A Gallery of Soft Modes: Theory and Experiment at a Ferromagnetic Quantum Phase Transition

TL;DR

This paper investigates the low-energy excitations near the ferromagnetic quantum critical point in LiHoF$_4$, combining microwave spectroscopy with theoretical modeling to understand soft modes, electronuclear interactions, and domain dynamics.

Contribution

It provides a detailed experimental and theoretical analysis of soft electronuclear modes and their interactions at a ferromagnetic quantum phase transition.

Findings

Identification of a soft mode at the quantum critical point.

Observation of competing infrared divergences from photons and electronuclear modes.

Detection of Walker modes related to ferromagnetic domain dynamics away from criticality.

Abstract

We examine the low-energy excitations in the vicinity of the quantum critical point in LiHoF$_4$, a physical realization of the Transverse Field Ising Model, focusing on the long-range fluctuations which soften to zero energy at the ferromagnetic quantum phase transition. Microwave spectroscopy in tunable loop-gap resonator structures identifies and characterizes the soft mode and higher-energy electronuclear states. We study these modes as a function of frequency and magnetic fields applied transverse and parallel to the Ising axis. These are understood in the context of a theoretical model of a soft electronuclear mode that interacts with soft photons as well as soft phonons. We identify competing infrared divergences at the quantum critical point, coming from the photons and the electronuclear soft mode. It is an incomplete cancellation of these divergences that leads to the muted but distinct signatures observed in the experiments. The application of a longitudinal magnetic field gaps the soft mode. Measurements well away from the quantum critical point reveal a set of ``Walker'' modes associated with ferromagnetic domain dynamics.