Coupled fields in one dimension : cross-over from a continuous symmetry to a discrete one

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Abstract
We study a pedagogical example of a one dimensional Landau Ginzburg functional with a symmetry-breaking perturbation which causes a cross-over behaviour from a continuous symmetry (n = 2) to a discrete one. The correlation length is shown to be governed by the lowest energy defects, which are solitons in the weak anisotropy limit. As the symmetry breaking term is introduced in the Hamiltonian the Goldstone mode transforms into solitons, the energy of which is proportional to the square root of the anisotropy — their characteristic localization length is inversely proportional to the square root of the anisotropy. Application to various physical cases is discussed : dilute magnet near the percolation threshold, three dimensional array of weakly coupled chains and structural transitions