Complex orders and chirality in the classical Kitaev-$\Gamma$ model

It is well-recognized that the low-energy physics of many Kitaev materials is governed by two dominant energy scales, the Ising-like Kitaev coupling $K$ and the symmetric off-diagonal $\Gamma$ coupling. An understanding of the interplay between these two scales is therefore the natural starting point toward a quantitative description that includes sub-dominant perturbations that are inevitably present in real materials. The present study focuses on the classical $K$-$\Gamma$ model on the honeycomb lattice, with a specific emphasis on the region $K<0$ and $\Gamma>0$, which is the most relevant for the available materials and which remains enigmatic in both quantum and classical limits, despite much effort. We employ large-scale Monte Carlo simulations on specially designed finite-size clusters and unravel the presence of a complex multi-sublattice magnetic orders in a wide region of the phase diagram, whose structure is characterized in detail. We show that this order can be quantified in terms of a coarse-grained scalar-chirality order, featuring a counter-rotating modulation on the two spin sublattices. We also provide a comparison to previous studies and discuss the impact of quantum fluctuations on the phase diagram.