Two-dimensional One-component Plasma Research Articles

Hyperuniformity of the two-dimensional one-component plasma

Hyperuniformity of the two-dimensional one-component plasma

Expanding the Fourier Transform of the Scaled Circular Jacobi beta Ensemble Density

The family of circular Jacobi beta ensembles has a singularity of a type associated with Fisher and Hartwig in the theory of Toeplitz determinants. Our interest is in the Fourier transform of the corresponding N rightarrow infty bulk scaled spectral density about this singularity, expanded as a series in the Fourier variable. Various integrability aspects of the circular Jacobibeta ensemble are used for this purpose. These include linear differential equations satisfied by the scaled spectral density for beta = 2 and beta = 4, and the loop equation hierarchy. The polynomials in the variable u=2/beta which occur in the expansion coefficents are found to have special properties analogous to those known for the structure function of the circular beta ensemble, specifically in relation to the zeros lying on the unit circle |u|=1 and interlacing. Comparison is also made with known results for the expanded Fourier transform of the density about a guest charge in the two-dimensional one-component plasma.

The boundary density profile of a Coulomb droplet. Freezing at the edge

We revisit the problem of computing the boundary density profile of a droplet of two-dimensional one-component plasma (2D OCP) with logarithmic interaction between particles in a confining harmonic potential. At a sufficiently low temperature, but still in the liquid phase, the density exhibits oscillations as a function of the distance to the boundary of the droplet. We obtain the density profile numerically using Monte-Carlo simulations of the 2D OCP. We argue that the decay and period of those oscillations can be explained within a picture of the Wigner crystallization near the boundary, where the crystal is gradually melted with the increasing distance to the boundary.

The effect of surface charge self-organization on gate-induced electron and hole two-dimensional systems

A model is proposed for describing the self-organization of localized charges and quantum scattering in undoped GaAs/AlGaAs structures in which a two-dimensional gas of electrons or holes is created by the corresponding gate voltage. We assume that in such a metal-dielectric-undoped semiconductor structure carrier scattering on surface charges localized at the interface between GaAs and the dielectric dominates. Proposed model considers these charges and the corresponding image charges in the metal gate as a closed system in a thermostat. The electrostatic self-organization for this system in thermodynamic equilibrium is studied numerically using the Metropolis algorithm in a wide temperature range. It is shown that, at T > 100 K, a simple formula derived from the theory of two-dimensional one-component plasma gives almost the same behavior of the structural factor at low wave numbers as the Monte Carlo calculation. The scattering times of gate-induced carriers are described by formulas in which the structural factor characterizes the frozen disorder in the given system. In these formulas, the behavior of the structural factor at small wave numbers is decisive. A calculation using these formulas with disorder corresponding to infinite T gives two to three times shorter scattering times than in the corresponding experiments. We found that the theory is consistent with experiment at a freezing point of disorder T ≈ 1000 K for a sample with a two-dimensional electron gas and T ≈ 700 K for a sample with a two-dimensional hole gas. The found values are an upper estimate of the freezing temperature in the studied structures, since the model ignores sources of disorder other than temperature.

Short-Distance Symmetry of Pair Correlations in Two-Dimensional Jellium

We consider the two-dimensional one-component plasma (jellium) of mobile pointlike particles with the same charge $e$, interacting pairwisely by the logarithmic Coulomb potential and immersed in a fixed neutralizing background charge density. Particles are in thermal equilibrium at the inverse temperature $\beta$, the only relevant dimensionless parameter is the coupling constant $\Gamma\equiv\beta e^2$. In the bulk fluid regime and for any value of the coupling constant $\Gamma=2\times{\rm integer}$, \v{S}amaj and Percus [J. Stat. Phys. {\bf 80}, 811--824 (1995)] have derived an infinite sequence of sum rules for the coefficients of the short-distance expansion of particle pair correlation function. In the context of the equivalent fractional quantum Hall effect, by using specific methods of quantum geometry Haldane [PRL {\bf 107}, 116801 (2011) and arXiv:1112.0990v2] derived a self-dual relation for the Landau-level guiding-center structure factor. In this paper, we establish the relation between the guiding-center structure factor and the pair correlation function of jellium particles. It is shown that the self-dual formula, which provides an exact relation between the pair correlation function and its Fourier component, comes directly from the short-distance symmetry of the bulk jellium. The short-distant symmetry of pair correlations is extended to the semi-infinite geometry of a rectilinear plain hard wall with a fixed surface charge density, constraining particles to a half-space. The symmetry is derived for the original jellium model as well as its simplified version with no background charge (charged wall surface with ``counter-ions only''). The obtained results are checked at the exactly solvable free-fermion coupling $\Gamma=2$.

Fourth Moment of the Charge Density Induced Around a Guest Charge in Two-Dimensional Jellium

The model under consideration is the classical two-dimensional one-component plasma (jellium) of pointlike particles with charge $e$, interacting pairwisely via the logarithmic Coulomb potential and immersed in a uniform neutralizing background charge density. The system is in thermal equilibrium at the inverse temperature $\beta$, its thermodynamics depends only on the coupling constant $\Gamma=\beta e^2$. We put into an infinite (homogeneous and translationally invariant) plasma a guest particle of charge $Ze$ with $Z>-2/\Gamma$ in order to prevent from the collapse of the jellium charges onto it. The guest particle induces a screening cloud (the excess charge density) in the plasma. The zeroth and second moments of this screening cloud were derived previously for any fluid value of $\Gamma$. In this paper, we propose a formula for the fourth moment of the screening cloud. The derivation is based on the assumption that the fourth moment is, similarly as the second moment, analytic in $Z$ around $Z=0$. An exact treatment of the limit $Z\to\infty$ shows that it is a finite (cube) polynomial in $Z$. The $\Gamma$-dependence of the polynomial coefficients is determined uniquely by considering the limits $Z\to 0$ and $Z\to\infty$, and the compressibility sum rule for $Z=1$. The formula for the fourth moment of screening cloud is checked in the leading and first correction orders of the Debye-H\"uckel limit $\Gamma\to 0$ and at the exactly solvable free-fermion point $\Gamma=2$. Sufficient conditions for sign oscillations of the induced charge density which follow from the second-moment and fourth-moment sum rules are discussed.

The two-dimensional Coulomb plasma: quasi-free approximation and central limit theorem

For the two-dimensional one-component Coulomb plasma, we derive an asymptotic expansion of the free energy up to order $N$, the number of particles of the gas, with an effective error bound $N^{1-\kappa}$ for some constant $\kappa > 0$. This expansion is based on approximating the Coulomb gas by a quasi-free Yukawa gas. Further, we prove that the fluctuations of the linear statistics are given by a Gaussian free field at any positive temperature. Our proof of this central limit theorem uses a loop equation for the Coulomb gas, the free energy asymptotics, and rigidity bounds on the local density fluctuations of the Coulomb gas, which we obtained in a previous paper.

Dynamical properties of magnetized two-dimensional one-component plasma

Molecular dynamics simulation are used to examine the effect of a uniform perpendicular magnetic field on a two-dimensional interacting electron system. In this simulation we include the effect of the magnetic field classically through the Lorentz force. Both the Coulomb and the magnetic forces are included directly in the electron dynamics to study their combined effect on the dynamical properties of the 2D system. Results are presented for the velocity autocorrelation function and the diffusion constants in the presence and absence of an external magnetic field. Our simulation results clearly show that the external magnetic field has an effect on the dynamical properties of the system.

Exact mean-energy expansion of Ginibre's gas for coupling constants Γ=2×(oddinteger).

Using the approach of a Vandermonde determinant to the power Γ=Q^{2}/k_{B}T expansion on monomial functions, a way to find the excess energy U_{exc} of the two-dimensional one-component plasma (2DOCP) on hard and soft disks (or a Dyson gas) for odd values of Γ/2 is provided. At Γ=2, the present study not only corroborates the result for the particle-particle energy contribution of the Dyson gas found by Shakirov [Shakirov, Phys. Lett. A 375, 984 (2011)10.1016/j.physleta.2011.01.004] by using an alternative approach, but also provides the exact N-finite expansion of the excess energy of the 2DOCP on the hard disk. The excess energy is fitted to the ansatz of the form U_{exc}=K^{1}N+K^{2}sqrt[N]+K^{3}+K^{4}/N+O(1/N^{2}) to study the finite-size correction, with K^{i} coefficients and N the number of particles. In particular, the bulk term of the excess energy is in agreement with the well known result of Jancovici for the hard disk in the thermodynamic limit [Jancovici, Phys. Rev. Lett. 46, 386 (1981)10.1103/PhysRevLett.46.386]. Finally, an expression is found for the pair correlation function which still keeps a link with the random matrix theory via the kernel in the Ginibre ensemble [Ginibre, J. Math. Phys. 6, 440 (1965)10.1063/1.1704292] for odd values of Γ/2. A comparison between the analytical two-body density function and histograms obtained with Monte Carlo simulations for small systems and Γ=2,6,10,... shows that the approach described in this paper may be used to study analytically the crossover behavior from systems in the fluid phase to small crystals.

Local Density for Two-Dimensional One-Component Plasma

We study the classical two-dimensional one-component plasma of $N$ positively charged point particles, interacting via the Coulomb potential and confined by an external potential. For the specific inverse temperature $\beta=1$ (in our normalization), the charges are the eigenvalues of random normal matrices, and the model is exactly solvable as a determinantal point process. For any positive temperature, using a multiscale scheme of iterated mean-field bounds, we prove that the equilibrium measure provides the local particle density down to the optimal scale of $N^{o(1)}$ particles. Using this result and the loop equation, we further prove that the particle configurations are rigid, in the sense that the fluctuations of smooth linear statistics on any scale are $N^{o(1)}$.

Onset of negative dispersion in one-component-plasma revisited

A simple approach to describe the long-wavelength dispersion of the longitudinal (plasmon) mode of the classical one-component-plasma (OCP), with the main objective to correctly capture the onset of negative dispersion, is discussed. The approach is applicable to both three-dimensional and two-dimensional OCP. The predicted onset of negative dispersion compares well with the available results from numerical simulations and more sophisticated theoretical models.

Large Deviations of Radial Statistics in the Two-Dimensional One-Component Plasma

The two-dimensional one-component plasma is an ubiquitous model for several vortex systems. For special values of the coupling constant $\beta q^2$ (where $q$ is the particles charge and $\beta$ the inverse temperature), the model also corresponds to the eigenvalues distribution of normal matrix models. Several features of the system are discussed in the limit of large number $N$ of particles for generic values of the coupling constant. We show that the statistics of a class of radial observables produces a rich phase diagram, and their asymptotic behaviour in terms of large deviation functions is calculated explicitly, including next-to-leading terms up to order 1/N. We demonstrate a split-off phenomenon associated to atypical fluctuations of the edge density profile. We also show explicitly that a failure of the fluid phase assumption of the plasma can break a genuine $1/N$-expansion of the free energy. Our findings are corroborated by numerical comparisons with exact finite-N formulae valid for $\beta q^2=2$.

Counterion density profile around a charged disk: From the weak to the strong association regime.

We present a comprehensive study of the two-dimensional one-component plasma in the cell model with charged boundaries. Starting from weak couplings through a convenient approximation of the interacting potential we were able to obtain an analytic formulation to the problem deriving the partition function, density profile, contact densities, and integrated profiles that compared well with the numerical data from Monte Carlo simulations. Additionally, we derived the exact solution for the special cases of Ξ=1,2,3,⋯, finding a correspondence between those from weak couplings and the latter. Furthermore, we investigated the strong-coupling regime taking into consideration the Wigner formulation. Elaborating on this, we obtained the profile to leading order, computed the contact density values as compared to those derived in an earlier work on the contact theorem. We formulated adequately the strong-coupling regime for this system that differed from previous formulations. Ultimately, we calculated the first-order corrections and compared them against numerical results from our simulations with very good agreement; these results compared equally well in the planar limit, whose results are well known.

Fluctuations in the two-dimensional one-component plasma and associated fourth-order phase transition.

We study the distribution of the mean radial displacement of charges of a two-dimensional (2D) one-component plasma in the thermodynamic limit N→∞ at finite temperature β>0. We compute explicitly the large deviation functions showing the emergence of a fourth-order phase transition as a consequence of a change of topology in the plasma distribution. This weak phase transition occurs exactly at the ground state of the plasma. These results have been compared with the integrable case (finite N) of plasma parameter βq2=2. In this case the problem can be mapped to the stationary properties of 2D Dyson Brownian particles and to a non-Hermitian matrix model.

Relations Among Two Methods for Computing the Partition Function of the Two-Dimensional One-Component Plasma

The two-dimensional one-component plasma ---2dOCP--- is a system composed by $n$ mobile particles with charge $q$ over a neutralizing background in a two-dimensional surface. The Boltzmann factor of this system, at temperature $T$, takes the form of a Vandermonde determinant to the power $\Gamma = q^2/(2\pi\varepsilon k_BT)$, where $\Gamma$ is the coupling constant of this Coulomb system. The partition function of the model has been computed exactly for the even values of the coupling constant $\Gamma$, and a finite number of particles $n$, by two means: 1) by recognizing that the Boltzmann factor is the square of a Jack polynomial and expanding it in an appropriate monomial base, and 2) by mapping the system onto a 1-dimensional chain of interacting fermions. In this work the connection among the two methods is derived, and some properties of the expansion coefficients for the power of the Vandermonde determinant are explored.

Exact and Asymptotic Features of the Edge Density Profile for the One Component Plasma in Two Dimensions

There is a well known analogy between the Laughlin trial wave function for the fractional quantum Hall effect, and the Boltzmann factor for the two-dimensional one-component plasma. The latter requires analytic continuation beyond the finite geometry used in its derivation. We consider both disk and cylinder geometry, and focus attention on the exact and asymptotic features of the edge density. At the special coupling \Gamma := q^2/k_BT=2 the system is exactly solvable. In particular the k-point correlation can be written as a k \times k determinant, allowing the edge density to be computed to first order in \Gamma - 2. A double layer structure is found, which in turn implies an overshoot of the density as the edge of the leading support is approached from inside the plasma. Asymptotic analysis shows that the deviation from the leading order (step function) value is different for into the plasma than for outside. For general \Gamma, a Gaussian fluctuation formula is used to study the large deviation form of the density for N large but finite. This asymptotic form involves thermodynamic quantities which we independently study, and moreover an appropriate scaling gives the asymptotic decay of the limiting edge density outside of the plasma.

Freezing of an unconventional two-dimensional plasma

We study an unconventional two-dimensional, two-component classical plasma on a sphere, with emphasis on detecting signatures of melting transitions. This system is relevant to Ising-type quantum Hall states, and is unconventional in the sense that it features particles interacting via two different two-dimensional Coulomb interactions. One species of particle in the plasma carries charge of both types $({Q}_{1},{Q}_{2})$, while the other species carries only charge of the second type $(0,\ensuremath{-}{Q}_{2})$. We find signatures of a freezing transition at ${Q}_{1}^{2}\ensuremath{\simeq}140$. This means that the species with charge of both types will form a Wigner crystal, whereas the species with charge of the second type also shows signatures of being a Wigner crystal, due to the attractive intercomponent interaction of the second type. Moreover, there is also a Berezinskii-Kosterlitz-Thouless phase transition at ${Q}_{2}^{2}\ensuremath{\simeq}4$, at which the two species of particles bind to form molecules that are neutral with respect to the second Coulomb interaction. These two transitions appear to be independent of each other, giving a rectangular phase diagram. As a special case, ${Q}_{2}=0$ describes the (conventional) two-dimensional one-component plasma. Our study is consistent with previous studies of this plasma, and sheds new light on the freezing transition of this system.

Proposed interpretation of the transverse magnetic field dependence of the melting temperature T m(B) of a two-dimensional one-component plasma driven by logarithmic interactions

We propose an approximate scaling property of the classical partition function of a two-dimenisional one-component plasma in a high magnetic field based on the molecular dynamics simulations of dubey and Gumbs

Expanded Vandermonde Powers and Sum Rules for the Two-Dimensional One-Component Plasma

The two-dimensional one-component plasma (2dOCP) is a system of $N$ mobile particles of the same charge $q$ on a surface with a neutralising background. The Boltzmann factor of the 2dOCP at temperature $T$ can be expressed as a Vandermonde determinant to the power $\Gamma=q^{2}/(k_B T)$. Recent advances in the theory of symmetric and anti-symmetric Jack polymonials provide an efficient way to expand this power of the Vandermonde in their monomial basis, allowing the computation of several thermodynamic and structural properties of the 2dOCP for $N$ values up to 14 and $\Gamma$ equal to 4, 6 and 8. In this work, we explore two applications of this formalism to study the moments of the pair correlation function of the 2dOCP on a sphere, and the distribution of radial linear statistics of the 2dOCP in the plane.

One-component plasma on a spherical annulus and a random matrix ensemble

The two-dimensional one-component plasma at the special couplingβ = 2 is known to be exactly solvable, for its free energy and all of its correlations,on a variety of surfaces and with various boundary conditions. Here we studythis system confined to a spherical annulus with soft wall boundary conditions,paying special attention to the resulting asymptotic forms from the viewpointof expected general properties of the two-dimensional plasma. Our study ismotivated by the realization of the Boltzmann factor for the plasma system withβ = 2, after stereographic projection from the sphere to the complex plane, by a certain randommatrix ensemble constructed out of complex Gaussian and Haar distributed unitarymatrices.