Unifying approach to Heisenberg models for spin distributions on curved surfaces of revolution with isothermal (conformal) metrics

Abstract

In the present article, a unifying approach to description of different Heisenberg models (i.e., XY model, isotropic case, easy‐plane, and easy‐axis regimes) for spin distributions on curved surfaces is discussed. After introduction of isothermal (conformal) metrics on surfaces of revolution, the axially symmetric solution of the Euler–Lagrange equation for the quasi‐one‐dimensional magnetic Hamiltonian has been obtained for all four discussed cases in the form of an incomplete elliptic integral of the third kind. It has been shown that the limiting case of XY model can be achieved from an easy‐plane regime when the parameter of anisotropy of interaction amongst spins is approaching −1, while the isotropic case can be achieved from both easy‐plane and easy‐axis regimes when is approaching 0. After the application of the corresponding boundary conditions, the soliton‐like solutions have been obtained that either cover the whole surface (in a one‐twist or multi‐twist manner) or can be arranged in the form of a periodic lattice structure on the curved surface. The general unifying scheme has been illustrated using the exemplary surfaces of revolution with zero (cylinder), negative (catenoid), and positive (sphere) Gaussian curvatures.