Two-dimensional Coulomb Research Articles

Structural transitions in laterally compressed two-dimensional Coulomb clusters

We model structural transitions of small-size Wigner crystals in laterally compressed two-dimensional traps. Ground and metastable configurations are calculated and their transformations are linked to conspicuous changes in the heat capacity of the system. We show that various types of structural transitions are reflected by characteristic features in the behavior of the heat capacity. For deeper understanding, results produced by the Monte Carlo numerical calculations are compared to predictions of simple one-dimensional models.

Infinite-dimensional stochastic differential equations related to random matrices

AbstractWe solve infinite-dimensional stochastic differential equations (ISDEs) describing an infinite number of Brownian particles interacting via two-dimensional Coulomb potentials. The equilibrium states of the associated unlabeled stochastic dynamics are the Ginibre random point field and Dyson’s measures, which appear in random matrix theory. To solve the ISDEs we establish an integration by parts formula for these measures. Because the long-range effect of two-dimensional Coulomb potentials is quite strong, the properties of Brownian particles interacting with two-dimensional Coulomb potentials are remarkably different from those of Brownian particles interacting with Ruelle’s class interaction potentials. As an example, we prove that the interacting Brownian particles associated with the Ginibre random point field satisfy plural ISDEs.

Packing and melting of mesoscopically confined two-dimensional Coulomb crystals in straight narrow channels

The microstructure and melting dynamics of the two-dimensional mesoscopic Coulomb crystal with 1/r-type mutual interaction force and parabolic transverse confining potential, under different degrees of incommensurability, are investigated through molecular-dynamics simulation. To tune the degree of incommensurability, N(a) extra particles are added into the commensurate uniform triangular lattice which has seven-layer structure and 40 particles in each layer with the periodic longitudinal boundary condition, until the system reaches another commensurate packing with eight-layer structure at N(a)=40. It is found that the increasing incommensurability with the increasing N(a) or 40-N(a) gradually deteriorates the structural order with the presence of intrinsic defects and the anisotropic bond-length distribution, except for the defect-free configurations at a few magic N(a)'s. The system prefers the seven- and the eight-layer single structures through the entire crystal for the low- and the high-N(a) regimes, respectively, and the configurations with seven- and eight-layer domain mixtures for 18≤N(a)≤24. The increasing strain or the worse local particle interlocking around the intrinsic defects with the increasing incommensurability also causes the easier structural rearrangement associated with the easier particle hopping and the earlier onset of melting transition. The transverse confinement suppresses the transverse motion, induces nonuniform melting, and sustains the layered structure after melting.

Electrostatics of Coulomb gas, lattice paths and discrete polynuclear growth

We study the partition function of a two-dimensional Coulomb gas on a circle, in the presence of external pointlike charges, in a double scaling limit where both the external charges and the number of gas particles are large. Our original motivation comes from studying amplitudes for multi-string emission from a decaying D-brane in the high-energy limit. We analyze the scaling limit of the partition function and calculate explicit results. We also consider applications to random matrix theory. The partition functions can be related to random scattering or to weights of lattice paths in certain growth models. In particular, we consider the discrete polynuclear growth model and use our results to compute the cumulative probability density for the height of long level-1 paths. We also obtain an estimate for an almost certain maximum height.

Nonlinear conductivity of two-dimensional Coulomb glasses

We have studied the nonlinear conductivity of two-dimensional Coulomb glasses. We have used a Monte Carlo algorithm to simulate the dynamic of the system under an applied electric field $E$. We found that in the nonlinear regime the site occupancy in the Coulomb gap follows a Fermi-Dirac distribution with an effective temperature $T_{\rm eff}$, higher than the phonon bath temperature $T$. The value of the effective temperature is compatible with that obtained for slow modes from the generalized fluctuation-dissipation theorem. The nonlinear conductivity for a given electric field and $T$ is fairly similar to the linear conductivity at the corresponding $T_{\rm eff}$. We found that the dissipated power and the effective temperature are related by an expression of the form $(T_{\rm eff}^\alpha-T^\alpha)T_{\rm eff}^{\beta-\alpha}$.

Dimensional Effects on Electric Potentials and Fields in High-permittivity Thin Films and Interfaces

An efiective Lagrange density for electromagnetic flelds in the presence of a high- permittivity thin fllm or interface yields an analytic expression for electromagnetic potentials which interpolates between the two-dimensional logarithmic Coulomb potential at short distances and the three-dimensional Coulomb potential at large distances. It is shown that the analytic expression for the dimensionally hybrid electromagnetic potential is a very good approximation to the inflnite series solution from image charges.

Optically driven Mott-Hubbard systems out of thermodynamic equilibrium

We have studied the nonlinear conductivity of two-dimensional Coulomb glasses. We have used a Monte Carlo algorithm to simulate the dynamic of the system under an applied electric field E. We have compared results for two different models: a regular square lattice with only diagonal disorder and a random array of sites with diagonal and off-diagonal disorder. We have found that for moderate fields the logarithm of the conductivity is proportional to reproducing experimental results. We have also found that in the nonlinear regime the site occupancy in the Coulomb gap follows a Fermi-Dirac distribution with an effective temperature Teff higher than the phonon bath temperature T.

Non-linear conductivity in Coulomb glasses

We have studied the nonlinear conductivity of two-dimensional Coulomb glasses. We have used a Monte Carlo algorithm to simulate the dynamic of the system under an applied electric field E. We have compared results for two different models: a regular square lattice with only diagonal disorder and a random array of sites with diagonal and off-diagonal disorder. We have found that for moderate fields the logarithm of the conductivity is proportional to √ E/T 2 , reproducing experimental results. We have also found that in the nonlinear regime the site occupancy in the Coulomb gap follows a Fermi-Dirac distribution with an effective temperature Teff higher than the phonon bath temperature T . c

String scattering amplitudes from Dyson gas

In this note we review the relation between the open string rolling tachyon background and a classical two-dimensional Coulomb gas on a circle – the Dyson gas. Open string n -point amplitudes directly relate to adding external charges on the Dyson gas ensemble. By calculating the grand canonical partition function of the Dyson with external charges we find an approximation formula for the string amplitude. This note is based on the work with M. Jarvinen and Esko Keski-Vakkuri, eprint arXiv:0806.1491 .

Ground state configurations and melting of two-dimensional non-uniformly chargedclassical clusters

We consider classical two-dimensional (2D) Coulomb clustersconsisting of two species containing five particles with chargeq1 and fivewith charge q2, respectively. Using Monte Carlo and molecular dynamics (MD) simulations,we investigated the ground state configurations as well as radial and angulardisplacements of particles as a function of temperature and their dependence on the ratioq = q2/q1. We found new configurations and a new multi-step melting behavior forq sufficiently different from the uniform charge limitq = 1.

Slow relaxation and equilibrium dynamics in a two-dimensional Coulomb glass: Demonstration of stretched exponential energy correlations

We have simulated energy relaxation and equilibrium dynamics in Coulomb glasses using the random energy lattice model. We show that in a temperature range where the Coulomb gap is already well developed $(T=0.03--0.1)$, the system still relaxes to an equilibrium behavior within the simulation time scale. For all temperatures $T$, the relaxation is slower than exponential. Analyzing the energy correlations of the system at equilibrium $C(\ensuremath{\tau})$, we find a stretched exponential behavior, $C(\ensuremath{\tau})={e}^{\ensuremath{-}{(\ensuremath{\tau}/{\ensuremath{\tau}}_{0})}^{\ensuremath{\gamma}}}$. We study the temperature dependence of ${\ensuremath{\tau}}_{0}$ and $\ensuremath{\gamma}$. ${\ensuremath{\tau}}_{0}$ is shown to increase faster than exponentially with decreasing $T$. $\ensuremath{\gamma}$ is proportional to $T$ at low temperature and approaches unity for high temperature. We define a time ${\ensuremath{\tau}}_{\ensuremath{\gamma}}$ from these stretched exponential correlations and show that this time corresponds well with the time required to reach equilibrium. From our data it is not possible to determine whether ${\ensuremath{\tau}}_{\ensuremath{\gamma}}$ diverges at any finite temperature, indicating a glass transition, or whether this divergence happens at zero temperature. While the time dependence of the system energy can be well fitted by a random walker in a harmonic potential for high temperatures $(T=10)$, this simple model fails to describe the long time scales observed at lower temperatures. Instead we present an interpretation of the configuration space as a structure with fractal properties and the time evolution as a random walk on this fractal-like structure.

On the Coulomb–dipole transition in mesoscopic classical and quantum electron–hole bilayers

We study the Coulomb-to-dipole transition which occurs when the separation d of an electron–hole bilayer system is varied with respect to the characteristic in-layer distances. An analysis of the classical ground state configurations for harmonically confined clusters with N ⩽ 30 electron–hole pairs reveals that the energetically most favourable state can differ from that of two-dimensional pure dipole or Coulomb systems. Performing a normal mode analysis for the N = 19 cluster it is found that the lowest mode frequencies exhibit drastic changes when d is varied. Furthermore, we present quantum-mechanical ground states for N = 6, 10 and 12 spin-polarized electrons and holes. We compute the single-particle energies and orbitals in self-consistent Hartree–Fock approximation over a broad range of layer separations and coupling strengths between the limits of the ideal Fermi gas and the Wigner crystal.

One- and two-dimensional Coulomb Green's function matrices in parabolic Sturmians basis

One- and two-dimensional operators which originate from the asymptotic form of the three-body Coulomb wave equation in parabolic coordinates are treated within the context of a square integrable basis set. The matrix representations of Green's functions corresponding to these operators are obtained.

Stripe width and nonlocal domain walls in the two-dimensional dipolar frustrated Ising ferromagnet

We describe a novel type of magnetic domain wall which, in contrast to Bloch or Neel walls, is non-localized and, in a certain temperature range, non-monotonic. The wall appears as the mean-field solution of the two-dimensional ferromagnetic Ising model frustrated by the long-ranged dipolar interaction. We provide experimental evidence of this wall delocalization in the stripe-domain phase of perpendicularly magnetized ultrathin magnetic films. In agreement with experimental results, we find that the stripe width decreases with increasing temperature and approaches a finite value at the Curie-temperature following a power law. The same kind of wall and a similar temperature dependence of the stripe width is expected in the mean-field approximation of the two-dimensional Coulomb frustrated Ising ferromagnet.

The partition function of a multi-component Coulomb gas on a circle

We study a two-dimensional Coulomb gas consisting of a mixture of particles carrying various positive multiple integer charges, confined on a unit circle. We consider the system in the canonical and grand canonical ensembles, and attempt to calculate the partition functions analytically, using Toeplitz and confluent Vandermonde determinants. Just like in the simple one-component system (Dyson gas), the partition functions simplify at special temperature β = 2, allowing us to find compact expressions for them.

Effects of the range of the potential on the structure and dynamics of two-dimensional Coulomb clusters

The structure, dynamics, and thermodynamics of finite systems of two-dimensional particles interacting mutually through a shielded Coulomb potential, V(r) = e (−κr)/r, and bound together by an isotropic parabolic trap, are investigated by means of global optimization and simulation methods. As the range of the potential decreases, or as κ increases, the number of distinct minima rapidly increases for a given size and the cluster generally shrinks. Looking more specifically at selected sizes below 50, non- monotonic cluster size effects seem to dominate the energetics, chaotic dynamics, and thermodynamics, even though the melting point is weakly affected by the range of the potential.

Renormalization group approach to two-dimensional Coulomb interacting Dirac fermions with random gauge potential

We argue that massless Dirac particles in two spatial dimensions with $1/r$ Coulomb repulsion and quenched random gauge field are described by a manifold of fixed points which can be accessed perturbatively in disorder and interaction strength, thereby confirming and extending the results of arXiv:0707.4171. At small interaction and small randomness, there is an infra-red stable fixed curve which merges with the strongly interacting infra-red unstable line at a critical endpoint, along which the dynamical critical exponent $z=1$.

Configurations of Coulomb clusters in plasma

The structures of Coulomb clusters formed by dust particles in a plasma are studied by numerical simulation. The present study reveals the presence of various types of self-organized structures of clusters confined in an electrostatic potential. The stable configurations depend on the number of dust particles in a cluster as well as on the form of the confining potential. One-dimensional strings, two-dimensional Coulomb lattices, and three-dimensional Coulomb crystals found in the simulation agree well with the observation in the laboratory dust plasma experiments. Elongated axisymmetric Coulomb clusters are found to be composed of a few fundamental configurations. The dynamical features of Coulomb clusters are also studied by the simulation.

Electromagnetic Coulomb gas with vector charges and “elastic” potentials: Renormalization group equations

We present a detailed derivation of the renormalization group equations for two-dimensional electromagnetic Coulomb gases whose charges lie on a triangular lattice (magnetic charges) and its dual (electric charges). The interactions between the charges involve both angular couplings and a new electromagnetic potential. This motivates the denomination of “elastic” Coulomb gas. Such elastic Coulomb gases arise naturally in the study of the continuous melting transition of two-dimensional solids coupled to a substrate, either commensurate or with quenched disorder.

Hidden symmetry and excitonic transitions in the quantum well

In this article it is shown that, Sommerfeld's coefficients for excitonic transitions in quantum wells are determined only with the principle quantum number within the framework of two-dimensional Coulomb potential. This is a consequence of hidden symmetry of two-dimensional Coulomb problem, conditioned by the existence of two-dimensional analog of the Runge–Lentz vector. For the narrow gap semiconductor quantum well with the non-parabolic dispersion law of electron and hole in the two-band Kane model it is shown that two-dimensional excitonic states are described in the frames of an analog of Klein–Gordon equation with the two-dimensional Coulomb potential. The non-stability of the ground state of the two-dimensional Kane's exciton is shown.