Anisotropic Spin Hamiltonians due to Spin-Orbit and Coulomb Exchange Interactions

TL;DR

This paper derives and analyzes anisotropic spin Hamiltonians in cuprate systems considering spin-orbit and Coulomb interactions, supported by analytical and numerical methods, revealing symmetry-dependent anisotropy effects.

Contribution

It provides a comprehensive derivation of anisotropic spin Hamiltonians including spin-orbit and Coulomb effects, with analytical proofs and numerical validation for various symmetries.

Findings

H is rotationally invariant without Coulomb exchange.

Presence of Coulomb exchange induces biaxial anisotropy.

Anisotropy order depends on symmetry and Coulomb interaction variation.

Abstract

This paper contains the details of Phys. Rev. Lett. 73, 2919 (1994) and, to a lesser extent, Phys. Rev. Lett. 72, 3710 (1994). We treat a Hubbard model which includes all the 3d states of the Cu ions and the 2p states of the O ions. We also include spin-orbit interactions, hopping between ground and excited crystal field states of the Cu ions, and rather general Coulomb interactions. Our analytic results for the spin Hamiltonian, H, are corroborated by numerical evaluations of the energy splitting of the ground manifold for two holes on either a pair of Cu ions or a Cu-O-Cu complex. In the tetragonal symmetry case and for the model considered, we prove that H is rotationally invariant in the absence of Coulomb exchange. When Coulomb exchange is present, each bond Hamiltonian has full biaxial anisotropy, as expected for this symmetry. For lower symmetry situations, the single bond spin Hamiltonian is anisotropic at order t**6 for constant U and at order t**2 for nonconstant U. (Constant U means that the Coulomb interaction between orbitals does not depend on which orbitals are involved.)