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.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.

arXiv:2603.10620Journal of the Physical Society of JapanJun 30, 20261 citation

Independent Operators for Interacting Electrons

Roger Haydock

Bulletin of the American Physical SocietyMar 7, 20180 citations

Symmetry-resolved fermionic correlations in iron-oxide ground spaces

Federico Petrovich, Daniel Staros, Daniel N. Blaschke +2 more

Symmetries of quantum systems provide a natural way to organize the eigenspaces of many-body Hamiltonians. When degeneracies exist, however, they play a more subtle role as different sets of symmetry operators can resolve the same energy subspace into states with very different correlation properties. We study this interplay using a sequence of FeO and Fe $_2$ O $_2$ models. For a Hubbard model of FeO, the ground space is 16-fold degenerate. Resolving it by local magnetizations gives a basis of Slater determinants, while total-spin quantum numbers select entangled and non-Gaussian states from the same ground manifold. Adding density-density Hund interactions breaks the independent spin rotations and reduces the ground space to a spin-reversal pair with maximal magnetization. These states are extremely close to Slater determinants in their natural occupations, yet have finite two-body cumulants and conserved-operator algebras whose orbit dimensions differ substantially from the Slater values. Fe $_2$ O $_2$ provides a larger example in which spatial mode entanglement can be substantial while fermionic non-Gaussianity remains weak, with the conserved-operator structure again distinguishing the state from the corresponding Slater transformation class. Together, these examples show that symmetry resolution, mode entanglement, reduced-density-matrix correlations, and conserved-operator structure provide complementary information about interacting and degenerate electronic ground states.Symmetries of quantum systems provide a natural way to organize the eigenspaces of many-body Hamiltonians. When degeneracies exist, however, they play a more subtle role as different sets of symmetry operators can resolve the same energy subspace into states with very different correlation properties. We study this interplay using a sequence of FeO and Fe$_2$O$_2$ models. For a Hubbard model of FeO, the ground space is 16-fold degenerate. Resolving it by local magnetizations gives a basis of Slater determinants, while total-spin quantum numbers select entangled and non-Gaussian states from the same ground manifold. Adding density-density Hund interactions breaks the independent spin rotations and reduces the ground space to a spin-reversal pair with maximal magnetization. These states are extremely close to Slater determinants in their natural occupations, yet have finite two-body cumulants and conserved-operator algebras whose orbit dimensions differ substantially from the Slater values. Fe$_2$O$_2$ provides a larger example in which spatial mode entanglement can be substantial while fermionic non-Gaussianity remains weak, with the conserved-operator structure again distinguishing the state from the corresponding Slater transformation class. Together, these examples show that symmetry resolution, mode entanglement, reduced-density-matrix correlations, and conserved-operator structure provide complementary information about interacting and degenerate electronic ground states.

quant-pharXiv:2610.36047Oct 28, 20260 citations

Symmetry of Many-Electron Systems

Elsevier eBooksJan 1, 197532 citations

Topological ground-state excitations and symmetry in the many-electron one-dimensional problem

J. M. P. Carmelo, Nuno M. R. Peres

We consider the Hubbard chain in a magnetic field and chemical potential. We introduce a pseudohole basis where all states are generated from a single reference vacuum. This allows the evaluation for all sectors of Hamiltonian symmetry of the model of the expression of the σ electron and hole operators at Fermi momentum ±kFσ and vanishing excitation energy in terms of pseudohole operators. In all sectors and to leading order in the excitation energy the electron and hole are constituted by one c pseudohole, one s pseudohole, and one topological momenton. These three quantum objects are confined in the electron or hole and cannot be separated. We find that the set of different pseudohole types which in pairs constitute the two electrons and two holes associated with the transitions from the (N↑,N↓) ground state to the (N↑+1,N↓), (N↑,N ↓ +1) and (N ↑ −1,N↓), (N↑,N ↓ −1) ground states, respectively, transform in the representation of the symmetry group of the Hamiltonian in the initial-ground-state sector of parameter space. We also find the pseudohole generators for the half-filling holon and zero-magnetic-field spinon. The pseudohole basis introduced in this paper is the only suitable for the extension of the present type of operator description to the whole Hilbert space. 1 I.We consider the Hubbard chain in a magnetic field and chemical potential. We introduce a pseudohole basis where all states are generated from a single reference vacuum. This allows the evaluation for all sectors of Hamiltonian symmetry of the model of the expression of the σ electron and hole operators at Fermi momentum ±kFσ and vanishing excitation energy in terms of pseudohole operators. In all sectors and to leading order in the excitation energy the electron and hole are constituted by one c pseudohole, one s pseudohole, and one topological momenton. These three quantum objects are confined in the electron or hole and cannot be separated. We find that the set of different pseudohole types which in pairs constitute the two electrons and two holes associated with the transitions from the (N↑,N↓) ground state to the (N↑+1,N↓), (N↑,N ↓ +1) and (N ↑ −1,N↓), (N↑,N ↓ −1) ground states, respectively, transform in the representation of the symmetry group of the Hamiltonian in the initial-ground-state sector of parameter space. We also find the pseudohole generators for the half-filling holon and zero-magnetic-field spinon. The pseudohole basis introduced in this paper is the only suitable for the extension of the present type of operator description to the whole Hilbert space. 1 I.

Jan 1, 19951 citation

Hidden symmetry of interacting electron systems

V. M. Zharkov

Theoretical and Mathematical PhysicsOct 1, 19881 citation

_2

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models. For a Hubbard model of FeO, the ground space is 16-fold degenerate. Resolving it by local magnetizations gives a basis of Slater determinants, while total-spin quantum numbers select entangled and non-Gaussian states from the same ground manifold. Adding density-density Hund interactions breaks the independent spin rotations and reduces the ground space to a spin-reversal pair with maximal magnetization. These states are extremely close to Slater determinants in their natural occupations, yet have finite two-body cumulants and conserved-operator algebras whose orbit dimensions differ substantially from the Slater values. Fe

_2

_2

provides a larger example in which spatial mode entanglement can be substantial while fermionic non-Gaussianity remains weak, with the conserved-operator structure again distinguishing the state from the corresponding Slater transformation class. Together, these examples show that symmetry resolution, mode entanglement, reduced-density-matrix correlations, and conserved-operator structure provide complementary information about interacting and degenerate electronic ground states.