Multi-modal spectroscopy of phase transitions

To understand a phase transition, independent measurement of the value and variation in each physical parameter of a material's Hamiltonian is vital. Conventional one-dimensional spectroscopy, which studies dynamical responses to fields, struggles to distinguish between different sources of noise. Multi-dimensional spectroscopy can avoid this issue and probe symmetry-specific Hamiltonian parameters by analyzing how the time delay between applied pulses ($\tau$) affects the response. In this work, we present a spectroscopic technique based on the multi-dimensional paradigm which can measure a quadrupolar interaction (inversion symmetric) even in the presence of large magnetic noise (inversion asymmetric). Inversion symmetric combinations of spin operators are found to give clear sinusoidal responses in $\tau$ due to periodic refocusing. The time-scale on which the magnetization partially decays in $\tau$ provides a direct measure of the distribution of interaction strengths, even when the average value of the interaction is zero. This method independently measures the distributions of different forms of disorder, helping elucidate which microscopic symmetry drives a phase transition.Comment: 14 pages, 8 figure