Phonon Hall Viscosity and the Intrinsic Thermal Hall Effect of $α$-RuCl$_3$

TL;DR

This work addresses whether phonons can intrinsically contribute to the thermal Hall effect in magnetic insulators. By measuring the acoustic Faraday effect in α-RuCl3, the authors extract a phonon Hall viscosity η_{xzyz}, linking phonon Berry curvature to a deflection of heat-carrying phonons. Their analysis shows κ_{xy} = (η_{xzyz}/ρ) C, yielding a phonon contribution to κ_{xy} on the order of κ_{xy}/κ_{xx} ~ 10^{-4}, compatible with experimental observations and suggesting phonons play a significant role alongside any spin-based excitations. The study thus establishes phonon Hall viscosity as a direct probe of phonon Berry curvature and proposes acoustic Faraday rotation as a powerful general tool for uncovering exotic states of matter where conventional probes fail.

Abstract

The thermal Hall effect has been observed in a wide variety of magnetic insulators, yet its origin remains controversial. While some studies attribute it to intrinsic origins -- such as heat carriers with Berry curvature -- others propose extrinsic origins -- such as heat carriers scattering off crystal defects. Even the nature of the heat carriers is unknown: magnons, phonons, and fractionalized spin excitations have all been proposed. These questions are significant for the study of quantum spin liquids and are particularly relevant for $α$-RuCl$_3$, where a quantized thermal Hall effect has been attributed to Majorana edge modes. Here, we use ultrasonic measurements of the acoustic Faraday effect to demonstrate that the phonons in $α$-RuCl$_3$ have Hall viscosity -- a non-dissipative viscosity that rotates phonon polarizations and deflects phonon heat currents. We show that phonon Hall viscosity produces an intrinsic thermal Hall effect that quantitatively accounts for a significant fraction of the measured thermal Hall effect in $α$-RuCl$_3$. More broadly, we demonstrate that the acoustic Faraday effect is a powerful tool for detecting phonon Hall viscosity and the associated phonon Berry curvature, offering a new way to uncover and study exotic states of matter that elude conventional experiments.

Figures (5)

• Figure 1: The thermal Hall and acoustic Faraday effects, and how they are related. Applying a magnetic field $B$ parallel to $z$ breaks time reversal symmetry and allows for both phonon Berry curvature, $\Omega_{xzyz}$, and phonon Hall viscosity, $\eta_{xzyz}$. Phonon Hall viscosity can be thought of as a long-wavelength manifestation of the microscopic Berry curvature---the double-headed arrow indicates that these two quantities are equivalent avronViscosityQuantumHall1995barkeshliDissipationlessPhononHall2012. On application of a temperature gradient along $x$ (indicated by the red-blue color gradient), phonon Berry curvature deflects phonons carrying heat along $x$ and generates a heat current along $y$ (the dashed line indicates the trajectory of a phonon in the presence of Berry curvature). Similarly, the polarization of acoustic phonons traveling along $z$ rotates at a rate that is proportional to the Hall viscosity ($\Phi$ indicates the angle of the polarization away from the $x$ axis, and the dashed line indicates the path of the polarization vector as the phonon propagates along $z$).
• Figure 2: Experimental geometry and sample characterization.a, A piezoelectric drive transducer (upper red circle) sends a pulse of traverse sound (red line) along the $c$ axis of a single-crystal $\alpha$-RuCl$_3$ sample (indicated schematically by the honeycomb). The pulse is detected by a second, receive transducer (lower red circle) polarized 45$^{\circ}$ relative to the drive transducer. b, Schematic temperature-field phase diagram of $\alpha$-RuCl$_3$ for different field orientations. The inset shows that $\theta$ is defined as the angle between the magnetic field $\bm B$ and the $c$ axis (i.e. rotation is in the $bc$ plane). We show the phase boundary for field along the $b$ axis ($\theta = 90^{\circ}$, sold line), field along the $c$ axis ($\theta = 0^{\circ}$, short dashed line), and field $45^{\circ}$ between the $b$ and $c$ axes (long dashed line). c, The relative change in sound velocity as a function of temperature across the ordering temperature of $T_{\rm{N}} = 7.5$ K. The sample exhibits a single magnetic phase transition, with no features near 14 K that would indicate a secondary structural and magnetic phase caoLowtemperatureCrystalMagnetic2016kimStructuralTransitionMagnetic2024. d, The relative change in sound velocity as a function of magnetic field across the critical field of $B_c = 8$ T for $B||b$, $B_c = 10$ T for $\theta = 45^{\circ}$, and for $B||c$ the sample remains in the ordered phase up to 12 tesla. The single phase transition indicates that the field is well-aligned with the $b$ axis balzFieldinducedIntermediateOrdered2021.
• Figure 3: Isolating the acoustic Faraday effect.a, The raw signal detected using the setup shown in \ref{['fig:geometry_coupling']}a, for $B||c = 10$ T and $T = 2$ K. Longitudinal and transverse pulses are identified based on their known speeds of sound hauspurgFractionalizedExcitationsProbed2024lebertAcousticPhononDispersion2022. The first longitudinal pulse arrives at $t_{\rm long}$, with echoes of this signal arriving at $3t_{\rm long}$ and $5t_{\rm long}$. The transverse pulse arrives at $t_{\rm trans}$ and is clearly separated in the time domain from the longitudinal signal. b, The field-symmetrized data show all longitudinal and transverse signals from 0 to 12 tesla. c, The field-antisymmetrized data contains only transverse signal because the Faraday effect rotates the transverse sound polarization in opposite directions for $\pm \bm B$, whereas it cannot rotate longitudinal polarization. d, The amplitude of transverse sound, normalized to its zero-field value, as a function of magnetic field for ${\bm B}||c$ at $T = 2$ K. The magnetic field suppresses the amplitude more for positive field than for negative field. e, This behaviour switches when the propagation direction is switched, as expected for a Faraday effect. f, The field-antisymmetrized amplitude for both propagation directions. The transverse signal shows the characteristics of a Faraday effect, going to zero at ${\bm B} = 0$, and switching sign when ${\bm k}\rightarrow-{\bm k}$. The longitudinal signal, in contrast, is zero at all fields for both propagation directions.
• Figure 4: The phonon Hall viscosity of $\alpha$-RuCl$_3$.a, The field-antisymmetrized Faraday signal normalized to the total zero-field signal, as a function of magnetic field for $B||c$, at $T = 2$ K, for $\pm k$. b, The Hall viscosity extracted from the antisymmetric signal using \ref{['eq:wave']}. The solid line is a guide to the eye. Panels c and d show the same data and analysis, but for magnetic field rotated $55^{\circ}$ toward the $b$ axis. The sharp feature slightly above $B_c$ in panel c is likely a result of the small absolute signal size and the rapid change in speed of sound near this field. We have truncated this feature from the viscosity in panel d.
• Figure 5: Temperature and field dependence of the phonon Hall viscosity.a, The phonon Hall viscosity as a function of temperature for $\mathbf{B}||c = 12$ T. Blue circles are measured data points, and the blue line is a guide to the eye. The quantity $\rho \kappa_{xy}/C$, calculated using $\kappa_{xy}$ from LeFrancois et al.lefrancoisEvidencePhononHall2022 and $C$ from Widman et al.widmannThermodynamicEvidenceFractionalized2019, is plotted in red for comparison. b, The phonon Hall viscosity measured with $\mathbf{B}$ rotated 55$^{\circ}$ from $c$ toward $b$, plotted as a function of in-plane magnetic field at $\it{T}=$ 2 K. The $\kappa_{xy}$ data are measured with $\mathbf{B}||a$ and are taken from Czajka et al.czajkaPlanarThermalHall2023.