Develop a Landau theory for the triangular-to-stripe transition

Develop a Landau theory framework for the structural order parameter governing the triangular-to-stripe transition in a dipolar quantum gas under dipole tilt.

Background

The paper experimentally observes competing triangular and stripe density-modulated states in a two-dimensional dipolar quantum gas and uses a structural order parameter, defined from the relative amplitudes of characteristic finite-wavevector peaks in the density power spectrum, to distinguish the two arrangements. The authors identify enhanced fluctuations in the intermediate tilt regime as evidence of a structural transition between triangular and stripe orders.

Despite introducing and experimentally analyzing this order parameter, the theoretical framework needed to characterize the transition has not been established. In particular, the authors state that no Landau theory currently describes the order parameter for the triangular-to-stripe transition under dipole tilt, leaving the symmetry-based theoretical characterization of this transition unresolved.