Symmetries and Exact Solutions in the Ginzburg–Landau Mean-Field Theory of the 3D XY Model

The XY model is one of the basic models of statistical mechanics. It is often used for the description of important physical systems like 4He films and liquid crystals. In the present work, we consider the Ginzburg–Landau mean-field approximation of the 3D XY model. We study the Lie-point symmetries of the corresponding Ginzburg–Landau Hamiltonian and the associated Euler–Lagrange equations for finite systems with film geometry and establish the variational symmetries among them in the presence of an external ordering field. Then, using the Noether theorem, we find conservation laws that allow us to obtain and present in analytic form, by means of Weierstrass elliptic, zeta, and sigma functions, the general solution of the so-called twisted boundary value problem in the absence of an external ordering field. It should be remarked that exact solutions in the absence of an external ordering field are only known within the so-called Ψ-theory and some other mean-field-like theories.