Composite spin and quadrupole wave in the ordered phase of Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$

TL;DR

This paper investigates the composite spin and quadrupole wave excitations in the hidden ordered state of Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$, revealing how pseudospin waves involve both magnetic and electric multipolar moments.

Contribution

It introduces a linear spin-wave theory calculation of pseudospin waves in the quadrupolar ordered phase of Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$, linking theory with observed neutron scattering spectra.

Findings

Pseudospin waves involve both magnetic and electric quadrupole moments.

Calculated spectra match low-energy excitations observed in experiments.

The study clarifies the nature of hidden order in this frustrated pyrochlore.

Abstract

The hidden ordered state of the frustrated pyrochlore oxide Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$ is possibly one of the two electric multipolar, or quadrupolar, states of the effective pseudospin-1/2 Hamiltonian derived from crystal-field ground state doublets of non-Kramers Tb$^{3+}$ ions. These long-range orders are antiparallel or parallel alignments of transverse pseudospin components representing electric quadrupole moments, which cannot be observed as magnetic Bragg reflections by neutron scattering. However pseudospin waves of these states are composite waves of the magnetic-dipole and electric-quadrupole moments, and can be partly observed by inelastic magnetic neutron scattering. We calculate these spin-quadrupole waves using linear spin-wave theory and discuss previously observed low-energy magnetic excitation spectra of a polycrystalline sample with $x = 0.005$ ($T_{\text{c}} = 0.5 $ K).