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arXiv:2604.27144

Unmasking Hidden Wigner's Symmetry from First Principles

Phong Dang, Daniel Langr, T. Dytrych, Jerry Paul Draayer, David Kekejian

Published Apr 29, 2026• 0 citations
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Abstract

We present quantitative evidence that high-quality internucleon forces derived from $χ$EFT exhibit a striking dominance of Wigner's supermultiplet symmetry, without invoking the large-$N_c$ limit of QCD or assumptions about specific nuclei. We trace the manifestation of this symmetry in nuclear structure using the ab initio Symmetry Adapted Model (SAM) and identify suppressed spin-isospin polarizability. Our calculations show that a majority of $\rm ^4He$, $\rm ^6Li$, and $\rm ^6He$ wave functions is concentrated in a few $\rm U(4)$ irreducible representations, without imposing any a priori constraints on the model space. This emergent feature points to a strategy for reducing explosive many-body bases of the NCSM while retaining physically important configurations needed to compute observables.

DOI
10.48550/arxiv.2604.27144

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We present quantitative evidence that high-quality internucleon forces derived from χχχEFT exhibit a striking dominance of Wigner's supermultiplet symmetry, without invoking the large-NcN_cNc​ limit of QCD or assumptions about specific nuclei. We trace the manifestation of this symmetry in nuclear structure using the ab initio Symmetry Adapted Model (SAM) and identify suppressed spin-isospin polarizability. Our calculations show that a majority of 4He\rm ^4He4He, 6Li\rm ^6Li6Li, and 6He\rm ^6He6He wave functions is concentrated in a few U(4)\rm U(4)U(4) irreducible representations, without imposing any a priori constraints on the model space. This emergent feature points to a strategy for reducing explosive many-body bases of the NCSM while retaining physically important configurations needed to compute observables.
Strong repelling interactions between a few fermions or bosons confined in two-dimensional circular traps lead to particle localization and formation of quantum Wigner molecules (QWMs) possessing definite point-group space symmetries. These point-group symmetries are "hidden" (or emergent), namely they cannot be traced in the circular single-particle densities (SPDs) associated with the exact many-body wave functions, but they are manifested as characteristic signatures in the ro-vibrational spectra. An example, among many, are the few-body QWM states under a high magnetic field or at fast rotation, which are precursor states for the fractional quantum Hall effect. The hidden geometric symmetries can be directly revealed by using spin-resolved conditional probability distributions, which are extracted from configuration-interaction (CI), exact-diagonalization wave functions. The hidden symmetries can also be revealed in the CI SPDs by reducing the symmetry of the trap (from circular to elliptic to quasi-linear). In addition the hidden symmetries are directly connected to the explicitly broken-symmetry (BS) solutions of mean-field approaches, such as unrestricted Hartree-Fock (UHF). A companion step of restoration of the broken symmetries via projection operators applied on the BS-UHF solutions produces wave functions directly comparable to the CI ones, and sheds further light into the role played by the emergence of hidden symmetries in the exact many-body wave functions. Illustrative examples of the importance of hidden symmetries in the many-body problem of few electrons in semiconductor quantum dots and of few ultracold atoms in optical traps (where unprecedented control of the interparticle interaction has been experimentally achieved recently) will be presented.
Microscopic models, which embody the simplicity and significance of a dynamical symmetry approach to nuclear structure, are reviewed. They can reveal striking features of atomic nuclei when a symmetry dominates and solutions in domains that may otherwise be unreachable. 1. Overview of algebraic fermion models. A theory that invokes group symmetries is driven by an expectation that the wave functions of the quantum mechanical system under consideration can be characterized by their invariance properties under the corresponding symmetry transformations. But even when the symmetries are not exact, if one can find near invariant operators, the associated symmetries can be used to help reduce the dimensionality of a model space to a tractable size. Throughout the years, group-theoretical approaches have identified fundamental symmetries in light to heavy nuclei and achieved a reasonable reproduction of experimental data (for a review of fermion models, see 1). In addition, they provide theoretical predictions for nuclear systems including heavy unstable
Background: Ab initio many-body methods have been developed over the past ten years to address mid-mass nuclei. In their best current level of implementation, their accuracy is of the order of a few percent error on the ground-state correlation energy. Recently implemented variants of these methods are operating a breakthrough in the description of medium-mass open-shell nuclei at a polynomial computational cost while putting state-of-the-art models of internucleon interactions to the test.Purpose: As progress in the design of internucleon interactions is made, and as questions one wishes to answer are refined in connection with increasingly available experimental data, further efforts must be made to tailor many-body methods that can reach an even higher precision for an even larger number of observable quantum states or nuclei. The objective of the present work is to contribute to such a quest by designing and testing a new many-body scheme.Methods: We formulate a truncated configuration-interaction method that consists of diagonalizing the Hamiltonian in a highly truncated subspace of the total NNN-body Hilbert space. The reduced Hilbert space is generated via the particle-number projected BCS state along with projected seniority-zero two- and four-quasiparticle excitations. Furthermore, the extent by which the underlying BCS state breaks U(1)\text{U}(1)U(1) symmetry is optimized in the presence of the projected two- and four-quasiparticle excitations. This constitutes an extension of the so-called restricted variation after projection method in use within the frame of multireference energy density functional calculations. The quality of the newly designed method is tested against exact solutions of the so-called attractive pairing Hamiltonian problem.Results: By construction, the method reproduces exact results for N=2N=2N=2 and N=4N=4N=4. For N=(8,16,20)N=(8,\phantom{\rule{0.16em}{0ex}}16,\phantom{\rule{0.16em}{0ex}}20)N=(8,16,20), the error in the ground-state correlation energy is less than (0.006%, 0.1%, 0.15%) across the entire range of internucleon coupling defining the pairing Hamiltonian and driving the normal-to-superfluid quantum phase transition. The presently proposed method offers the advantage of automatic access to the low-lying spectroscopy, which it does with high accuracy.Conclusions: The numerical cost of the newly designed variational method is polynomial (N6{N}^{6}N6) in system size. This method achieves unprecedented accuracy for the ground-state correlation energy, effective pairing gap, and one-body entropy as well as for the excitation energy of low-lying states of the attractive pairing Hamiltonian. This constitutes a sufficiently strong motivation to envision its application to realistic nuclear Hamiltonians in view of providing a complementary, accurate, and versatile ab initio description of mid-mass open-shell nuclei in the future.
The extension of ab initio quantum many-body theory to higher accuracy and larger systems is intrinsically limited by the handling of large data objects in form of wave-function expansions and/or many-body operators. In this work we present matrix factorization techniques as a systematically improvable and robust tool to significantly reduce the computational cost in many-body applications at the price of introducing a moderate decomposition error. We demonstrate the power of this approach for the nuclear two-body systems, for many-body perturbation theory calculations of symmetric nuclear matter, and for non-perturbative in-medium similarity renormalization group simulations of finite nuclei. Establishing low-rank expansions of chiral nuclear interactions offers possibilities to reformulate many-body methods in ways that take advantage of tensor factorization strategies.
Background: Entanglement plays a central role in a diverse array of increasingly important research areas, including quantum computation, simulation, measurement, sensing, and communication. Extensive suites of investigations have been performed to better understand entanglement in atomic and molecular quantum many-body systems, while the exploration of entanglement in the structure of nuclei and their reactions is presently in its infancy.Purpose: The goal of this work is to begin investigating the entanglement properties of nuclei from first-principles nuclear many-body calculations. We attempt to identify common features and emergent structures of entanglement that could ultimately lead to new and natural many-body schemes. With an eye toward quantum accelerators in future hybrid-supercomputers, criteria for partitioning nuclear many-body calculations into quantum and classical components may provide advantages in future large-scale computations. Along the way we look for explanations of the relative success of phenomenological models such as the nuclear shell model, and for better ways to match to low-energy nuclear effective field theories and lattice QCD calculations to nuclear many-body techniques that are based upon entanglement.Method: We explore the entanglement between single-particle states in 4He^{4}\mathrm{He}4He and 6He^{6}\mathrm{He}6He. The patterns of entanglement emerging from different single-particle bases are compared, and possible links with the convergence of observables are explored, in particular, ground-state energies. The nuclear wave functions are obtained by performing active-space no-core configuration-interaction calculations using a two-body nucleon-nucleon interaction derived from chiral effective field theory. Entanglement measures within single-particle bases exhibiting different degrees of complexity are determined, in particular, harmonic oscillator (HO), Hartree-Fock (HF), natural (NAT) and variational natural (VNAT) bases. Specifically, single-orbital entanglement entropy, two-orbital mutual information, and negativity are studied.Results: The entanglement structures in 4He^{4}\mathrm{He}4He and 6He^{6}\mathrm{He}6He are found to be more localized within NAT and VNAT bases than within a HO basis for the optimal HO parameters we have worked with. In particular a core-valence structure clearly emerges from the full no-core calculation of 6He^{6}\mathrm{He}6He. The two-nucleon mutual information shows that the VNAT basis, which typically exhibits good convergence properties, effectively decouples the active and inactive spaces.Conclusions: Measures of one- and two-nucleon entanglement are found to be useful in analyzing the structure of nuclear wave functions, in particular the efficacy of basis states, and may provide useful metrics toward developing more efficient schemes for ab initio computations of the structure and reactions of nuclei, and quantum many-body systems more generally.
In this work ground and excited electronic states of Heisenberg cluster models, in the form of con- figuration interaction many-body wave functions, are characterized within the spin-adapted Graph- ical Unitary Group Approach framework, and relying on a novel combined unitary and symmetric group approach. Finite-size cluster models of well-defined point-group symmetry and of general local-spin Slocal ≥ 21 are presented, including J1–J2 triangular and tetrahedral clusters, which are often used to describe magnetic interactions in biological and bio-mimetic polynuclear transition metal clusters with unique catalytic activity, such as nitrogen fixation and photosynthesis. We show that a unique block-diagonal structure of the underlying Hamiltonian matrix in the spin-adapted basis emerges when an optimal lattice site ordering is chosen that reflects the internal symmetries of the model investigated. The block-diagonal structure is bound to the commutation relations between cumulative spin operators and the Hamiltonian operator, that in turn depend on the geometry of the cluster investigated. The many-body basis transformation, in the form of the orbital/site reorder- ing, exposes such commutation relations. These commutation relations represent a rigorous and formal demonstration of the block-diagonal structure in Hamiltonian matrices and the compression of the corresponding spin-adapted many-body wave functions. As a direct consequence of the block- diagonal structure of the Hamiltonian matrix it is possible to selectively optimize electronic excited states without the overhead of calculating the lower-energy states by simply relying on the initial ansatz for the targeted wave function. Additionally, more compact many-body wave functions are obtained. In extreme cases, electronic states are precisely described by a single configuration state function, despite the curse of dimensionality of the corresponding Hilbert space. These findings are crucial in the electronic structure theory framework, for they offer a conceptual route towards wave functions of reduced multi-reference character, that can be optimized more easily by approximated eigensolvers and are of more facile physical interpretation. They open the way to study larger ab initio and model Hamiltonians of increasingly larger number of correlated electrons, while keeping the computational costs at their lowest.