A System Exhibiting Toroidal Order

TL;DR

This paper explores a two-dimensional disc system with spins, revealing a sequence of phase transitions involving toroidal and orientational order, including a gauge toroid phase and symmetry-breaking at low temperatures.

Contribution

It introduces a novel model exhibiting toroidal order and gauge symmetry breaking, with detailed analysis of phase transitions and symmetry considerations.

Findings

Identification of a gauge toroid phase with broken time reversal symmetry.

Demonstration of symmetry breaking and order development at low temperatures.

Proposal of a new definition for the toroidal magnetic moment.

Abstract

A two dimensional system of discs upon which a triangle of spins are mounted 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 charge 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. A definition of the toroidal magnetic moment $T_i$ of the $i$th plaquette is proposed such that the magnetostatic interaction between plaquettes $i$ and $j$ is proportional to $T_iT_j$. 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.