Grgur Palle, University of Illinois, Urbana-Champaign

Consistent Treatment of Multiple Vestigial Orders within Saddle-Point Approximations

When a primary order breaks multiple symmetries, partially ordered phases, known as vestigial phases, may arise which breaks only some of them at a higher temperature. Yet for a given primary order, there are generically multiple possible vestigial orders, each one described by a composite order parameter (OP) that is quadratic in the multi-component OP of the primary order. Which vestigial order prevails is dictated by the quartic self-interactions of the primary OP. However, within saddle-point analyses of this problem, there is a large ambiguity in the results. Because the different vestigial channels feed into each other, as specified by the Fierz identities, one can apparently enhance or eliminate altogether the condensation tendency in any vestigial channel. Here, we resolve this ambiguity. By carefully introducing a large-N limit which respects both the initial group structure and the Fierz identities, we deduce a unique prescription for how the saddle-point approximation (SPA) must be carried out. Compared to previous SPA, we find non-trivial competition between the different vestigial channels, as specified by the symmetrized quartic coefficients. In addition, we establish agreement of our SPA to mean-field variational and self-consistent Hartree-Fock approximations at weak coupling. We illustrate our prescription on three examples: X-point charge-density waves in primitive tetragonal systems, M-point spin-density waves in hexagonal systems, and multi-component superconductivity in cubic systems.