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1. Spatial Entanglement Between Electrons Confined to Rings

   https://doi.org/10.3390/sym16121662  

   Ciftja, Orion; Batle, Josep; Abdel-Aty, Mahmoud; Hafez, Mohamed Ahmed; Alkhazaleh, Shawkat (December 2024, Symmetry)

   We study systems of two and three electrons confined to circular rings. The electrons are considered spinless, and we assume that one electron occupies a single ring. We use the framework of such a model to calculate the linear entropy and, thus, the spatial entanglement between the confined electrons. The geometry of the problem for the case of two electrons incorporates situations in which the planes of the two rings form an arbitrary angle with each other. The resulting Schrödinger’s equation is solved numerically with very high accuracy by means of the exact diagonalization method. We compute the ground state energy and entanglement for all configurations under consideration. We also study the case of three electrons confined to identical, parallel and concentric rings which are located in three different equidistant planes. The vertically separated system of rings is allowed to gradually merge into a single ring geometry, which would represent the equivalent system of a ring with three electrons. It is observed that the system of three electrons gives rise to a richer structure, as the three rings merge into a single one. 

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2. Deformation of the Fermi surface of a spinless two-dimensional electron gas in presence of an anisotropic Coulomb interaction potential

   https://doi.org/10.1038/s41598-021-82564-y  

   Ciftja, Orion (February 2021, Scientific Reports)

   Abstract We consider the stability of the circular Fermi surface of a two-dimensional electron gas system against an elliptical deformation induced by an anisotropic Coulomb interaction potential. We use the jellium approximation for the neutralizing background and treat the electrons as fully spin-polarized (spinless) particles with a constant isotropic (effective) mass. The anisotropic Coulomb interaction potential considered in this work is inspired from studies of two-dimensional electron gas systems in the quantum Hall regime. We use a Hartree–Fock procedure to obtain analytical results for two special Fermi liquid quantum electronic phases. The first one corresponds to a system with circular Fermi surface while the second one corresponds to a liquid anisotropic phase with a specific elliptical deformation of the Fermi surface that gives rise to the lowest possible potential energy of the system. The results obtained suggest that, for the most general situations, neither of these two Fermi liquid phases represent the lowest energy state of the system within the framework of the family of states considered in this work. The lowest energy phase is one with an optimal elliptical deformation whose specific value is determined by a complex interplay of many factors including the density of the system. 

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3. Observation of spontaneous ferromagnetism in a two-dimensional electron system

   https://doi.org/10.1073/pnas.2018248117  

   Hossain, M. S.; Ma, M. K.; Rosales, K. A.; Chung, Y. J.; Pfeiffer, L. N.; West, K. W.; Baldwin, K. W.; Shayegan, M. (December 2020, Proceedings of the National Academy of Sciences)

   null (Ed.)

   What are the ground states of an interacting, low-density electron system? In the absence of disorder, it has long been expected that as the electron density is lowered, the exchange energy gained by aligning the electron spins should exceed the enhancement in the kinetic (Fermi) energy, leading to a (Bloch) ferromagnetic transition. At even lower densities, another transition to a (Wigner) solid, an ordered array of electrons, should occur. Experimental access to these regimes, however, has been limited because of the absence of a material platform that supports an electron system with very high quality (low disorder) and low density simultaneously. Here we explore the ground states of interacting electrons in an exceptionally clean, two-dimensional electron system confined to a modulation-doped AlAs quantum well. The large electron effective mass in this system allows us to reach very large values of the interaction parameter r s , defined as the ratio of the Coulomb to Fermi energies. As we lower the electron density via gate bias, we find a sequence of phases, qualitatively consistent with the above scenario: a paramagnetic phase at large densities, a spontaneous transition to a ferromagnetic state when r s surpasses 35, and then a phase with strongly nonlinear current-voltage characteristics, suggestive of a pinned Wigner solid, when r s exceeds ≃ 38 . However, our sample makes a transition to an insulating state at r s ≃ 27 , preceding the onset of the spontaneous ferromagnetism, implying that besides interaction, the role of disorder must also be taken into account in understanding the different phases of a realistic dilute electron system. 

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4. Statistical Interaction Description of Pauli Crystals in 2D Systems of Harmonically Confined Fermions

   https://doi.org/10.1002/andp.201900075  

   Ciftja, Orion; Batle, Josep (August 2019, Annalen der Physik)

   Abstract It is conjectured that the Pauli exclusion principle alone may be responsible for a particular geometric arrangement of confined systems of identical fermions even when there is no interaction between them. These geometric structures, called Pauli crystals, are predicted for a two‐dimensional (2D) system of free fermions under harmonic confinement. In this work, the possibility of this outcome is pursued and a theoretical model is considered that may capture both qualitatively and quantitatively, the key features of the abovementioned setup. The results for and 6 particles show that the minimum energy configuration corresponds to and is in good quantitative agreement with the reported values of Pauli crystals seen in single‐shot imaging data obtained via the configuration density technique. Numerical results for larger systems of and 30 particles show that the crystalline configurations observed are not the same as the classical Wigner crystal structures that emerge should the confined charged particles interact with a Coulomb potential. An important question floated is whether such crystalline structures do really exist in a quantum system or whether they are artifacts of the methods used to analyze them. 

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5. Correspondence between Electrostatics and Contact Mechanics with Further Results in Equilibrium Charge Distributions

   https://doi.org/10.1002/andp.202300269  

   Batle, Josep; Vlasiuk, Oleksandr; Ciftja, Orion (November 2023, Annalen der Physik)

   Abstract It is common in mesoscopic systems to find instances where several charges interact among themselves. These particles are usually confined by an external potential that shapes the symmetry of the equilibrium charge configuration. In the case of classical charges moving on a plane and repelling each other via the Coulomb potential, they possess a ground state à la Thomson or Wigner crystal. As the number of particles N increases, the number of local minima grows exponentially and direct or heuristic optimization methods become prohibitively costly. Therefore the only feasible approximation to the problem is to treat the system in the continuum limit. Since the underlying framework is provided by potential theory, we shall by‐pass the corresponding mathematical formalism and list the most common cases found in the literature. Then we prove a (albeit known) mathematical correspondence that will enable us to re‐discover analytical results in electrostatics. In doing so, we shall provide different methods for finding the equilibrium surface density of charges, analytical and numerical. Additionally, new systems of confined charges in three‐dimensional surfaces will be under scrutiny. Finally, we shall highlight exact results regarding a modified power‐law Coulomb potential in thed‐dimensional ball, thus generalizing the existing literature. 

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