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arXiv:2609.36356 · Journal of Physics Condensed Matter

Artificial versus natural atoms: the uncanny capability of the many-body Schrödinger equation to produce emergent behavior

Constantine Yannouleas

Published Aug 28, 2026• 0 citations
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Abstract

The paper reviews the theoretical and experimental progress achieved in the last 25 years in understanding the novel physics of artificial atoms and molecules as arising from the formation of Wigner molecules (WMs) of localized (to a stronger or lesser extent) fermionic or bosonic particles, which are finite quantum analogs of the more familiar bulk Wigner crystal (WC). The term artificial atoms, as used here, encompasses a broad range of recently fabricated quantum nanodevices and experimental apparatuses consisting of a finite number of mutually repelling confined particles, including two-dimensional semiconductor and moiré transition metal dichalcogenide quantum dots, as well as trapped ultracold neutral atoms or ions. These nano-sized or micro-sized artificial devices and apparatuses (in single well or multi-well of variable-shape arrangements) hold a great promise for technological applications in the field of quantum information and quantum computers, as well as for advances in fundamental many-body physics. Prominent quantum effects of Wigner molecularization (WM) are the strong quenching of the spectral energy gaps, the appearance of rovibational spectra (in analogy with natural molecules), entanglement, and pinning due to an external perturbation. In high magnetic fields or at rapid rotation, Wigner molecules (WMs) provide an alternative theory to the fractional quantum Hall effect. The physics of WMs is shown to derive from the solutions of the many-body Schrödinger equation (MBSE) in the regime of strong interparticle correlations arising from the dominance of the potential over the kinetic energy, or from a high magnetic field, as well as from a rapid rotation. With the help of a hierarchical scheme of computational approaches involving group-theoretical projection techniques beyond the mean field and exact configuration interaction, in parallel to experimental investigations, WMs are shown to provide an ideal platform for investigating the interplay between symmetry-preserving (stationary, referred to as rotating or sliding WMs) and broken-symmetry (superposition-necessitating, referred to as pinned or static WMs) solutions of the MBSE. In the process, a germane view of the phenomenon of symmetry breaking, based exclusively on finite systems and referred to as emergent symmetry breaking, is developed as a replacement to the formalistic spontaneous symmetry breaking that requires invocation of a singular thermodynamic limit. Attention is drawn to the counterintuitive fact that the very MBSE, which was inspired by de Broglie's undulatory matter waves and successfully explained the shell structure of delocalized electrons in atomic physics, is nevertheless capable of yielding contrasting solutions, which relate to corpuscular geometries of localized particles. This unforeseen behavior classifies the recent developments concerning WM and WC as an example of weak emergence, and thus of an arrow of reductionist explanation according to Weinberg's reasoning. Finally, this review, demonstrating the unexpected mathematical effectiveness beyond original expectations of the Schrödinger equation, is dedicated as a tribute to its 100 year anniversary.

DOI
10.1088/1361-648x/ae8c9e

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The paper reviews the theoretical and experimental progress achieved in the last 25 years in understanding the novel physics of artificial atoms and molecules as arising from the formation of Wigner molecules (WMs) of localized (to a stronger or lesser extent) fermionic or bosonic particles, which are finite quantum analogs of the more familiar bulk Wigner crystal (WC). The term artificial atoms, as used here, encompasses a broad range of recently fabricated quantum nanodevices and experimental apparatuses consisting of a finite number of mutually repelling confined particles, including two-dimensional semiconductor and moiré transition metal dichalcogenide quantum dots, as well as trapped ultracold neutral atoms or ions. These nano-sized or micro-sized artificial devices and apparatuses (in single well or multi-well of variable-shape arrangements) hold a great promise for technological applications in the field of quantum information and quantum computers, as well as for advances in fundamental many-body physics. Prominent quantum effects of Wigner molecularization (WM) are the strong quenching of the spectral energy gaps, the appearance of rovibational spectra (in analogy with natural molecules), entanglement, and pinning due to an external perturbation. In high magnetic fields or at rapid rotation, Wigner molecules (WMs) provide an alternative theory to the fractional quantum Hall effect. The physics of WMs is shown to derive from the solutions of the many-body Schrödinger equation (MBSE) in the regime of strong interparticle correlations arising from the dominance of the potential over the kinetic energy, or from a high magnetic field, as well as from a rapid rotation. With the help of a hierarchical scheme of computational approaches involving group-theoretical projection techniques beyond the mean field and exact configuration interaction, in parallel to experimental investigations, WMs are shown to provide an ideal platform for investigating the interplay between symmetry-preserving (stationary, referred to as rotating or sliding WMs) and broken-symmetry (superposition-necessitating, referred to as pinned or static WMs) solutions of the MBSE. In the process, a germane view of the phenomenon of symmetry breaking, based exclusively on finite systems and referred to as emergent symmetry breaking, is developed as a replacement to the formalistic spontaneous symmetry breaking that requires invocation of a singular thermodynamic limit. Attention is drawn to the counterintuitive fact that the very MBSE, which was inspired by de Broglie's undulatory matter waves and successfully explained the shell structure of delocalized electrons in atomic physics, is nevertheless capable of yielding contrasting solutions, which relate to corpuscular geometries of localized particles. This unforeseen behavior classifies the recent developments concerning WM and WC as an example of weak emergence, and thus of an arrow of reductionist explanation according to Weinberg's reasoning. Finally, this review, demonstrating the unexpected mathematical effectiveness beyond original expectations of the Schrödinger equation, is dedicated as a tribute to its 100 year anniversary.
Investigations of emergent symmetry breaking phenomena occurring in small finite-size systems are reviewed, with a focus on the strongly correlated regime of electrons in two-dimensional semiconductor quantum dots and trapped ultracold bosonic atoms in harmonic traps. Throughout the review we emphasize universal aspects and similarities of symmetry breaking found in these systems, as well as in more traditional fields like nuclear physics and quantum chemistry, which are characterized by very different interparticle forces. A unified description of strongly correlated phenomena in finite systems of repelling particles (whether fermions or bosons) is presented through the development of a two-step method of symmetry breaking at the unrestricted Hartree-Fock level and of subsequent symmetry restoration via post Hartree-Fock projection techniques. Quantitative and qualitative aspects of the two-step method are treated and validated by exact diagonalization calculations.Strongly-correlated phenomena emerging from symmetry breaking include the following.Chemical bonding, dissociation and entanglement (at zero and finite magnetic fields) in quantum dot molecules and in pinned electron molecular dimers formed within a single anisotropic quantum dot, with potential technological applications to solid-state quantum-computing devices.Electron crystallization, with particle localization on the vertices of concentric polygonal rings, and formation of rotating electron molecules (REMs) in circular quantum dots. Such electron molecules exhibit ro-vibrational excitation spectra, in analogy with natural molecules.At high magnetic fields, the REMs are described by parameter-free analytic wave functions, which are an alternative to the Laughlin and composite-fermion approaches, offering a new point of view of the fractional quantum Hall regime in quantum dots (with possible implications for the thermodynamic limit).Crystalline phases of strongly repelling bosons. In rotating traps and in analogy with the REMs, such repelling bosons form rotating boson molecules (RBMs). For a small number of bosons, the RBMs are energetically favored compared with the Gross-Pitaevskii solutions describing vortex formation.We discuss the present status concerning experimental signatures of such strongly correlated states, in view of the promising outlook created by the latest experimental improvements that are achieving unprecedented control over the range and strength of interparticle interactions.
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.
Quantum Wigner molecules (WMs) in strongly interacting few-body fermionic moir\'e quantum dots in twisted bilayers of transition metal dichalcogenide (TMD) materials are uncovered via full configuration interaction calculations, going beyond the Aufbau principle of natural atoms and Hubbard modeling. Nested polygonal sliding rings of localized fermions, hidden in the exact particle densities, are revealed through wavefunction correlation analysis [see (3,9) and (1,6,10) electron WMs in 2D semiconductors], broadening the WM portfolio to include the TMD trilobal symmetry as an added resource in twistronics, and providing benchmarks for AI-based many-body computations.
It is experimentally established that there is no ground triplet state of the natural He atom. There is also no exact analytical solution to the Schrödinger equation corresponding to this state. For a two-dimensional two-electron ‘artificial atom’ or a semiconductor quantum dot in a magnetic field, as described by the Schrödinger–Pauli equation, we provide theoretical proof of the existence of a ground triplet state by deriving an exact analytical correlated wave function solution to the equation. The state exists in the Wigner high-electron-correlation regime. We further explain that the solution satisfies all requisite symmetry and electron coalescence constraints of a triplet state. Since, due to technological advances, such a Wigner crystal quantum dot can be created, we propose an experimental search for the theoretically predicted ground triplet-state spectral line. We note that there exists an analytical solution to the Schrödinger–Pauli equation for a ground singlet state in the Wigner regime for the same value of the magnetic field. The significance to quantum mechanics of the probable experimental observation of the ground triplet state for an ‘artificial atom’ is discussed.
A constructive theoretical platform for the description of quantum space-time crystals uncovers for NNN interacting and ring-confined rotating particles the existence of low-lying states with proper space-time crystal behavior. The construction of the corresponding many-body trial wave functions proceeds first via symmetry breaking at the mean-field level followed by symmetry restoration using projection techniques. The ensuing correlated many-body wave functions are stationary states and preserve the rotational symmetries, and at the same time they reflect the point-group symmetries of the mean-field crystals. This behavior results in the emergence of sequences of select magic angular momenta Lm{L}_{m}Lm​. For angular-momenta away from the magic values, the trial functions vanish. Symmetry breaking beyond the mean-field level can be induced by superpositions of such good-Lm{L}_{m}Lm​ many-body stationary states. We show that superposing a pair of adjacent magic angular momenta states leads to formation of special broken-symmetry states exhibiting quantum space-time-crystal behavior. In particular, the corresponding particle densities rotate around the ring, showing undamped and nondispersed periodic crystalline evolution in both space and time. The experimental synthesis of such quantum space-time-crystal wave packets is predicted to be favored in the vicinity of ground-state energy crossings of the Aharonov-Bohm-type spectra accessed via an externally applied, natural or synthetic, magnetic field. These results are illustrated here for Coulomb-repelling fermionic ions and for a lump of contact-interaction attracting bosons.
It is demonstrated that exact diagonalization of the microscopic many-body Hamiltonian via systematic full configuration-interaction (FCI) calculations is able to predict the spectra as a function of detuning of three-electron hybrid qubits based on GaAs asymmetric double quantum dots (QDs). It is further shown that, as a result of strong inter-electron correlations, these spectroscopic patterns, including avoided crossings between states associated with different electron occupancies of the left and right wells, are inextricably related to the formation of Wigner molecules (WMs). These physical entities cannot be captured by the previously employed independent-particle or Hubbard-type theoretical modeling of the hybrid qubit. We report remarkable agreement with recent experimental results. Moreover, the present FCI methodology for multi-well QDs can be straightforwardly extended to treat Si/SiGe hybrid qubits, where the central role of WMs was recently experimentally confirmed as well.