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The problem of finding equilibrium configurations of one-component charged particles, induced by externalelectrostaticfieldsinplanarsystems,isasubjectofactivestudiesinfundamentalaswellinexperimental investigations. In this paper the results of numerical analysis of the equilibrium configurations of charged particles (electrons), confined in a circular region by an infinite external potential at its boundary are presented. Equilibrium configurations with minimal energy are searched by means of special calculation scheme. This computational scheme consists of the following steps. First, the configuration of the system with the energy as close as possible to the expected energy value in the ground equilibrium state is found using a model of stable configurations. Next, classical Newtonian molecular dynamics is used using viscous friction to bring the system into equilibrium with a minimum energy. With a sufficient number of runs, we obtain a stable configuration with an energy value as close as possible to the global minimum energy value for the ground stable state for a given number of particles. Our results demonstrate a significant efficiency of using the method of classical molecular dynamics (MD) when using the interpolation formulas in comparison with algorithms based on Monte Carlo methods and global optimization. This approach makes it possible to significantly increase the speed at which an equilibrium configuration is reached for an arbitrarily chosen number of particles compared to the Metropolis annealing simulation algorithm and other algorithms based on global optimization methods

ABC-stacked trilayer graphene on boron nitride (ABC-TLG/hBN) moiré superlattices provides a tunable platform for exploring Wigner crystal states in which the electron correlation can be controlled by electric and magnetic fields. Here we report the observation of magnetic field-stabilized Wigner crystal states in a ABC-TLG/hBN. We show that correlated insulating states emerge at multiple fractional and integer fillings corresponding to ν = 1/3, 2/3, 1, 4/3, 5/3, and 2 electrons per moiré lattice site under a magnetic field. These correlated insulating states can be attributed to generalized Mott states for the integer fillings and generalized Wigner crystal states for the fractional fillings. The generalized Wigner crystal states are stabilized by a vertical magnetic field and are strongest at one magnetic flux quantum per three moiré superlattices. The ν = 2 insulating state persists up to 30 T, which can be described by a Mott-Hofstadter transition at a high magnetic field.

Quasi-two-dimensional systems play an important role in the manufacture of various devices for the needs of nanoelectronics. Obviously, the functional efficiency of such systems depends on their structure, which can change during phase transitions under the influence of external conditions (e.g., temperature). Until now, the main attention has been focused on the search for signals of phase transitions in continuous two-dimensional systems. One of the central issues is the analysis of conditions for the nucleation of the hexatic phase in such systems, which is accompanied by the appearance of defects in the Wigner crystalline phase at a certain temperature. However, both practical and fundamental questions arise about the critical number of electrons at which the symmetry of the crystal lattice in the system under consideration will begin to break and, consequently, the nucleation of defects will start. The dependences of the orientational order parameter and the correlation function, which characterize topological phase transitions, as functions of the number of particles at zero temperature have been studied. The calculation results allow us to establish the precursors of the phase transition from the hexagonal phase to the hexatic one for N = 92, 136, and 187 considered as an example. [ABSTRACT FROM AUTHOR]

Abstract We study the ground-state properties of quasi-one-dimensional paramagnetic electron fluid using variational quantum Monte Carlo method. The electrons are transversally confined using the harmonically regularized Coulomb potential with long range order. In this work we have investigated the crossover from liquid to crystalline phase in the finite width paramagnetic quantum wire. The calculated pair correlation function shows the oscillations of period $$2r_\text {s} a_0$$ 2 r s a 0 , where $$a_0$$ a 0 is the Bohr’s radius and $$r_\text {s}$$ r s is Wigner Seitz radius or electron-density parameter which corresponds to a $$k=4k_\text {F}$$ k = 4 k F peak with $$k_\text {F}$$ k F being Fermi wave vector in the static structure factor at finite electron density. It is a signature of the onset of Wigner crystallization. It is found that the cross-over from liquid to a quasi-Wigner phase is due to enhancement of the electron correlation effects as $$r_\text {s}$$ r s increases or as the wire width b decreases.

We study multi-valley electron gases in the low density (rs ≫ 1) limit. Here the ground-state is always a Wigner crystal (WC), with additional pseudo-spin order where the pseudo-spins are related to valley occupancies. Depending on the symmetries of the host semiconductor and the values of the parameters such as the anisotropy of the effective mass tensors, we find a striped or chiral pseudo-spin antiferromagnet, or a time-reversal symmetry breaking orbital loop-current ordered pseudo-spin ferromagnet. Our theory applies to the recently-discovered WC states in AlAs and in mono and bilayer transition metal dichalcogenides. We identify a set of interesting electronic liquid crystalline phases that could arise by continuous quantum melting of such WCs. [ABSTRACT FROM AUTHOR]

Many quantum phenomena responsible for key applications in material science and quantum chemistry arise in the strongly correlated regime. This is at the same time, a costly regime for computer simulations. In the limit of strong correlation analytic solutions exist, but as we move away from this limit numerical simulation are needed, and accurate quantum solutions do not scale well with the number of interacting particles. In this work we propose to use few-particle harmonic traps in combination with twisted light as a quantum emulator to investigate the transition into a strongly-correlated regime. Using both analytic derivations and numerical simulations we generalize previous findings on 2 Coulomb interacting fermions trapped in a one-dimensional harmonic trap to the case of 3 fermions. The 4 signatures of strong correlation we have identified in the one-dimensional harmonic trap are: (i) the ground state density is highly localized around N equilibrium positions, where N is the number of particles, (ii) the symmetric and antisymmetric ground state wavefunctions become degenerate, (iii) the von Neumann entropy grows, (iv) the energy spectrum is fully characterized by N normal modes or less. Our findings describe the low-energy behavior of electrons in quantum wires and ions in Paul traps. Similar features have also been reported for cold atoms in optical lattices.

The problem of finding equilibrium configurations of similarly charged particles (ions) induced by external electrostatic fields in planar systems is of great interest for fundamental and applied research. In this paper, the results of numerical analysis of equilibrium configurations of negatively charged particles (electrons) confined in a circular region by an infinite external potential at its boundary are presented. A hybrid numerical algorithm is developed to seek stable configurations with minimal energies. The algorithm is based on interpolation formulas derived using the analysis of equilibrium configurations obtained by means of the variational principle of the minimum energy for an arbitrary finite number of particles in a circular model. The solution of nonlinear equations of the given model make it possible to predict structure formation in the form of rings (shells) filled with electrons, the number of which decreases when passing from the external ring to the internal one. The number of rings depends on the total number of charged particles. The obtained interpolation formulas of the distribution of the total number of electrons over the rings are used as initial configurations for the molecular-dynamics method. Our results demonstrate the significant efficiency of the application of the classical molecular dynamics (MD) method when using interpolation formulas in comparison with algorithms based on Monte Carlo methods and global optimization. For an arbitrarily chosen number of particles in the considered system, the proposed method makes it possible to enhance the speed at which the stable configuration with the minimum energy is reached by several orders in comparison with the classical-molecular-dynamics method. [ABSTRACT FROM AUTHOR]

Self-organizing neural networks are used to analyze uncorrelated white noises of different distribution types (normal, triangular, and uniform). The artificially generated noises are analyzed by clustering the measured time signal sequence samples without its preprocessing. Using this approach, we analyze, for the first time, the current noise produced by a sliding "Wigner-crystal"-like structure in the insulating phase of a 2D electron system in silicon. The possibilities of using the method for analyzing and comparing experimental data obtained by observing various effects in solid-state physics and numerical data simulated using theoretical models are discussed. [ABSTRACT FROM AUTHOR]

Metastable states appear in many areas of physics as a result of symmetry-breaking phase transitions. An important challenge is to understand the microscopic mechanisms which lead to the formation of the energy barrier separating a metastable state from the ground state. In this paper, we describe an experimental example of the hidden metastable domain state in 1T-TaS2, created by photoexcitation or carrier injection. The system is an example of a charge density wave superlattice in the Wigner crystal limit displaying discommensurations and domain formation when additional charge is injected either through contacts or by photoexcitation. The domain walls and their crossings in particular display interesting, topologically entangled structures, which have a crucial role in the metastability of the system. We model the properties of experimentally observed thermally activated dynamics of topologically protected defects—dislocations—whose annihilation dynamics can be observed experimentally by scanning tunnelling microscopy as emergent phenomena described by a doped Wigner crystal. The different dynamics of trivial and non-trivial topological defects are quite striking. Trivial defects appear to annihilate quite rapidly at low temperatures on the timescale of the experiments, while non-trivial defects annihilate rarely, if at all. [ABSTRACT FROM AUTHOR]

The energy of a two-dimensional electron crystal in the low-density limit is studied. The first correction to the electrostatic energy of a crystal is shown to be 1.62733 r s − 3 / 2 Ry. The answer is obtained using the method of numerical integration. [ABSTRACT FROM AUTHOR]

We examine a charge lattice coupled to a one-dimensional asymmetric potential in the presence of an applied magnetic field, which induces gyrotropic effects in the charge motion. This system could be realized for Wigner crystals in nanostructured samples, dusty plasmas, or other classical charge-ordered states where gyrotropic motion and damping can arise. For zero magnetic field, an applied external ac drive can produce a ratchet effect in which the particles move along the easy flow direction of the substrate asymmetry. The zero field ratchet effect can only occur when the ac drive is aligned with the substrate asymmetry direction; however, when a magnetic field is added, the gyrotropic forces generate a Hall effect that leads to a variety of new behaviors, including a transverse ratchet motion that occurs when the ac drive is perpendicular to the substrate asymmetry direction. We show that this system exhibits commensuration effects as well as reversals in the ratchet effect and the Hall angle of the motion. The magnetic field also produces a nonmonotonic ratchet efficiency when the particles become localized at high fields.

We consider simulations of Wigner crystals in solid state systems interacting with random quenched disorder in the presence of thermal fluctuations. When quenched disorder is absent, there is a well defined melting temperature determined by the proliferation of topological defects, while for zero temperature, there is a critical quenched disorder strength above which topological defects proliferate. When both thermal and quenched disorder are present, these effects compete, and the thermal fluctuations can reduce the effectiveness of the quenched disorder, leading to a reentrant ordered phase in agreement with the predictions of Nelson (1983 Phys. Rev. B 27 2902–14). There are two competing theories for the low temperature behavior, and our simulations show that both capture aspects of the actual response. The critical disorder strength separating ordered from disordered states remains finite as the temperature goes to zero, as predicted by Cha and Fertig (1995 Phys. Rev. Lett. 74 4867–70), instead of dropping to zero as predicted by Nelson. At the same time, the critical disorder strength decreases with decreasing temperature, as predicted by Nelson, instead of remaining constant, as predicted by Cha and Fertig. The onset of the reentrant phase can be deduced based on changes in the transport response, where the reentrant ordering appears as an increase in the mobility or the occurrence of a depinning transition. We also find that when the system is in the ordered state and thermally melts, there is an increase in the effective damping or pinning. This produces a drop in the electron mobility that is similar to the peak effect phenomenon found in superconducting vortices, where thermal effects soften the lattice or break down its elasticity, allowing the particles to better adjust their positions to take full advantage of the quenched disorder.

Inspired by recent studies of the two-dimensional Wigner crystals for electrons, we study the time evolution of a gas of particles interacting with a Dirac pair potential. We find that the gas crystalizes into a square lattice for low free-particle quantum numbers.

Metastable states appear in many areas of physics as a result of symmetry-breaking phase transitions. An important challenge is to understand the microscopic mechanisms which lead to the formation of the energy barrier separating a metastable state from the ground state. In this paper, we describe an experimental example of the hidden metastable domain state in 1T-TaS2, created by photoexcitation or carrier injection. The system is an example of a charge density wave superlattice in the Wigner crystal limit displaying discommensurations and domain formation when additional charge is injected either through contacts or by photoexcitation. The domain walls and their crossings in particular display interesting, topologically entangled structures, which have a crucial role in the metastability of the system. We model the properties of experimentally observed thermally activated dynamics of topologically protected defects—dislocations—whose annihilation dynamics can be observed experimentally by scanning tunnelling microscopy as emergent phenomena described by a doped Wigner crystal. The different dynamics of trivial and non-trivial topological defects are quite striking. Trivial defects appear to annihilate quite rapidly at low temperatures on the timescale of the experiments, while non-trivial defects annihilate rarely, if at all.

Multielectron bubbles (MEBs) are cavities in liquid helium that contain electrons and possibly vapour. They provide a versatile platform for exploring properties of interacting electrons in two dimensions in a regime of densities that has not been previously available. Since the electrons are pushed against the deformable curved inner surface of the bubble, MEBs also provide an environment to study the roles of curvatures. In this review, we describe the current understanding of MEBs, with a focus on their stability issues, and highlight recent experimental studies of this interesting system. [ABSTRACT FROM AUTHOR]

The molecular dynamics method is used to study the melting of two-dimensional clusters that can form in electronic systems above the surface of superfluid helium with the help of triangular, square, hexagonal and round holding electrodes. Clusters with a fixed surface density of 108 cm−2, but different numbers of particles (from 3 to 406), are considered. It is shown that in defect-free clusters, the transition to a disordered state occurs abruptly and the melting point, Тmelt, is easy to determine. The dependence of Тmelt on the cluster size is observed for clusters with N < 400 particles. In square clusters with N < 50, the ordering of particles is imposed by the boundary conditions, whereas for N > 100, a triangular lattice with defects near the boundary is observed. In round clusters, electrons begin to move easily in the angular direction at rather low temperatures, but the "radial melting" temperature is consistent with the melting point of small square clusters. In the presence of defects, the transition to a disordered state begins from the defect region and gradually spreads to the entire cluster. There is no clear definition of the melting point in this scenario. [ABSTRACT FROM AUTHOR]

The molecular dynamics method is used to study the structure of two-dimensional clusters that can be created in electronic systems above the surface of superfluid helium, using triangular, square, hexagonal and round holding electrode geometries. Clusters with a fixed surface density of 108 cm−2, but different numbers of particles (from 3 to 406) are considered. The spatial configurations of particles in clusters are calculated corresponding to a temperature of 10−2 K. The shape of the cluster significantly affects the arrangement of the electrons inside. If the symmetry of the cluster is consistent with the triangular symmetry of the electronic crystal, then for some numbers of particles, clusters with ideal defect-free packing are possible; otherwise, the clusters contain defects. An additional external field applied to the cluster significantly changes the configuration of its particles. [ABSTRACT FROM AUTHOR]

The Cohn-Kumar conjecture states that the triangular lattice in dimension 2, the E8 lattice in dimension 8, and the Leech lattice in dimension 24 are universally minimizing in the sense that they minimize the total pair interaction energy of infinite point configurations for all completely monotone functions of the squared distance. This conjecture was recently proved by Cohn-Kumar-Miller-Radchenko-Viazovska in dimensions 8 and 24. We explain in this note how the conjecture implies the minimality of the same lattices for the Coulomb and Riesz renormalized energies as well as jellium and periodic jellium energies, hence settling the question of their minimization in dimensions 8 and 24. [ABSTRACT FROM AUTHOR]

We investigate non-perturbative corrections to the large N expansion of the homogeneous electron gas. These are associated with instanton solutions to the effective action of the plasmon field. We show that, although the large field behavior of that action dominates the quadratic bare Coulomb term, there are no solutions at large field, and consequently none at large density. We argue that solutions would exist at low density if the large N theory had a Wigner crystal (WC) phase. However, we argue that this is not the case. Together with the implied convergence of the large N expansion, this implies that the homogeneous electron gas with N component spins and a Coulomb interaction scaling like 1 ∕ N can only have a WC phase below a curve in the plane of N and density, which asymptotes to zero density at infinite N. We argue that for systems with a semi-classical expansion for order parameter dynamics, and a first order quantum transition between fluid and crystal phases, there are instantons associated with the decays of meta-stable fluid and crystal phases in the appropriate regions of the phase diagram. We argue that the crystal will decay into one or more colloidal or bubble phases (Kivelson and Spivak, 2006) rather than directly into the fluid. It is possible that the bubble phases remain stable all the way to the density where the crystal solution disappears. The bubbles of fluid will expand as the density is raised and gradually convert the system into a fluid filled with chunks of crystal. The transition to a translationally invariant phase is likely to be second order. Unfortunately, the HEG does not have a crystal phase at large N , where these semi-classical ideas could be examined in detail. We suggest that the evidence for negative dielectric function at intermediate densities for N = 2 is an indicator of this second order transition. While the infinite N limit does not have a negative dielectric function at any density, it is possible that the closed large N equation for the plasmon two point function, derived in Banks (2018) might capture at least the qualitative features of the second order transition. [ABSTRACT FROM AUTHOR]

A two-dimensional Wigner crystal formed by electrons located above a rectangular electrode with uniformly distributed positive charge has been studied by molecular dynamics methods. The authors have analyzed not only possible electron configurations over a narrow electrode region at a fixed electron density, depending on the width of the electrode, but also patterns of transitions from one configuration to another. In addition, this paper considers an electron configuration consisting of several chains. Using the Fourier transform for time dependencies of average electron coordinates during modeling, the authors have determined plasma oscillation spectra. Frequencies observed during modeling have been compared with theoretical frequencies for one and two chains. [ABSTRACT FROM AUTHOR]

Transport results from measuring ultra-clean two-dimensional systems, containing tunable carrier densities from 7 × 10 8 cm − 2 to ∼ 1 × 10 10 cm − 2 , reveal a strongly correlated liquid up to r s ≈ 40 where a Wigner crystallization is anticipated. A critical behavior is identified in the proximity of the metal-to-insulator transition. The nonlinear DC responses for r s > 40 captures hard pinning modes that likely undergo a first order transition into an intermediate phase in the course of melting. [ABSTRACT FROM AUTHOR]

The simplest possible structural transition that an electronic system can undergo is Wigner crystallization. The aim of this short review is to discuss the main aspects of three recent experimets on the one-dimensional Wigner molecule, starting from scratch. To achieve this task, the Luttinger liquid theory of weakly and strongly interacting fermions is briefly addressed, together with the basic properties of carbon nanotubes that are required. Then, the most relevant properties of Wigner molecules are addressed, and finally the experiments are described. The main physical points that are addressed are the suppression of the energy scales related to the spin and isospin sectors of the Hamiltonian, and the peculiar structure that the electron density acquires in the Wigner molecule regime.

This study sets out to describe the interaction of charged confined particles with a gravitational field. The system consisted of charged particles interacting electrostatically between themselves and with a confining box whose walls have a linear charge distribution. We studied numerically several configurations including the introduction of a tilt, that takes into account the Earth's gravitational field. In our analysis, we included the α parameter for describing the position of the center of mass and its evolution according to the influence of the box size. We also studied in detail the competition between the gravitational field and electrostatic forces over the system. Such a competition determines the shape and size of the distributed charges. When the electrostatic force is dominant, the charges are distributed almost symmetrically. Whereas, when the gravitational field is preponderant, the distribution of particles tends to have a hive-like shape and then as the tilt angle increases the distribution is flattened. This effect is exacerbated when the particles have bigger masses and the inclination is higher. From a physical analysis, we obtained an expression to determine the center of mass of the distributed particles, depending on the box's size and other physical parameters. Finally, based on the data analysis, we obtained fitting equations for the gravitational (logarithmic) and electrostatic (exponential) energies as a function of the enveloping area. • A detailed analysis of charged particles confined in a box with linear distributed charged walls is carried out. • The effects of gravitational field were taken into account by means of the box tilt and particle mass. • Based on physical factors such as gravitational field, electrostatic force and box size, an equation for the distributions' center of mass was established. • The study of the competition between the gravitational and electrostatic energies enabled the determination of the sizes and shapes of the enveloping areas. • From data analysis, exponential and logarithmic fitting equations for the gravitational and electrostatic energies were obtained respectively. [ABSTRACT FROM AUTHOR]

Transport results from measuring ultra-clean two-dimensional systems, containing tunable carrier densities from 7 × 10 8 cm − 2 to ∼ 1 × 10 10 cm − 2 , reveal a strongly correlated liquid up to r s ≈ 40 where a Wigner crystallization is anticipated. A critical behavior is identified in the proximity of the metal-to-insulator transition. The nonlinear DC responses for r s > 40 captures hard pinning modes that likely undergo a first order transition into an intermediate phase in the course of melting.