[Paper Review] Exchange interaction between $J$-multiplets

Naoya Iwahara, Liviu F. Chibotaru|arXiv (Cornell University)|Mar 9, 2015

Rare-earth and actinide compounds4 citations

TL;DR

This paper derives analytical expressions for exchange interactions between J-multiplets in magnetic ions with unquenched orbital angular momentum, using a complete microscopic model that includes relativistic effects. It demonstrates that the commonly used isotropic Heisenberg form $\mathcal{J}\mathbf{J}_1\cdot\mathbf{J}_2$ and the $1/U$ approximation are invalid for such systems, revealing instead a complex, anisotropic exchange Hamiltonian governed by electronic matrix elements from ab initio calculations.

ABSTRACT

Analytical expressions for the exchange interaction between $J$-multiplets of interacting metallic centers are derived on the basis of a complete electronic model. A common belief that this interaction can be approximated by an isotropic form $\propto {\mathbf{J}}_1\cdot{\mathbf{J}}_2$ (or $\propto {\mathbf{J}}_1\cdot{\mathbf{S}}_2$ in the case of interaction with an isotropic spin) is found to be ungrounded. It is also shown that the often used "1/U approximation" for the description of the kinetic contribution of the exchange interaction is not valid in the case of $J$-multiplets. The developed theory can be used for microscopic description of exchange interaction in materials containing lanthanides, actinides and some transition metal ions.

Motivation & Objective

• To provide a rigorous microscopic derivation of exchange interaction between J-multiplets in magnetic ions with unquenched orbital momentum.
• To challenge the widespread use of the isotropic Heisenberg Hamiltonian $\mathcal{J}\mathbf{J}_1\cdot\mathbf{J}_2$ for lanthanide and actinide systems.
• To demonstrate the failure of the $1/U$ approximation in describing kinetic exchange contributions for J-multiplets.
• To establish a framework for computing exchange parameters from electronic structure calculations, enabling accurate modeling of magnetic materials with strong spin-orbit coupling.

Proposed method

• Derives the exchange Hamiltonian from a complete electronic model including spin-orbit coupling and relativistic effects on metallic centers.
• Uses second-order perturbation theory to express the exchange interaction in terms of electronic matrix elements between intermediate states.
• Applies the Wigner-Eckart theorem to reduce matrix elements to reduced matrix elements and fractional parentage coefficients.
• Constructs the exchange Hamiltonian in the basis of total angular momentum eigenstates $|J,M\rangle$ for both interacting centers.
• Performs numerical diagonalization of the full exchange Hamiltonian to compare with simplified models.
• Validates results using ab initio-derived matrix elements and compares eigenstates with those from the Heisenberg and $1/U$ approximations.

Experimental results

Research questions

• RQ1 Is the isotropic Heisenberg Hamiltonian $\mathcal{J}\mathbf{J}_1\cdot\mathbf{J}_2$ a valid approximation for exchange interaction between J-multiplets in lanthanides and actinides?
• RQ2 Does the $1/U$ approximation accurately describe the kinetic contribution to exchange interaction in systems with unquenched orbital momentum?
• RQ3 How do the eigenstates of the true exchange Hamiltonian differ from those predicted by the Heisenberg and $1/U$ models?
• RQ4 What is the role of higher-order terms in the total angular momentum $\mathbf{J}$ in determining the exchange interaction?
• RQ5 Can the exchange parameters be reliably computed from electronic structure calculations using the derived formalism?

Key findings

• The isotropic Heisenberg form $\mathcal{J}\mathbf{J}_1\cdot\mathbf{J}_2$ is not justified for J-multiplets and leads to significant errors in predicting low-energy eigenstates.
• The $1/U$ approximation fails to describe the kinetic exchange contribution for J-multiplets, as it neglects essential matrix element dependencies.
• The true exchange Hamiltonian exhibits strong anisotropy and non-bilinear terms in $\mathbf{J}_1$ and $\mathbf{J}_2$, deviating fundamentally from the Heisenberg form.
• Numerical comparison shows that eigenstates of the full exchange Hamiltonian differ significantly from those of the Heisenberg and $1/U$ models, especially in degenerate or near-degenerate manifolds.
• The exchange parameters are expressed in terms of ab initio-computable matrix elements, enabling direct microscopic modeling of lanthanide and actinide compounds.
• For the N₂²⁻-bridged Dy₃⁺ dimer, the full exchange Hamiltonian yields energy levels and state compositions that differ substantially from predictions of the Heisenberg and $1/U$ models, confirming the breakdown of these approximations.