Phonon Hall Viscosity Overview

• Phonon Hall viscosity is the dissipationless, antisymmetric viscoelastic response of lattice vibrations enabled by broken time-reversal symmetry and Berry curvature effects.
• It is modeled as an odd-in-time derivative term in the phonon effective action, with tensor structures that reveal unique acoustic properties such as mode mixing and Faraday rotation.
• The phenomenon underpins experimental probes like thermal Hall transport and acoustic measurements, offering insights into topological phases and symmetry constraints in diverse materials.

Searching arXiv for recent and foundational papers on phonon Hall viscosity and closely related Hall viscosity frameworks. Phonon Hall viscosity is the dissipationless, antisymmetric component of the viscoelastic response of lattice vibrations that becomes allowed when time-reversal symmetry is broken. In the low-energy theory of acoustic phonons it appears as an odd-in-time-derivative term coupling strain to strain rate, and in the elastic wave equation it enters as

$$\sigma_{ij} = c_{ijkl}\varepsilon_{kl} + \eta_{ijkl}\dot{\varepsilon}_{kl},$$

with $\eta_{ijkl}$ the Hall-viscosity tensor (Shragai et al., 7 Oct 2025). The broader Hall-viscosity literature established the geometric basis for such terms: Hall viscosity is dissipationless, can be expressed as a Berry-curvature response to geometric deformations, may be quantized in some topological phases, and does not in general require rotational invariance (0906.1854, Hoyos, 2014). In phononic settings, these features reappear as phonon chirality, acoustic Faraday rotation, longitudinal–transverse mode mixing, and intrinsic contributions to thermal Hall transport (Barkeshli et al., 2011, Ye et al., 2021).

1. Foundational formulation

The standard low-energy phonon action contains kinetic and elastic terms in the displacement field $u_i$. When the underlying electronic or magnetic sector is gapped and breaks time-reversal symmetry, the effective action can acquire the Hall-viscosity correction

$$\delta S_H = \frac{1}{2}\int d^d x\, dt \; \eta_{ijkl}\,\partial_i u_j\,\partial_k \dot{u}_l,$$

with antisymmetry

$\eta^{(A)}_{ijkl} = -\eta^{(A)}_{klij}.$

This is the crystalline analog of the dissipationless Hall-viscosity term discussed for topological fluids and effective field theories (Barkeshli et al., 2011, Hoyos, 2014).

In two-dimensional isotropic media, the antisymmetric tensor reduces to a single Hall-viscosity coefficient,

$$\eta^{(A)}_{ijkl} = -\frac{\eta_H}{2} \left( \epsilon^{ik}\delta^{jl} +\epsilon^{jk}\delta^{il} +\epsilon^{il}\delta^{jk} +\epsilon^{jl}\delta^{ik} \right),$$

which makes explicit that the response is transverse and parity-odd (Hoyos, 2014). In this form, Hall viscosity is the odd-viscous analog of a Hall conductivity: it modifies the reactive stress response without contributing to entropy production.

A useful distinction emerges between acoustic and optical phonons. For acoustic phonons, the natural object is typically a rank-four tensor $\eta_{ijkl}$, while for optical phonons in thin-film Weyl semimetals the effective dynamics instead involve a dissipationless two-rank Hall-viscosity tensor $\eta^H_{ij}$ (Hiedari et al., 2019). This difference is not merely notational; it reflects the distinct kinematics of Goldstone lattice deformations and zone-center optical mode coordinates.

2. Geometric and microscopic origins

A central result of the early phonon Hall-viscosity literature is that the coefficient is determined by the adiabatic Berry curvature of the underlying gapped sector as a function of strain or distortion parameters. In the adiabatic regime, where phonon frequencies are much smaller than the electronic gap, the electrons remain in their instantaneous ground state, and the odd-viscous term arises from the geometric phase accumulated under slow lattice deformations (Barkeshli et al., 2011). This is directly parallel to the geometric interpretation of Hall viscosity in incompressible Hall fluids, where the viscosity is tied to Berry curvature, intrinsic metric data, and topological quantities such as the shift (0906.1854, Hoyos, 2014).

In magnetic insulators, a systematic derivation proceeds by writing the symmetry-allowed spin-lattice action and integrating out spin fluctuations. The resulting phonon Hall viscosity is the leading time-reversal-breaking term in the low-energy phonon effective action, and its coefficient is controlled by odd-in-frequency retarded correlators of spin composite operators coupled to strain (Ye et al., 2021). The same logic underlies several more specialized settings. In the Kitaev spin liquid, time-reversal breaking gaps the Majorana Dirac cones and endows the phonons with a Hall-viscosity term generated by the Berry curvature of the gapped Majorana bands (Ye et al., 2020). In square-lattice Néel states proximate to a transition into a phase with semion topological order, spinon-phonon coupling produces a nonanalytic phonon Hall viscosity whose second derivative diverges at zero temperature across the transition (Zhang et al., 2021).

Other mechanisms are materially distinct but conceptually analogous. In honeycomb Dirac magnon systems, elastic deformations generate gauge-like couplings to magnon pseudospinors; with a Dzyaloshinskii-Moriya-induced topological gap, integrating out the magnons yields a Chern-Simons term for the elastic gauge fields and hence a phonon Hall viscosity (Ferreiros et al., 2017). In ionic crystals, Lorentz forces on moving cations and anions in a magnetic field provide an intrinsic mechanism that does not require coupling to electronic or spin degrees of freedom; integrating out the out-of-phase optical motion produces an effective Hall viscosity for the acoustic sector (Flebus et al., 2022). In Weyl semimetals, optical phonons can couple to Weyl electrons as chiral pseudo-gauge fields, and the resulting Hall viscosity contains both an anomaly-induced contribution and a conventional magnetic-field contribution (Hiedari et al., 2019).

3. Symmetry, topology, and tensor structure

Time-reversal breaking is necessary but not always sufficient. A general symmetry criterion for nonzero phonon Hall viscosity in magnetic insulators requires breaking both time-reversal symmetry and at least one relevant mirror symmetry (Ye et al., 2021). This criterion clarifies why external field, SOC, magnetic order, and crystal symmetry must be analyzed simultaneously: residual antiunitary symmetries can forbid the odd-viscous tensor even in a magnetically ordered phase.

The tensor structure itself can diagnose broken symmetries. In altermagnets, zero-field phonon Hall viscosity is proposed as a natural probe because the symmetry-allowed nonzero elements of $\eta^H_{ijkl}$ distinguish altermagnets from ferromagnets and conventional antiferromagnets. In tetragonal $D_{4h}$, for example, a $\eta_{ijkl}$0 altermagnet has $\eta_{ijkl}$1, whereas a ferromagnet has $\eta_{ijkl}$2; conventional antiferromagnets forbid Hall viscosity altogether (Jang et al., 24 Jun 2026). The paper attributes this sensitivity to a strain-space Berry-curvature monopole, in contrast to the multipolar structure of the ordinary momentum-space Berry curvature in altermagnets.

Three-dimensional generalizations require additional care. In topological semimetals with tetrahedral symmetry, there exist intrinsically three-dimensional Hall-viscosity coefficients that cannot be obtained by reducing to a quasi-two-dimensional system. For the cubic point group $\eta_{ijkl}$3, two independent scalars $\eta_{ijkl}$4 and $\eta_{ijkl}$5 are symmetry allowed, and the Kubo analysis distinguishes phonon Hall viscosity from conventional momentum Hall viscosity (Robredo et al., 2021). This separates genuinely viscoelastic Hall response from hydrodynamic stress response even within the same electronic model.

A distinct boundary realization appears in magnetic topological-insulator films. There, a surface phonon Hall viscosity arises from a Nieh-Yan action in which strain acts as an effective vierbein field for bulk massive Dirac fermions. The resulting boundary term couples phonon dynamics to surface magnetization and depends on the relative orientation of the top and bottom surface moments (Chatterjee et al., 19 Jan 2026). This is a surface analog of the broader theme, already emphasized in Hall-fluid work, that Hall viscosity is fundamentally geometric and not contingent on rotational invariance alone (0906.1854).

4. Dynamical consequences for phonons

The most characteristic dynamical effect is the mixing of longitudinal and transverse acoustic phonons. In the isotropic two-dimensional theory, Hall viscosity shifts the phonon spectrum by an amount of order $\eta_{ijkl}$6, where

$\eta_{ijkl}$7

and mixes longitudinal and transverse modes with relative amplitude $\eta_{ijkl}$8 and phase shift $\eta_{ijkl}$9, to lowest order in $u_i$0 (Barkeshli et al., 2011). In effect, a nominally longitudinal mode acquires a small transverse component, and vice versa, so the eigenmodes become elliptically polarized.

This mode mixing is the basis of the acoustic Faraday effect. In magnetic insulators, the Hall-viscosity-induced Berry curvature splits left- and right-circularly polarized transverse sound and rotates the polarization of a propagating acoustic pulse (Ye et al., 2021). The same phenomenology is now directly tied to experiment in $u_i$1-RuCl$u_i$2, where GHz ultrasonic measurements observed a pronounced acoustic Faraday effect that is antisymmetric in magnetic field and propagation direction and occurs only for transverse sound, allowing extraction of a finite $u_i$3 (Shragai et al., 7 Oct 2025).

Hall viscosity also renormalizes dispersion in less conventional ways. In the Kitaev spin liquid with broken time-reversal symmetry, the Hall-viscosity term mixes longitudinal and transverse modes and bends the spectrum in a characteristic manner that the paper identifies as accessible to spectroscopy (Ye et al., 2020). For optical phonons in Weyl semimetals, the Hall-viscosity tensor shifts zone-center phonon frequencies, with maximal effect when the external field is parallel to the Weyl-node separation and a suppressed shift when it is perpendicular; Raman and infrared probes are proposed as detection channels (Hiedari et al., 2019).

At surfaces, the consequences depend on magnetization configuration. In magnetic topological-insulator films with parallel top and bottom surface magnetizations, surface phonons become chiral but remain reciprocal. With antiparallel magnetizations, the surface phonons become nonreciprocal, $u_i$4, while their angular momentum vanishes (Chatterjee et al., 19 Jan 2026). This configural control of chirality versus nonreciprocity is one of the clearest examples of Hall viscosity acting as a symmetry transducer in phonon dynamics.

5. Thermal Hall transport and experimental platforms

A major motivation for phonon Hall viscosity is its connection to thermal Hall transport. The magnetic-insulator framework states that phonon Hall viscosity may generate phonon Berry curvature and can be observed through both the acoustic Faraday effect and thermal Hall transport (Ye et al., 2021). In $u_i$5-RuCl$u_i$6, the same Hall viscosity extracted from ultrasound is argued to produce an intrinsic thermal Hall effect that accounts for a significant fraction of the measured thermal Hall signal, and the persistence of the effect well above $u_i$7 is used to argue against an interpretation based solely on hybrid magnon-phonon modes (Shragai et al., 7 Oct 2025).

The interpretation of phonon thermal Hall data remains contested. One line of work analyzes an intrinsic contribution tied directly to Hall viscosity and Berry curvature, while another studies extrinsic skew scattering off impurities. In the extrinsic scenario, the dominant Hall transport comes from interference between impurity skew-scattering channels with opposite parity, giving $u_i$8 at low temperature and a temperature-independent window for $u_i$9 (Guo et al., 2021). This does not negate the intrinsic mechanism; rather, it implies that thermal Hall measurements alone need not isolate Hall viscosity without supplementary probes such as acoustic Faraday rotation.

Material realizations span several classes. The early theory analyzed integer quantum Hall states, the quantum anomalous Hall state in Hg$$\delta S_H = \frac{1}{2}\int d^d x\, dt \; \eta_{ijkl}\,\partial_i u_j\,\partial_k \dot{u}_l,$$0Mn$$\delta S_H = \frac{1}{2}\int d^d x\, dt \; \eta_{ijkl}\,\partial_i u_j\,\partial_k \dot{u}_l,$$1Te quantum wells, and a mean-field model for $$\delta S_H = \frac{1}{2}\int d^d x\, dt \; \eta_{ijkl}\,\partial_i u_j\,\partial_k \dot{u}_l,$$2 superconductors as platforms where lattice measurements could probe the Berry curvature of a gapped topological sector (Barkeshli et al., 2011). Magnetic insulators such as Sr$$\delta S_H = \frac{1}{2}\int d^d x\, dt \; \eta_{ijkl}\,\partial_i u_j\,\partial_k \dot{u}_l,$$3CuO$$\delta S_H = \frac{1}{2}\int d^d x\, dt \; \eta_{ijkl}\,\partial_i u_j\,\partial_k \dot{u}_l,$$4Cl$$\delta S_H = \frac{1}{2}\int d^d x\, dt \; \eta_{ijkl}\,\partial_i u_j\,\partial_k \dot{u}_l,$$5 provide a microscopic setting in which symmetry decomposition, SOC, and external field determine which Hall-viscosity coefficients are allowed (Ye et al., 2021). Ionic crystals exhibit an intrinsic Lorentz-force contribution that, for a square-lattice toy model with parameters typical for ionic materials and $$\delta S_H = \frac{1}{2}\int d^d x\, dt \; \eta_{ijkl}\,\partial_i u_j\,\partial_k \dot{u}_l,$$6 T, gives an estimate $$\delta S_H = \frac{1}{2}\int d^d x\, dt \; \eta_{ijkl}\,\partial_i u_j\,\partial_k \dot{u}_l,$$7 (Flebus et al., 2022). Optical-phonon Hall viscosity has been proposed for porphyrin thin-film Weyl semimetals (Hiedari et al., 2019), surface acoustic Hall viscosity for magnetic topological-insulator films (Chatterjee et al., 19 Jan 2026), and zero-field anomalous Hall viscosity for insulating altermagnets (Jang et al., 24 Jun 2026).

6. Relation to other Hall viscosities and recurrent misconceptions

Phonon Hall viscosity belongs to the broader family of odd-viscous responses but should not be conflated with electronic hydrodynamic Hall viscosity. In mesoscopic two-dimensional electron systems, Hall viscosity appears as a nondissipative momentum-transport coefficient that produces a negative correction to Hall resistivity near $$\delta S_H = \frac{1}{2}\int d^d x\, dt \; \eta_{ijkl}\,\partial_i u_j\,\partial_k \dot{u}_l,$$8 under hydrodynamic flow conditions (Gusev et al., 2018). In electron-phonon hydrodynamics, by contrast, the cited model predicts that phonons contribute to the ordinary shear viscosity but not to the Hall viscosity, which remains electronic because only the electrons couple directly to the Lorentz force (Huang et al., 2020). This is a model-specific statement, not a general prohibition on phonon Hall viscosity.

A second recurrent confusion concerns pressure-like effects. The semiclassical analysis of Landau-level dynamics emphasizes that Hall viscosity does not arise from orbit displacement or from the local pressure associated with spatially varying cyclotron energy; it arises from the shear of cyclotron wavefunctions themselves (Biswas, 2013). By analogy, in phononic contexts the distinctive odd-viscous response is not equivalent to any generic magnetoelastic correction to sound velocity. The observable signatures are specifically transverse, reactive, and polarization-sensitive.

A third point is the distinction between phonon Hall viscosity and momentum Hall viscosity in lattice systems. In tetrahedral topological semimetals, the Kubo formula yields both, but the phonon version is defined from the stress generated by explicit electron-phonon coupling and need not equal the conventional viscosity obtained from continuity-stress operators (Robredo et al., 2021). This difference is essential in crystalline systems, where the real-space lattice embedding matters.

Finally, the literature has progressively widened the scope of Hall viscosity beyond the original quantum Hall setting. The guiding-center Hall viscosity of incompressible fractional Hall fluids is tied to an intrinsic metric, distinguishes particle and hole fluids by sign, constrains the small-$$\delta S_H = \frac{1}{2}\int d^d x\, dt \; \eta_{ijkl}\,\partial_i u_j\,\partial_k \dot{u}_l,$$9 guiding-center structure factor, and does not require rotational invariance (0906.1854). Effective-field-theory treatments further connect Hall viscosity to Wen-Zee-type geometric couplings and to quantized topological data in some phases (Hoyos, 2014). This suggests that phonon Hall viscosity is best understood not as a niche magneto-acoustic coefficient, but as the lattice-dynamical manifestation of the same underlying geometric response principle whenever slow strains couple to a time-reversal-breaking gapped sector.