Ladislav Šamaj

Thermodynamic Properties of the Two-Dimensional Two-Component Plasma

The model under consideration is a two-dimensional two-component plasma, stable against collapse for the dimensionless coupling constant β<2. The combination of a technique of renormalized Mayer expansion with the mapping onto the sine-Gordon theory provides the full thermodynamics of the plasma in the whole stability range of β. The explicit forms of the density–fugacity relationship and of the specific heat (at constant volume) per particle are presented.

The statistical mechanics of the classical two-dimensional Coulomb gas is exactly solved

The model under consideration is a classical 2D Coulomb gas of pointlike positive and negative unit charges, interacting via a logarithmic potential. In the whole stability range of temperatures, the equilibrium statistical mechanics of this fluid model is exactly solvable via an equivalence with the integrable 2D sine-Gordon field theory. The exact solution includes the bulk thermodynamics, special cases of the surface thermodynamics and the large-distance asymptotic behaviour of the two-body correlation functions.

Wigner-crystal formulation of strong-coupling theory for counterions near planar charged interfaces.

We present a new analytical approach to the strong electrostatic coupling regime (SC) that can be achieved equivalently at low temperatures, high charges, low dielectric permittivity, etc. Two geometries are analyzed in detail: one charged wall first, and then two parallel walls at small distances that can be likely or oppositely charged. In all cases, only one type of mobile counterions is present, and ensures electroneutrality (salt-free case). The method is based on a systematic expansion around the ground state formed by the two-dimensional Wigner crystal(s) of counterions at the plate(s). The leading SC order stems from a single-particle theory, and coincides with the virial SC approach that has been much studied in the last 10 years. The first correction has the functional form of the virial SC prediction, but the prefactor is different. The present theory is free of divergences and the obtained results, both for symmetrically and asymmetrically charged plates, are in excellent agreement with available data of Monte Carlo simulations under strong and intermediate Coulombic couplings. All results obtained represent relevant improvements over the virial SC estimates. The present SC theory starting from the Wigner crystal and therefore coined Wigner SC, sheds light on anomalous phenomena like the counterion mediated like-charge attraction, and the opposite-charge repulsion.

The Sixth-Moment Sum Rule for the Pair Correlations of the Two-Dimensional One-Component Plasma: Exact Result

The system under consideration is a two-dimensional one-component plasma in the fluid regime, at density n and arbitrary coupling Γ=βe2 (e=unit charge, β=inverse temperature). The Helmholtz free energy of the model, as the generating functional for the direct pair correlation c, is treated in terms of a convergent renormalized Mayer diagrammatic expansion in density. Using specific topological transformations within the bond-renormalized Mayer expansion, we prove that the nonzero contributions to the regular part of the Fourier component of c up to the k2-term originate exclusively from the ring diagrams (unable to undertake the bond-renormalization procedure) of the Helmholtz free energy. In particular, ĉ(k)=−Γ/k2+Γ/(8πn)−k2/[96(πn)2]+O(k4). This result fixes via the Ornstein–Zernike relation, besides the well-known zeroth-, second-, and fourth-moment sum rules, the new sixth-moment condition for the truncated pair correlation h, n(πΓn/2)3 ∫ r6h(r) dr=3(Γ−6)(8−3Γ)/4.

Another derivation of a sum rule for the two-dimensional two-component plasma

In a two-dimensional two-component plasma, the second moment of the number density correlation function has the simple value {12π[1−(Γ/4)]2}−1, where Γ is the dimensionless coupling constant. This result is derived directly by using diagrammatic methods.

The t-expansion study of critical phenomena in quantum systems

For quantum systems in the ground state, the t-expansion technique is used to construct a canonical sequence of classical theories. The consequent application of the coherent-anomaly method provides an estimation of the non-classical values of critical indices related to the singular behaviour of the ground-state energy. The accuracy of the results is tested on the Ising model with a transverse field, formulated on two- and three-dimensional lattices.

Critical phenomena and phase sequence in a classical bilayer Wigner crystal at zero temperature

We study the ground-state properties of a system of identical classical Coulombic point particles, evenly distributed between two equivalently charged parallel plates at distance

Translation Symmetry Breaking in the One-Component Plasma on the Cylinder

d

Thermodynamic properties of the two-dimensional Coulomb gas in the low-density limit

; the system as a whole is electroneutral. It was previously shown that upon increasing d from 0 to infinity, five different structures of the bilayer Wigner crystal become energetically favored, starting from a hexagonal lattice (phase I, d=0) and ending at a staggered hexagonal lattice (phase V, d -> infinity). In this paper, we derive new series representations of the ground-state energy for all five bilayer structures. The derivation is based on a sequence of transformations for lattice sums of Coulomb two-particle potentials plus the neutralizing background, having their origin in the general theory of Jacobi theta functions. The new series provide convenient starting points for both analytical and numerical progress. Its convergence properties are indeed excellent: Truncation at the fourth term determines in general the energy correctly up to 17 decimal digits. The accurate series representations are used to improve the specification of transition points between the phases and to solve a controversy in previous studies. In particular, it is shown both analytically and numerically that the hexagonal phase I is stable only at d=0, and not in a finite interval of small distances between the plates as was anticipated before. The expansions of the structure energies around second-order transition points can be done analytically, which enables us to show that the critical behavior is of the Ginzburg-Landau type, with a mean-field critical index beta=1/2 for the growth of the order parameters.

Counter-ions at charged walls: Two-dimensional systems

The two-dimensional one-component plasma, i.e. the system of point-like charged particles embedded in a homogeneous neutralizing background, is studied on the surface of a cylinder of finite circumference, or equivalently in a semiperiodic strip of finite width. The model has been solved exactly by Choquardet al. at the free-fermion couplingΓ = 2: in the thermodynamic limit of an infinitely long strip, the particle density turns out to be a nonconstant periodic function in space and the system exhibits long-range order of the Wigner-crystal type. The aim of this paper is to describe, qualitatively as well as quantitatively, the crystalline state for a larger set of couplingsΓ = 2γ (γ = 1,2,..., a positive integer) when the plasma is mappable onto a one-dimensional fermionic theory. The fermionic formalism, supplemented by some periodicity assumptions, reveals that the density profile results from a hierarchy of Gaussians with a uniform variance but with different amplitudes. The number and spatial positions of these Gaussians within an elementary cell depend on the particular value of γ. Analytic results are supported by the exact solution at γ = 1 (Γ = 2) and by exact finite-size calculations at γ = 2,3.