[Paper Review] Phonon Hall Viscosity in Magnetic Insulators

Mengxing Ye, L. Savary+1 more|arXiv (Cornell University)|Mar 7, 2021

Physics of Superconductivity and Magnetism4 citations

TL;DR

This paper presents a systematic theoretical framework for calculating phonon Hall viscosity (PHV) in magnetic insulators induced by spin-lattice coupling under an external magnetic field. Using symmetry analysis and microscopic field theory, it derives a general criterion for non-zero PHV, showing that PHV arises when time-reversal and specific mirror symmetries are broken, and explicitly computes PHV coefficients in the cuprate Sr2CuO2Cl2, linking them to spin correlators and enabling experimental detection via acoustic Faraday rotation and thermal Hall effect.

ABSTRACT

The Phonon Hall Viscosity is the leading term evincing time-reversal symmetry breaking in the low energy description of lattice phonons. It may generate phonon Berry curvature, and can be observed experimentally through the acoustic Faraday effect and thermal Hall transport. We present a systematic procedure to obtain the phonon Hall viscosity induced by phonon-magnon interactions in magnetic insulators under an external magnetic field. We obtain a general symmetry criterion that leads to non-zero Faraday rotation and Hall conductivity, and clarify the interplay between lattice symmetry, spin-orbit-coupling, external magnetic field and magnetic ordering. The symmetry analysis is verified through a microscopic calculation. By constructing the general symmetry-allowed effective action that describes the spin dynamics and spin-lattice coupling, and then integrating out the spin fluctuations, the leading order time-reversal breaking term in the phonon effective action, i.e. the phonon Hall viscosity, can be obtained. The analysis of the square lattice antiferromagnet for a cuprate Mott insulator, Sr$_2$CuO$_2$Cl$_2$, is presented explicitly, and the procedure described here can be readily generalized to other magnetic insulators.

Motivation & Objective

• To establish a general theoretical framework for phonon Hall viscosity (PHV) in magnetic insulators driven by magnetoelastic coupling.
• To identify the symmetry conditions under which PHV becomes non-zero, particularly in the presence of external magnetic fields and spin order.
• To connect PHV to measurable quantities such as acoustic Faraday rotation and thermal Hall conductivity.
• To provide a microscopic derivation of PHV coefficients in the square-lattice antiferromagnet Sr2CuO2Cl2 using effective field theory and spin-1/S expansion.

Proposed method

• Construct a general symmetry-allowed effective action for spin dynamics and spin-lattice coupling in magnetic insulators.
• Integrate out spin degrees of freedom to derive the leading-order time-reversal-odd term in the phonon effective action, corresponding to PHV.
• Apply group theory to classify irreducible representations of strain and spin operators, identifying allowed PHV channels.
• Use 1/S expansion to compute PHV coefficients in terms of spin correlators, such as ⟨m_y n_z⟩.
• Analyze both out-of-plane and in-plane magnetic field configurations, identifying symmetry-protected vanishing PHV terms.
• Derive explicit expressions for PHV terms in momentum space, linking them to phonon Berry curvature and experimental observables.

Experimental results

Research questions

• RQ1 Under what symmetry conditions does phonon Hall viscosity (PHV) become non-zero in magnetic insulators under an external magnetic field?
• RQ2 How does spin-orbit coupling and magnetic ordering influence the emergence of PHV in the presence of magnetoelastic coupling?
• RQ3 What is the microscopic origin of PHV in the cuprate Sr2CuO2Cl2, and how can it be computed from spin correlators?
• RQ4 How do in-plane versus out-of-plane magnetic fields affect the selection rules for PHV?
• RQ5 What is the relationship between PHV and experimentally measurable effects such as acoustic Faraday rotation and phonon thermal Hall conductivity?

Key findings

• The phonon Hall viscosity (PHV) coefficient η^H_A1Ex is proportional to h⟨m_y n_z⟩, with similar dependence for other channels, showing direct link to spin correlators.
• PHV vanishes when time-reversal and mirror symmetries (e.g., σ_h and σ_vx) are preserved, establishing a clear symmetry criterion for non-zero PHV.
• For in-plane magnetic fields along y, the effective magnetic group preserves T C2z symmetry, which forces η^H_ExEy = 0, suppressing certain PHV channels.
• The leading-order PHV coefficients in the 1/S expansion are determined by spin correlators such as ⟨m_y n_z⟩, which are non-zero in the presence of magnetic order and external field.
• The analysis confirms that PHV can be generated via magnetoelastic coupling in the absence of intrinsic spin-orbit coupling, provided time-reversal and specific mirror symmetries are broken.
• The derived PHV terms are directly linked to the phonon Berry curvature and can be probed via acoustic Faraday rotation and thermal Hall transport, providing a route to experimental detection.