Thermodynamic Spectroscopy of Emergent Excitations in Quantum Spin Ice
Quantum spin ice (QSI) is a three-dimensional quantum spin liquid where fractionalized spinons interact with emergent photons. The XXZ model on the pyrochlore lattice realizes such a $U(1)$ quantum spin liquid and a number of pyrochlore magnets have been investigated as candidate materials, but the detection of emergent excitations has been a major challenge. The specific heat is expected to show the higher-energy spinon excitations and a lower-energy anomaly at the ring-exchange scale, which sets both the photon bandwidth and the energy of the emergent magnetic monopoles (or visons). In reality, the lower-energy peak is obscured by the nuclear Schottky anomaly in non-Kramers Pr-based systems, while it is not clearly resolved from the spinon contributions in Ce-based dipolar-octupolar systems. In this work, we propose a novel thermodynamic probe of the elusive ring-exchange energy scale. We show that in the presence of a weak perturbation coupled to the transverse component of the pseudospin degrees of freedom, the temperature derivative of an observable conjugate to such a weak perturbing force is highly sensitive to the ring-exchange energy scale. Using this scheme, it is shown that the difference between thermal expansion coefficients along the [100] and [010] directions should show a peak at the ring-exchange energy scale for non-Kramers QSI. Similarly, the temperature derivative of the magnetization, $dM/dT$, of dipolar-octupolar pyrochlores under a weak magnetic field can also detect the same signal. Moreover, the sign of these signatures distinguishes the zero-flux and $\pi$-flux QSI states.