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Spiral-spin-liquid behaviors and persistent reciprocal kagomé structure in frustrated van der Waals magnets and beyond

We study classical $J_1$-$J_2$ models with distinct spin degrees of freedom on a honeycomb lattice. For the XY and Heisenberg spins, the system develops a spiral spin liquid (SSL) that is a thermal cooperative paramagnetic regime with spins fluctuating around the spiral contours in the momentum space, and at low temperatures supports a vector spin-chirality order despite the absence of long-range magnetic order. In a strong contrast, for the Ising moments, the low-temperature spin correlation forms a reciprocal "kagomé" structure in the momentum space that resembles the SSL behaviors and persists for a range of exchange couplings. The unexpected emergence and persistence of the reciprocal "kagomé" are attributed to the stiffness of the Ising moments and the frustration. At higher temperatures when the thermal fluctuations is strong and the spin correlation is not fully melted, the reciprocal structures evolve from the "kagomé" towards the ones demanded by the soft spin limit. This contrasts strongly with the behaviors of the spiral contours in the SSL regime for the continuous spins. We suggest various experimentally relevant systems including van der Waals magnets such as the transition metal phosphorus trichalcogenides TMPX$_3$, Cr$_2$Ge$_2$Te$_6$, the rare-earth chalcohalides (like HoOF, ErOF and DyOF) and other isostructural systems to realize the SSL-like behaviors and/or the reciprocal kagomé structure.