A system exhibiting toroidal order

This paper treats the dipolar interactions of a two-dimensional system of discs upon which a triangle of spins is mounted. We obtain the leading term of the multipole expansion of the interaction energy of discs on which is mounted a regular n-gon of spins. A definition of the toroidal magnetic moment T-i of the ith plaquette is proposed such that the magnetostatic interaction between plaquettes i and j is proportional to TiTj. The system for n=3 is shown to undergo a sequence of interesting phase transitions as the temperature is lowered. We are mainly concerned with the "solid" phase in which bond-orientational order but not positional order is long ranged. As the temperature is lowered in the solid phase, the first phase transition involving the orientation or toroidal magnetism of the discs is into a "gauge toroid" phase in which the product of a magnetic toroidal parameter and an orientation variable (for the discs) orders but due to a local gauge symmetry these variables themselves do not individually order. Finally, in the lowest temperature phase the gauge symmetry is broken and toroidal order and orientational order both develop. In the "gauge toroidal" phase time-reversal invariance is broken and in the lowest temperature phase inversion symmetry is also broken. In none of these phases is there long-range order in any Fourier component of the average spin. Symmetry considerations are used to construct the magnetoelectric free energy and thereby to deduce which coefficients of the linear magnetoelectric tensor are allowed to be nonzero. In none of the phases does symmetry permit a spontaneous polarization.