This paper investigates interaction-induced symmetry breaking in circular quantum dots. We start by explaining what is known about symmetry breaking in quantum dots, pointing out that the anisotropic "static Wigner molecule" ground states frequently observed in simulations are created by interference effects that occur even in the non-interacting limit. They have nothing in common with the interaction-driven crystallization of the uniform electron gas described by Wigner. This leads us to define the term Wigner molecule more carefully via a finite analog of the spontaneous symmetry breaking that arises in the homogeneous electron gas when the interactions are strong. According to this definition, the charge density patterns characteristic of true interaction-induced Wigner molecules can only be seen if a small symmetry-breaking perturbation is applied to a strongly interacting quantum dot. A simple argument based on separation of variables into center-of-mass and internal coordinates shows that the strength of the perturbation required to produce a finite effect on the density tends to zero in the limit as the strength of the interaction tends to infinity. We confirm computationally that interaction-induced Wigner molecules satisfying this definition exist. The neural-network variational Monte Carlo method used in our simulations proves more accurate than the coupled cluster and diffusion Monte Carlo methods employed in previous benchmark calculations of quantum dots at small to intermediate interaction strengths. For high interaction strengths, our neural-network variational Monte Carlo energies agree very well with existing fixed-node diffusion Monte Carlo benchmarks, proving ∼0.01% better for small values of the total spin projection Sz but ∼0.01% worse for fully spin-polarized systems.
This paper investigates interaction-induced symmetry breaking in circular quantum dots. We start by explaining what is known about symmetry breaking in quantum dots, pointing out that the anisotropic "static Wigner molecule" ground states frequently observed in simulations are created by interference effects that occur even in the non-interacting limit. They have nothing in common with the interaction-driven crystallization of the uniform electron gas described by Wigner. This leads us to define the term Wigner molecule more carefully via a finite analog of the spontaneous symmetry breaking that arises in the homogeneous electron gas when the interactions are strong. According to this definition, the charge density patterns characteristic of true interaction-induced Wigner molecules can only be seen if a small symmetry-breaking perturbation is applied to a strongly interacting quantum dot. A simple argument based on separation of variables into center-of-mass and internal coordinates shows that the strength of the perturbation required to produce a finite effect on the density tends to zero in the limit as the strength of the interaction tends to infinity. We confirm computationally that interaction-induced Wigner molecules satisfying this definition exist. The neural-network variational Monte Carlo method used in our simulations proves more accurate than the coupled cluster and diffusion Monte Carlo methods employed in previous benchmark calculations of quantum dots at small to intermediate interaction strengths. For high interaction strengths, our neural-network variational Monte Carlo energies agree very well with existing fixed-node diffusion Monte Carlo benchmarks, proving ∼0.01% better for small values of the total spin projection Sz but ∼0.01% worse for fully spin-polarized systems.
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.
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.
We present and compare several many-body methods as applied to two-dimensional quantum dots with circular symmetry. We calculate the approximate ground state energy using a harmonic oscillator basis optimized by Hartree-Fock (HF) theory and further improve the ground state energy using two post-HF methods: in-medium similarity renormalization group and coupled cluster with singles and doubles. With the application of quasidegenerate perturbation theory or the equations-of-motion method to the results of the previous two methods, we obtain addition and removal energies as well. Our results are benchmarked against full configuration interaction and diffusion Monte Carlo where available. We examine the rate of convergence and perform extrapolations to the infinite basis limit using a power-law model.
Exact-diagonalization calculations for N=3 electrons in anisotropic quantum dots, covering a broad range of confinement anisotropies and strength of interelectron repulsion, are presented for zero and low magnetic fields. The excitation spectra are analyzed as a function of the strength of the magnetic field and for increasing quantum-dot anisotropy. Analysis of the intrinsic structure of the many-body wave functions through spin-resolved two-point correlations reveals that electrons tend to localize forming Wigner molecules. For certain ranges of dot parameters (mainly at strong anisotropy), the Wigner molecules acquire a linear geometry, and the associated wave functions with a spin projection Sz=1∕2 are similar to the representative class of strongly entangled states referred to as the W states. For other ranges of parameters (mainly at intermediate anisotropy), the Wigner molecules exhibit a more complex structure consisting of two mirror isosceles triangles. This latter structure can be viewed as an embryonic unit of a zigzag Wigner crystal in quantum wires. The degree of entanglement in three-electron quantum dots can be quantified through the use of the von Neumann entropy.
The strongly correlated phases of the homogeneous electron gas constitute the vocabulary of many-body condensed matter physics and find a natural realization in semiconductors. In this setting, recent neural-network variational Monte Carlo calculations discovered an unexpected quantum phase of matter in wide quantum wells: an electronic quasicrystal formed by a bilayer Wigner crystals with a 30-degrees twist. This state defies classical expectations and emerges in a regime dominated by quantum fluctuations. Here, we develop an analytical framework to reveal its origin. By computing zero-point energy corrections to bilayer Wigner crystal configurations, we show that quantum fluctuations qualitatively reshape the energetic landscape, destabilizing the classical honeycomb state and selecting the 30-degrees quasicrystalline ground state over a broad parameter range. Our results identify zero-point motion as the mechanism stabilizing the electronic quasicrystal and establish a route to spontaneous moiré physics driven by many-body quantum effects.
This thesis investigates the use of wave-function methods for the study of quantum-dot systems. It investigates single dots, using quantum Monte Carlo techniques in a wide range of magnetic field values, and a double-dot system, using the exact-diagonalization method.\n\nThe thesis proposes simple yet accurate many-particle wave functions for various angular-momentum and spin states, for both weak and strong magnetic fields. Using these trial wave functions, it evaluates various properties of dots and studies Wigner crystallization and spin polarization for the weak-field limit. For strong magnetic fields, the thesis investigates ground states of different spin polarizations as a function of the magnetic field and tests the commonly used lowest-Landau-level approximation. The results are compared to calculations from the density-functional theory.\n\nFinally, the thesis presents a method that combines the accuracy of the exact-diagonalization method and the scalability of Monte Carlo methods. The Monte Carlo-based diagonalization is a promising tool for use in situations that cannot be handled with a simple trial wave function, and have too many particles for an exact-diagonalization treatment. For quantum dots, methods for efficiently evaluating the high-magnetic-field basis functions and their gradients are demonstrated.