Orientational ordering in two-dimensional dipole systems

The first step in studying the orientation ordering of two-dimensional dipole systems consists in the analysis of the ground state. If the orientation of rigid dipoles is described by two-dimensional unit vectors er lying in the lattice plane, then the ground state corresponds to the minimum of the system Hamiltonian 

The star of the L wave vectors k, corresponds to the degenerate (at L 1) minimum eigenvalue Pj (k,). 

Luttinger-Tisza method is burdened by independent minimization variables, while analysis of the values of the Fourier components F k) makes it possible to immediately exclude no less than half of the variable set and to obtain a result much more quickly. Degeneracy of the ground state occurs either due to coincidence of minimal values of Vt (k) at two boundary points of the first Brillouin zone k = b]/2 and k = b2/2, or as a result of the equality Fj (k) = F2 (k) at the same point k = h/2. The natural consequence of the ground state degeneracy is the presence of a Goldstone mode in the spectrum of orientational vibrations.53 

Let us present the ground state characteristics of dipoles (interacting as defined by Eq. (2.2.2)) on square, triangular, rectangular, and rhombic lattices. The ground state of a square dipole lattice was first determined by the Luttinger-Tisza method in 

The expansion of Fourier components of the dipole interaction tensor in the vicinity of the minimum point at the boundary of the first Brillouin zone, with the Cartesian axes Ox and Oy respectively chosen along bi and b2 (see Fig. 2.9b), has the form 

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In the present book, we aim at the unified description of ground states and collective excitations in orientationally structured adsorbates based on the theory of two-dimensional dipole systems. Chapter 2 is concerned with the discussion of orientation ordering in the systems of adsorbed molecules. In Section 2.1, we present a concise review on basic experimental evidence to date which demonstrate a variety of structures occurring in two-dimensional molecular lattices on crystalline dielectric substrates and interactions governing this occurrence. 

Ferromagnetic ordering of two-dimensional systems with dipole-dipole and exchange interactions in the approximation of a spherical model was examined in Ref. 73. The main simplifying assumption of the spherical model is the replacement of the condition er = 1 on the orientation vectors with the weaker condition... 

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