Functional Renormalization Group treatment of the spiral phase

Spiral magnetic order is a natural instability of the doped two-dimensional Hubbard and t–J models, and has been proposed as a candidate for the incommensurate phases observed in high-temperature cuprate superconductors. With doping, for a wide coupling range and finite next-to-nearest hopping amplitude, the Nèel antiferromagnet becomes unstable as spins rearrange to maximize carrier kinetic energy, leading to an incommensurate spiral state with ordering vector $Q \neq (\pi/a,\pi/a). This state fully breaks SU(2) spin symmetry and gives rise to three Goldstone modes: two from out-of-plane and one from in-plane spin fluctuations.At finite temperature, magnetic fluctuations of the spiral order become strong enough to destroy the long-range order, as stated by the Mermin-Wagner theorem. A remaining short-range fluctuating spiral order competes or coexists with charge and pairing instabilities and is regarded as a promising candidate for the normal state of cuprates under strong magnetic fields.

In this project, we investigate the spiral phase and its interplay with superconducting and charge-order instabilities using the functional renormalization group (fRG). To this end, we recast the fRG formulation within the single-boson exchange (SBE) representation, which not only provides a transparent physical interpretation in terms of collective bosonic fluctuations but also yields significant computational advantages.

A central element of our framework is the explicit treatment of the flow of the order parameter, which is essential to preserve the Goldstone theorem.