Noether's theorem and low symmetry aspects concerning the crystal (ligand) field Hamiltonians and spin Hamiltonians

Abstract

This review presents a concise summary of major findings arising from our recent studies concerning the symmetry properties of crystal/ligand field (CF/LF) Hamiltonians and spin Hamiltonians (SHs). First we provide a bird's-eye view of these studies. Then we overview (i) the pertinent basic concepts and notations, (ii) the algebraic symmetry (AS) of Hamiltonians for continuous rotational symmetry (CRS) cases, and (iii) the concepts of the rotational invariants and moments of CF Hamiltonians. This enables a new look from the point of view of the Noether's theorem on the properties of CF/LF Hamiltonians and SHs invariant under CRS, i.e. hexagonal II, tetragonal II, trigonal II, monoclinic, and triclinic ones. An important theorem and a conjecture on the conserved quantities stipulated by Noether's theorem for the Hamiltonians in question formulated by us helps to elucidate the interrelationships and deeper meaning of the concepts involved. Implications of the existence of the conserved quantities for interpretation of experimental CF parameter (CFP) datasets are encapsulated in five corollaries. These considerations reveal that the feasibility of determination of CFPs from fitting experimental spectra and the reduction of the existing higher-order rotational invariants for hexagonal type II and cubic symmetry require reinterpretation. This novel approach enables adoption of better fitting strategies utilizing welldefined conserved quantities, which are invariant under CRS. The advantages of this approach are illustrated using the CFP datasets reported in literature for RE3+ (4fN) ions in LiYF4. This review deals also with the fundamental intricate aspects, hitherto not fully understood, concerning the CF Hamiltonians for the 'low symmetry' cases, including the CRS cases as well as orthorhombic ones. This includes: (1) selection of the axis systems, (2) types of CF parameters and their properties, (3) introduction of a new notion of a nominal axis system for the fitted CFP datasets, (4) implications of the Noether's theorem and the AS of CF Hamiltonians, (5) correlation properties among CFP datasets, (6) the rotational degrees of freedom and the reduction of the number of independent CFPs, and (7) extension of the multiple correlated fitting technique. Clarification of these intricate aspects enables us to provide a general framework aimed at achieving an increased compatibility and reliability of CFP datasets for transition ions at low symmetry sites in crystals. The usefulness of this framework is illustrated by reanalysis of the CFP datasets for Nd3+ (Pr3+ ) in NdGaO3 (PrGaO3) and RNiO3.