Competing structures in two dimensions: Squaretohexagonal transition
Abstract
We study a system of particles in two dimensions interacting via a dipolar longrange potential D /r^{3} and subject to a squarelattice substrate potential V (r ) with amplitude V and lattice constant b . The isotropic interaction favors a hexagonal arrangement of the particles with lattice constant a , which competes against the square symmetry of the underlying substrate lattice. We determine the minimalenergy states at fixed external pressure p generating the commensurate density n =1 /b^{2}=(4^{/3 ) 1 /2}/a^{2} in the absence of thermal and quantum fluctuations, using both analytical techniques based on the harmonic and continuum elastic approximations as well as numerical relaxation of particle configurations. At large substrate amplitude V >0.2 e_{D}, with e_{D}=D /b^{3} the dipolar energy scale, the particles reside in the substrate minima and hence arrange in a square lattice. Upon decreasing V , the square lattice turns unstable with respect to a zoneboundary shear mode and deforms into a perioddoubled zigzag lattice. Analytic and numerical results show that this perioddoubled phase in turn becomes unstable at V ≈0.074 e_{D} towards a nonuniform phase developing an array of domain walls or solitons; as the density of solitons increases, the particle arrangement approaches that of a rhombic (or isosceles triangular) lattice. At a yet smaller substrate value estimated as V ≈0.046 e_{D}, a further solitonic transition establishes a second nonuniform phase which smoothly approaches the hexagonal (or equilateral triangular) lattice phase with vanishing amplitude V . At small but finite amplitude V , the hexagonal phase is distorted and hexatically locked at an angle of φ ≈3 .8^{∘} with respect to the substrate lattice. The squaretohexagonal transformation in this twodimensional commensurateincommensurate system thus involves a complex pathway with various nontrivial lattice and modulated phases.
 Publication:

Physical Review B
 Pub Date:
 August 2016
 DOI:
 10.1103/PhysRevB.94.054110
 arXiv:
 arXiv:1605.08262
 Bibcode:
 2016PhRvB..94e4110G
 Keywords:

 Condensed Matter  Statistical Mechanics;
 Condensed Matter  Materials Science
 EPrint:
 30 pages, 25 figures