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 $d$; the system as a whole is electroneutral. It was previously shown that upon increasing $d$ from 0 to $\infty$, 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, $d\to \infty$). In this paper, we derive 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 series proposed in this manuscript 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.

Figure 1

Ground-state structures I–V of counterions on two parallel equivalently and homogeneously charged plates. Open and filled symbols correspond to particle positions on the opposite surfaces. The primitive translation vectors of the Bravais lattice on one of the surfaces are denoted by ${\mathit{a}}_1$ and ${\mathit{a}}_2$. For structures I–III, we define the aspect ratio as $\Delta =|{\mathit{a}}_1|/|{\mathit{a}}_2|$, so that $\Delta =\sqrt{3}$ with structure I, $1<\Delta <\sqrt{3}$ for structure II, and $\Delta =1$ for structure III. The dashed circle for structure IV is a guide to the eye for identifying those points that are equidistant to the ion in the circle center. For a more detailed description of the structures, see the text.

Figure 2

The difference between the dimensionless energies $[E(\Delta ,\eta )-E(\sqrt{3},\eta )]/\left(e^2\sqrt{n}\right)$ versus $\epsilon =\sqrt{3}-\Delta$, calculated numerically by using Eq. (24) for two small values of $\eta$: (a) $\eta ={10}^{-3}$ and (b) $\eta ={10}^{-2}$. For the range of aspect ratios chosen, these curves are indistinguishable from the analytical prediction $-0.5833{\eta }^2\epsilon +0.0408{\epsilon }^2$, stemming from Eqs. (29) and (30). The values of ${\epsilon }^{*}$, which provide the energy minimum in the asymptotic limit $\eta \to 0$ according to (32), are depicted by the vertical dashed lines for comparison.

Figure 3

Going from phase I to II: The test of the asymptotic relation (32) (dashed line) against numerical minimization of the energy (24) (solid curve) for small and intermediary values of $\eta$ in the logarithmic scale.

Figure 4

Transition between phases II and III: The test of the asymptotic relation (37) (dashed line) against numerical minimization of the energy (24) (solid curve) in the logarithmic scale.

Figure 5

The stability range of phase II: The plot of the lattice aspect ratio ${\Delta }^{*}$ versus $\eta$, obtained by the numerical minimization of the energy (24), is pictured by the solid curve. Numerical data of Ref. 18 are presented by open circles. The asymptotic relations (32) for $\eta \to 0$ and (37) for $\eta \to {\eta }_2^c$ are depicted by dashed curves.

Figure 6

Transition between phases III and IV: The test of the asymptotic relation (51) (dashed line) against a numerical minimization of the energy (46) (solid curve), in the logarithmic scale. The content of upper and lower insets is commented in the text.

Figure 7

Stability range of phase IV: The plot of the angle parameter ${\delta }^{*}$ versus $\eta$, obtained by the numerical minimization of the energy (46), is shown by the solid curve. Numerical data of Ref. 18 are presented by open circles. The asymptotic relation (51) for $\eta \to {\eta }_3^c$ is depicted by the dashed curve.

Figure 8

Summary of phase transition scenario. The rounded off values of the thresholds are mentioned. Structure I is realized at $\eta =0$ only (vanishing interplate separation).