Finite-$q$ antiferrotoroidal and ferritoroidal order in a distorted kagome structure

A highly geometrically frustrated lattice structure such as a distorted kagome (or quasikagome) structure enriches physical phenomena through coupling with the electronic structure, topology, and magnetism. Recently, it has been reported that an intermetallic HoAgGe exhibits two distinct magnetic structures with the finite magnetic vector $q=(1/3,1/3,0)$: One is the partially ordered state in the intermediate-temperature region, and the other is the kagome spin ice state in the lowest-temperature region. We theoretically elucidate that the former is characterized by antiferrotoroidal ordering, while the latter is characterized by ferritoroidal ordering based on the multipole representation theory, which provides an opposite interpretation to previous studies. We also show how antiferrotoroidal and ferritoroidal orderings are microscopically formed by quantifying the magnetic toroidal moment activated in a multiorbital system. As a result, we find that the degree of distortion for the kagome structure plays a significant role in determining the nature of antiferrotoroidal and ferritoroidal orderings, which brings about the crossover between the antiferro-type and the ferri-type distributions of the magnetic toroidal dipole. We confirm such a tendency by evaluating the linear magnetoelectric effect. Our analysis can be applied irrespective of lattice structures and magnetic vectors without annoying the cluster origin.