Codimension, an integer number that corresponds to the reduced dimensions of a submanifold relative to the complete vector space, has been proven as a powerful tool in describing geometry-related physical laws. Its applications span from the Ads/CFT correspondence in string theory (see e.g. PRD 2020) to the high-order Weyl semimetals in condensed matter physics (see e.g. PRL 2020). In a recent review paper by X.-P. Yao et al (Front. Phys. 16, 53303), the concept of codimension was applied to the classification of spiral spin-liquids, which are exotic correlated paramagnetic states induced by magnetic frustration. In such states, spins fluctuate collectively as spirals, and their propagation vectors, q, form a continuous submanifold in reciprocal space, called the spiral surface. According to the codimension classification, all the previously established spiral spin-liquids in real materials fall in the category of codimension one, where the dimension of the spiral surface is reduced by one compared to that of the hosting lattice. The identification of spiral spin-liquids with a higher codimension may shed lights on many novel phenomena related to spiral spin-liquids, including the order-by-disorder transition that has been elusive till now.

In our recent post, arXiv2405.18973, we study the effective honeycomb-lattice compound Cs3Fe2Cl9 by neutron scattering and demonstrate the emergence of a codimension-two spiral spin-liquid. Through analysis of the inelastic neutron scattering spectra, we establish the spin Hamiltonian in Cs3Fe2Cl9, revealing comparable intralayer and interlayer interactions that stabilize spiral spin-liquids. Through diffuse neutron scattering experiments on a single crystal sample, we directly observe the one-dimensional spiral surface in this three-dimensional lattice compound and demonstrate how does its unique codimension two affect the scattering intensity over the spiral surface.

Besides the codimension-two spiral spin-liquid phase, our study also clarifies the magnetic long-range ordered phases at low temperatures. Previous transport characterizations revealed a rich phase diagram in field (PRB 2021), but the structures and origins of the field-induced phases remain unknown. By combining single-crystal neutron diffraction experiments and classical Monte Carlo simulations, we determined the magnetic structures of the eight field-induced phases, whose origins are attributed to the competition among thermal fluctuations, magnetic frustration, and single-ion anisotropy. Most importantly, in the presence of a magnetic field, the long-range ordered phase at temperatures just below the spiral spin-liquid phase is observed to exhibit a propagation vector that is the same as that predicted by the order-by-disorder transition. This observation contrasts the previously known spiral spin-liquid hosts where the propagation vector of the magnetic long-range ordered phase is determined by perturbative interactions. Our work thus establishes Cs3Fe2Cl9 as a spiral spin-liquid host that may realize the elusive order-by-disorder transition.

This work utilizes JuliaSCGA.jl, a package developed in our lab for the calculations of equal-time spin correlations in classical spin systems.