Abstract
Since the logarithm function is the solution of Poisson's equation in two dimensions, it appears as the Coulomb interaction in two dimensions, the interaction between Abrikosov flux lines in a type II superconductor, or between line defects in elastic media, and so on. Lattices of lines interacting logarithmically are, therefore, a subject of intense research due to their manifold applications. The solution of the Poisson equation for such lattices is known in the form of an infinite sum since the late 1990's. In this article we present an alternative analytical solution, in closed form, in terms of the Jacobi theta function.
Highlights
As intriguing as topological defects might be, crystals made out of them appear to be even more exotic, like the soliton lattice that forms in doped polyacetylene [1], for instance
Examples of topological defect lattices abound in Condensed Matter Physics where one might find lattices of parallel screw dislocations in solids [2], vortex lattices in rotating superfluids [3] and in Bose-Einstein condensates [4], as well as the much studied magnetic flux lattices in type II superconductors [5]
The most fascinating topological defect lattices are found in the realm of chiral liquid crystals [8] where skyrmions [9], hopfions (3D skyrmions) [10], merons [11] and even knots [12] may form regular arrays
Summary
As intriguing as topological defects might be, crystals made out of them appear to be even more exotic, like the soliton lattice that forms in doped polyacetylene [1], for instance. The most fascinating topological defect lattices are found in the realm of chiral liquid crystals [8] where skyrmions [9], hopfions (3D skyrmions) [10], merons (halfskyrmions) [11] and even knots [12] may form regular arrays Off this planet one might have magnetic flux tube lattices in neutron stars [13] and crystals of cosmic strings or of cosmic domain walls, which have been considered as possible candidates for solid dark matter models [14, 15]. In this work we move a step forward and obtain for the solution of Poisson equation a closed form for the logarithmic sum involving Jacobi theta functions These functions are special functions of complex variables which appear in the theory of elliptic functions which are ubiquitous in mathematical physics. Previous results for finite lattices with periodic boundary conditions have been reported [30,31,32,33], to the best of our knowledge, this is the first time where a closed form for the logarithmic sum is achieved for the infinite lattice
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