Topological solitons of the Lawrence–Doniach model as equilibrium Josephson vortices in layered superconductors

Abstract

We present a complete, exact solution of the problem of the magnetic properties of layered superconductors with an infinite number of superconducting layers in parallel fields H>0. Based on a new exact variational method, we determine the type of all stationary points of both the Gibbs and Helmholtz free-energy functionals. For the Gibbs free-energy functional, they are either points of strict, strong minima or saddle points. All stationary points of the Helmholtz free-energy functional are those of strict, strong minima. The only minimizers of both the functionals are the Meissner (0-soliton) solution and soliton solutions. The latter represent equilibrium Josephson vortices. In contrast, nonsoliton configurations (interpreted in some previous publications as “isolated fluxons” and “fluxon lattices”) are shown to be saddle points of the Gibbs free-energy functional: They violate the conservation law for the flux and the stationarity condition for the Helmholtz free-energy functional. For stable solutions, we give a topological classification and establish a one-to-one correspondence with Abrikosov vortices in type-II superconductors. In the limit of weak interlayer coupling, exact, closed-form expressions for all stable solutions are derived: They are nothing but the “vacuum state” and topological solitons of the coupled static sine-Gordon equations for the phase differences. The stable solutions cover the whole field range 0⩽H<∞ and their stability regions overlap. Soliton solutions exist for arbitrary small transverse dimensions of the system, provided the field H is sufficiently high. Aside from their importance for weak superconductivity, the new soliton solutions can find applications in different fields of nonlinear physics and applied mathematics.