Theoretical Proof of and Proposed Experimental Search for the Ground Triplet State of a Wigner-Regime Two-Electron ‘Artificial Atom’ in a Magnetic Field
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.
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.
We prove the Wave Function Identity (WFI), a new symmetry constraint on the solutions of the Schrödinger-Pauli equation for the Coulombically interacting N = 3 electron 2-dimensional harmonically bound ‘artificial atom’ or quantum dot in a uniform magnetic field. The symmetry is comprised of an interchange of the spatial coordinates of two electrons whilst keeping their spin moments unchanged, followed by an inversion of all three electron coordinates. The proof is achieved by first deriving the general form of the exact correlated wave function in the high-electron-correlation Wigner regime and then proving the WFI employing this form for a quartet state. The satisfaction of the Pauli principle (PP) and the odd parity of this state are also proved. An exact closed-form analytical expression for a quartet state in the Wigner regime is derived, and the properties of the PP, WFI, parity and other properties exhibited diagrammatically. Together with our prior proof of the WFI for the N = 2 electron case, we propose ad hoc the validity of the WFI for all bound-state N ≥ 2 electron systems for arbitrary symmetric binding potential, arbitrary electron-interaction of the form ω(|r – r’|) and arbitrary dimensionality.
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.
Moiré superlattices can bind several electrons to each potential minimum, forming few-electron "moiré atoms." Using neural quantum states that obey magnetic periodic boundary conditions, include full Landau-level mixing, and treat both spin sectors, we map the ground state of two-electron moiré atoms in a superlattice as a function of moiré depth and magnetic field strength. At strong field each atom is a spin-triplet maximum-density droplet with canonical angular momentum ⟨Lz⟩=−1 and a persistent circulating current: the rotating counterpart of the "Wigner molecular crystal" recently imaged in moiré superlattices. Deepening the wells drives a singlet-triplet transition to non-rotating spin-singlet atoms whose only current is a weak diamagnetic counter-circulation. The transition moves to deeper wells as the field grows, so depth and field together open a window for the formation of rotors. This is the collective analog of the singlet-triplet transition of two-electron quantum dots. At the transition, the orbital magnetic moments of the atoms reverse sign as their spin polarization switches, a joint signature accessible to magnetometry. If the atoms are instead polarized by other means, such as magnetic proximity coupling, the rotors lock at weak field into a spontaneously ordered crystal of dimers in a pinwheel pattern.
The nature of the ground state of a few-electron system in a parabolic twodimensional (2D) quantum dot (QD) is theoretically investigated on the basis of a variational many-body approach within a second quantized formulation. The resulting analytical expressions lead to various phase transitions: maximum density droplet, Wigner phase, etc., in a broad range of magnetic fields. It is observed that in the case of relatively strong QD confinement and weak magnetic fields, a rotationally symmetric spin-singlet state is the ground state of the system. However, in a strong magnetic field and for the same QD confinement, a broken-symmetry spinsinglet state appears to be energetically favoured over the symmetric spin-singlet state. The method developed is generalized for the investigation of possible novel phases for electron-hole interacting systems.
A hydrogen atom in a constant magnetic field is a problem whose Schrödinger equation has been considered to be analytically unsolvable. However, it has been found for the case of two-dimensional space some specific values of the magnetic field for which the problem has analytical solutions. In this paper, we give a full theory for these analytical solutions. First, we construct an equation to calculate of exact-solution-points, the values of magnetic field which allow exact solutions for the Schrödinger equation. The explicit forms of the wave functions corresponding to these analytical solutions are also given. Second, we analyse the exact-solution-points and discover some important rules. (i) There exist (n − 1)2 different values of the magnetic field so that the Schrödinger equation of a two-dimensional hydrogen atom in such field has exact analytical solutions with the principal quantum number n. (ii) For more details, among these (n − 1)2 exact analytical solutions, there are nr=n−|m|−1 solutions with the same magnetic quantum number m. Both properties are also proved mathematically in this paper. For illustration, some concrete values are shown for the cases of quantum number n up to 8.
We propose a ground-state trial wave function for a two-dimensional Wigner crystal in a strong perpendicular magnetic field. The wave function includes Laughlin-Jastrow correlations between electron pairs, and may be interpreted as a crystal state of composite fermions or composite bosons. Treating the power m of the Laughlin-Jastrow factor as a variational parameter, we use quantum Monte Carlo simulations to compute the energy of these new states. We find that our wave functions have lower energy than existing crystalline wave functions in the lowest Landau level. Our results are consistent with experimental observations of the filling factor at which the transition between the fractional quantum Hall liquid and the Wigner crystal occurs for electron systems. Exchange contributions to the wave functions are estimated quantitatively and shown to be negligible for sufficiently small filling factors.