaces. Then by coupling them we arrive at the spin-$\frac{1}{2}$ 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.

Comments
18 pages, 1 figure

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Werner Kutzelnigg

The density operator (density matrix) of a quantum mechenical system can be decomposed into operators which transform as irreducible representations of the symmetry group in coordinate and spin space. Each of these components has a physical meaning connected with the expectation values of certain operators. The reduced density matrices can be decomposed in a completely analogous way. The symmetry properties of the total wave function give rise to degeneracies of the eigenvalues of the reduced density matrices. These degeneracies can be removed by requiring that the natural spin orbitals (NSO, defined as the eigenfunctions of the first-order density matrix), as well as the natural spin geminals (NSG, the eigenfunctions of the second-order density matrix) and their spinless counterparts transform as irreducible representations of the symmetry group and are eigenfunctions of S/sup 2/ and S/sub z/. In many cases this requirement is compatible with the original definition of the NSO, the NSG, etc., e.g., when there is no spatial degeneracy of the total wave function and when the Z- component of the total spin vanishes. When these conditions are not fulfilled an alternative definition of the NSO and the NSG is proposed. (auth)

•Oct 1, 1963•0 citations
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.
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.
Dual Magnetic and Electric Dipole Symmetry: Pseudo Angular Momentum in Parity Space and the Electric Landé ggg-Factor
Electric dipole moments (EDMs) are sensitive probes of fundamental symmetries and central to searches for physics beyond the Standard Model. We present a symmetry-based, Zeeman-analogue operator framework that places magnetic and electric dipole physics on parallel footing under electromagnetic duality, and introduce a polar-sector pseudo-angular-momentum degree of freedom in parity space together with an associated electric Landé factor that organizes induced orbital dipoles. Following Ohanian's effective-current formulation of the Zeeman effect, we construct its electric dual: the wavefunction's microscopic polarization admits an equivalent effective magnetic probability-current representation, providing a field-equivalence description of parity-mixed charge displacement. In this notation the total EDM expectation takes the unified form ⟨d⃗^tot⟩=dB(gE ⟨J⃗^p⟩ℏ+gEe ⟨S⃗^⟩ℏ)\langle \hat{\vec d}_{\rm tot}\rangle= d_B(g_E\,\frac{\langle \hat{\vec J}_p\rangle}{\hbar}+ g_E^{e}\,\frac{\langle \hat{\vec S}\rangle}{\hbar})⟨d^tot​⟩=dB​(gE​ℏ⟨J^p​⟩​+gEe​ℏ⟨S^⟩​), with gEe=2dintdBg_E^{e}=\frac{2d_{\rm int}}{d_B}gEe​=dB​2dint​​, where J⃗^p\hat{\vec J}_pJ^p​ captures Stark-induced pseudo-angular momentum and S⃗^\hat{\vec S}S^ encodes any intrinsic (spin-aligned) EDM dintd_{\rm int}dint​ from symmetry-violating interactions. We define a natural electric dipole unit (the ``Bohr EDM'') as dB≡ea0=2μBcαd_B \equiv e a_0=\frac{2μ_B}{cα}dB​≡ea0​=cα2μB​​ (a0a_0a0​ the Bohr radius and μBμ_BμB​ the Bohr magneton). As a canonical analytic benchmark, we show in the hydrogenic problem that a static electric field couples within a fixed nnn manifold through the scaled Runge--Lenz structure, yielding a compact Landé-like description and reproducing the Stark doublet (e.g.\ ∣⟨dorb⟩∣=3dB|\langle d_{\rm orb}\rangle|=3d_B∣⟨dorb​⟩∣=3dB​ for the 2s2s2s--2pm=02p_{m=0}2pm=0​ mixing).
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.
Symmetry-resolved monopoles and stability criteria for an Nf=12N_f=12Nf​=12 Dirac spin liquid on the maple-leaf lattice
The U(1)\mathrm{U}(1)U(1) Dirac spin liquid provides a useful organizing framework for frustrated magnets: it offers an algebraic parent state from which competing orders, confinement patterns, and low-energy spectral features can be understood. Whether such a state can occur as a stable ground state of a two-dimensional spin Hamiltonian remains an open question, because monopole operators of the compact gauge field can proliferate and confine the spinons. Here we determine the symmetry quantum numbers of the fundamental monopoles of a maple-leaf-lattice Dirac spin liquid whose generic six-node regime realizes QED3_33​ with Nf=12N_f=12Nf​=12 two-component Dirac fermions. The charge-one monopoles span a 924924924-dimensional complex representation. In the spin-singlet, zero-momentum, zero-C6C_6C6​-angular-momentum sector we find five complex charge-one directions, of which three are mapped to their Hermitian conjugates under time reversal. Their real and imaginary Hermitian combinations therefore supply six independent monopole perturbations that are trivial under translations, C6C_6C6​, time reversal, and spin rotation. The Dirac spin liquid phase is thus not protected by symmetry exclusion of charge-one monopoles: its stability depends on whether these allowed perturbations are relevant in the renormalization group sense. Existing large-NfN_fNf​ and numerical estimates of the charge-one monopole scaling dimension lie close to the relevance threshold, making the maple-leaf lattice a concrete platform for testing the stability of compact QED3_33​ in a quantum magnet. Our complete monopole classification also identifies the spin, momentum, and rotation sectors in which monopole excitations should appear, providing symmetry-resolved benchmarks for exact-diagonalization and variational tests of the Nf=12N_f=12Nf​=12 Dirac spin liquid.
Abstract We examine and compare ways of dividing into subspaces the space whose elements are density matrices or other operators for the class of model problems defined by a finite one‐particle basis set. One method of decomposition makes the significance of the subspaces apparent. We show that this decomposition is also complete, in the group‐theoretic sense, for the group of unitary transformations of the set of one‐electron basis functions. The irreducible subspaces are labeled by particle number and by an additional integer we call the reduction index. For spaces of particle‐number‐conserving operators, all subspaces with the same reduction index are isomorphic, and an analogous isomorphism exists for non‐particle‐number‐conserving cases. The general linear group also plays a key role, and we introduce the term “canonical superoperators” to characterize those superoperators which commute with this group. When an appropriate basis set is chosen for the matrix spaces, the supermatrices corresponding to these superoperators have a particularly simple form: a block structure with the only nonzero blocks being multiples of unit matrices. The superoperators of interest can be constructed in terms of two operators, , and these two have been expressed simply in terms of creation and annihilation operators. When only real orthogonal transformations of the basis are considered, a further decomposition is possible. We have introduced superoperators associated with this decomposition.
The density operator (density matrix) of a quantum mechenical system can be decomposed into operators which transform as irreducible representations of the symmetry group in coordinate and spin space. Each of these components has a physical meaning connected with the expectation values of certain operators. The reduced density matrices can be decomposed in a completely analogous way. The symmetry properties of the total wave function give rise to degeneracies of the eigenvalues of the reduced density matrices. These degeneracies can be removed by requiring that the natural spin orbitals (NSO, defined as the eigenfunctions of the first-order density matrix), as well as the natural spin geminals (NSG, the eigenfunctions of the second-order density matrix) and their spinless counterparts transform as irreducible representations of the symmetry group and are eigenfunctions of S/sup 2/ and S/sub z/. In many cases this requirement is compatible with the original definition of the NSO, the NSG, etc., e.g., when there is no spatial degeneracy of the total wave function and when the Z- component of the total spin vanishes. When these conditions are not fulfilled an alternative definition of the NSO and the NSG is proposed. (auth)