Dynamics of two-dimensional dipole systems

Using a combined analytical/molecular dynamics approach, we study the current fluctuation spectra and longitudinal and transverse collective mode dispersions of the classical two-dimensional (point) dipole system (2DDS) characterized by the ${\varphi }_D\left(r\right)={\mu }^2/r^3$ repulsive interaction potential; $\mu$ is the electric dipole strength. The interest in the 2DDS is twofold. First, the quasi-long-range $1/r^3$ interaction makes the system a unique classical many-body system, with a remarkable collective mode behavior. Second, the system may be a good model for a closely spaced semiconductor electron-hole bilayer, a system that is in the forefront of current experimental interest. The longitudinal collective excitations, which are of primary interest for the liquid phase, are acoustic at long wavelengths. At higher wave numbers and for sufficiently high coupling strength, we observe the formation of a deep minimum in the dispersion curve preceded by a sharp maximum; this is identical to what has been observed in the dispersion of the zero-temperature bosonic dipole system, which in turn emulates so-called roton-maxon excitation spectrum of the superfluid ${}^4\text{H}\text{e}$. The analysis we present gives an insight into the emergence of this apparently universal structure, governed by strong correlations. We study both the liquid and the crystalline solid state. We also observe the excitation of combination frequencies, resembling the roton-roton, roton-maxon, etc. structures in ${}^4\text{H}\text{e}$.

Figure 1

Longitudinal collective mode dispersion curves generated from classical molecular dynamics (MD) simulations (squares) at ${\Gamma }_D=28$ and from the Refs. [20, 21]. zero-temperature quantum Monte Carlo (QMC) simulations (dots) at $r_D=28.4$. ${\omega }_D=\sqrt{2\pi n{\mu }^2/m{a}^3}$ is a characteristic dipole oscillation frequency.

Figure 2

MD pair distribution function for ${\Gamma }_D=7$, 15, 30, and 60.

Figure 3

(Color online) Longitudinal $\left(L\right)$ current fluctuation spectra for ${\Gamma }_D=60$; $a=1/\sqrt{\pi n}$ is the 2D Wigner-Seitz radius and ${\omega }_D=\sqrt{2\pi n{\mu }^2/m{a}^3}$ is a characteristic dipole oscillation frequency. The thick arrows point in the direction of increasing $ka$.

Figure 4

(Color online) Transverse $\left(T\right)$ current fluctuation spectra for ${\Gamma }_D=60$.

Figure 5

Longitudinal dispersion curves for ${\Gamma }_D=7$, 15, 30, and 60. The vertical lines indicate the widths of the spectra. The shaded region in (d) should be regarded as a ghostlike dispersion depicting the emergence of a faint maxon-maxon $\left(\text{M}+\text{M}\right)$ harmonic.

Figure 6

Transverse shear mode dispersion curve for ${\Gamma }_D=60$. Note the wave number cutoff $k^{\ast }a$ as a function of ${\Gamma }_D$ marking the $ka$ value where $\omega \left(ka\right)=0$.

Figure 7

QLCA Longitudinal $\left(L\right)$ and transverse $\left(T\right)$ dispersion curves for ${\Gamma }_D=7$, 15, 30, and 60. Note the evolution of the roton minimum with increasing ${\Gamma }_D$.

Figure 8

MD and QLCA longitudinal dispersion curves for ${\Gamma }_D=7$, 15, 30, and 60.

Figure 9

MD and QLCA transverse dispersion curves for ${\Gamma }_D=60$.

Figure 10

Electric dipole hexagonal lattice eigenmodes and corresponding mode polarizations for angular directions (with respect to the nearest neighbor direction) $\phi =0–{30}^0$ in ${10}^0$ steps. The measured sound velocity $s_L=1.284a{\omega }_D$.

Figure 11

2DDS lattice dispersion. Angle-averaged longitudinal and transverse lattice modes shown with liquid-phase MD data and the QLCA dispersion curves at ${\Gamma }_D=60$.

Figure 12

(Color online) Illustration of the appearance of combination frequencies in the longitudinal current-current correlation functions $L\left(k=k_x,\omega \right)$, measured along the direction of the nearest neighbor in a hexagonal lattice simulated at ${\Gamma }_D=1000$. The center color map of the $L\left(k=k_x,\omega \right)$ fluctuation spectra shows the strong primary dispersion and the ghosts of the combination frequencies; thin white lines are the dispersion curves from both $L\left(k=k_x,\omega \right)$ and $T\left(k=k_x,\omega \right)$. The four small panels show vertical cross-sections taken for the selected group of wave numbers, where the appearance of the combination frequencies is the most manifest.

Figure 13

(Color online) Same as Fig. 12 for wave numbers perpendicular to the nearest neighbor direction, $L\left(k=k_y,\omega \right)$.