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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.

We present a theory of spontaneous Fermi surface deformations for half-filled Landau levels (filling factors of the form$$\nu =2 \, n+1/2$$$\nu =2n+1/2$). We assume the half-filled level to be in a compressible, Fermi liquid state with a circular Fermi surface. The Landau level projection is incorporated via a modified effective electron-electron interaction and the resulting band structure is described within the Hartree-Fock approximation. We regulate the infrared divergences in the theory and probe the intrinsic tendency of the Fermi surface to deform through Pomeranchuk instabilities. We find that the corresponding susceptibility never diverges, though the system is asymptotically unstable in the$$n \rightarrow \infty $$$n\to \infty$limit.

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.

We study a structure consisting of two electrostatically interacting objects, a uniformly charged square nanoplate and a uniformly charged nanowire. A straightforward motivation behind this work is to introduce a model that allows a classical description of a finite two-dimensional quantum Hall system of few electrons when the Landau gauge is imposed. In this scenario, the uniformly charged square nanoplate would stand for the neutralizing background of the system while a uniformly charged nanowire would represent the resulting quantum striped state of the electrons. A second important feature of this model is that it also applies to hybrid charged nanoplate-nanowire systems in which the dominant interaction has electrostatic origin. An exact analytical expression for the electrostatic interaction potential between the uniformly charged square nanoplate and coplanar nanowire is obtained by using a special mathematical method adept for this geometry. It is found that the resulting interaction potential is finite, monotonic and slowly-varying for all locations of the nanowire inside the nanoplate.

We explain a general mathematical method that allows one to calculate the electrostatic potential created by a uniformly charged rectangular plate with arbitrary length and width at an arbitrary point in space. Exact analytical results for the electrostatic potential due to a uniformly charged finite rectangular plate are shown for special cases in order to illustrate the implementation of the formalism. Results of this nature are very important to various problems in physical sciences, applied mathematics, and potential theory.

Calculation of the Coulomb self-energy of a solid hemisphere with uniform volume charge density represents a very challenging task. This system is an interesting example of a body that lacks spherical symmetry though it can be conveniently dealt with in spherical coordinates. In this work, we explain how to calculate the Coulomb self-energy of a solid hemisphere with uniform volume charge density by using a method that relies on the expansion of the Coulomb potential as an infinite series in terms of Legendre polynomials. The final result for the Coulomb self-energy of a uniformly charged solid hemisphere turns out to be quite simple.

An almost ideal two-dimensional system of electrons can now be easily created in semiconductor heterojunctions. The quantum Hall effect state of the electrons is induced via the application of a strong perpendicular magnetic under specific quantum conditions. The most robust integer and/or fractional quantum Hall states already observed show the expected characteristic magnetoresistance for such systems. However, anisotropic patterns and features in transport properties have been seen for a few other peculiar cases. The origin of such anisotropic patterns may have various mechanisms or may also be due the specific details of the system and material such as the isotropic or anisotropic nature of the effective mass of electrons, the nature of the host substrate parameters, the nature of the interaction potentials, as well as other subtler effects. The interplay between all these factors can lead to many outcomes. In this work we consider small quantum Hall states of electrons at filling factor 1/6 and study the appearance of such anisotropic patterns as a result of some form of innate interaction anisotropy in the system.