Systematic Derivation of Spin Hamiltonian with Higher-Order Terms by Spin S Representation

It is known that complex magnetism emerge from competing Heisenberg interaction and higher-order spin interactions [1]. In addition, previously unknown interactions, such as the Chiral-Chiral interaction due to topological orbital magnetism, have been found to be essential in explaining magnetism [2]. It is beginning to be recognized that the higher order terms of such spin interactions depend not only on the number of sites of magnetic ions, but also on the local spin magnitude S. However, a method to uniquely derive a spin Hamiltonian that captures the spin interaction for a given system has not yet been established. In this presentation, we focus on the algebraic nature of the spin operator and give an exact spin Hamiltonian of the isotropic spin interaction for a given total local spin S and the number of magnetic sites. The derivation of the spin Hamiltonian is obtained from operators with the spin S representation depending on the spin magnitude. By organizing the algebraic structure followed by the general spin operators, we discuss the construction of the exact spin Hamiltonian, especially for S=1/2 and 1, including the higher-order terms of the interaction. Higher-order terms of scalar spin chirality obtained by this algebraic method will also be discussed. These results have applications not only to magnetism in solids, but also to nuclei or cold atom systems.

[1] M. Hoffmann et al., Phys. Rev. B 101, 024418 (2020).

[2] S. Grytsiuk et al., Nat. Commun. 11, 511 (2020).