Surface Phonon Hall Viscosity

• Surface phonon Hall viscosity is a non-dissipative, antisymmetric response localized at material boundaries that break time-reversal symmetry, driving chiral phonon propagation.
• It bridges viscoelasticity with topological band theory, arising from electron–phonon and magnon–phonon couplings in systems like topological insulators and Weyl semimetals.
• Experimental probes such as Raman spectroscopy, surface-acoustic-wave measurements, and neutron scattering reveal its nonreciprocal effects and enable control via magnetic fields and engineered strain.

Surface phonon Hall viscosity is a non-dissipative, antisymmetric component of the stress–strain rate response localized at the surfaces or interfaces of solids that break time-reversal symmetry (TRS), often by magnetization or topological boundary conditions. It is responsible for chiral or nonreciprocal propagation of surface-acoustic or optical phonon modes, and links the field of viscoelasticity, topological band theory, and modern condensed matter physics. Surface phonon Hall viscosity arises from microscopic mechanisms including topological electronic structure (e.g., in topological insulators or Weyl semimetals), electron–phonon or magnon–phonon couplings, and is formally connected to topological field theory via the Nieh–Yan torsional response or axionic actions.

1. Fundamental Definition and Theoretical Framework

Hall viscosity generally refers to a time-reversal odd, non-dissipative rank-4 viscosity tensor $\eta_H^{ijkl}$ (or lower ranks in lower dimensions/symmetry) that enters the effective action for the elastic (phonon) degrees of freedom via terms of the form

$$S_{\rm PHV} = \frac{1}{2}\int dt\, d^d\mathbf{x}\ \eta_H^{ijkl} \, \partial_t u_i\, \partial_j u_k$$

where $u_i(\mathbf{x},t)$ is the phonon displacement, and $\eta_H^{ijkl}$ is antisymmetric under $i \leftrightarrow j$ (or other index permutations determined by symmetry).

In systems with a boundary, surface-localized phonon modes inherit an effective two-dimensional Hall viscosity upon projecting the bulk action onto evanescent or Rayleigh-like states. In the presence of broken TRS, this gives rise to a term in the 2D action

$$S_{\rm surf} = \int dt\, d^2x_\perp \Big\{ \frac{\rho_s}{2} \dot{u}_i^2 - \frac{1}{2} C^{ijkl}_s \partial_i u_j \partial_k u_l + \frac{1}{2} \eta_{H,s}^{ijkl}\, \partial_t u_i\, \partial_j u_k + ... \Big\}$$

with the surface-projected $\eta_{H,s}^{ijkl}$ determined by integrating the bulk $\eta_H$ over the evanescent profile (Ye et al., 2021).

Topological insulators (TIs), Weyl semimetals, and magnetic insulators all realize forms of surface phonon Hall viscosity due to their peculiar electronic or magnetic orderings. In 3D TIs, the surface PHV can be traced to the torsional Nieh–Yan term in the effective action (Chatterjee et al., 19 Jan 2026); in magnetic insulators, it is generated via nontrivial spin-lattice couplings in a magnetically ordered background (Ye et al., 2021).

2. Microscopic Origins: Topological Insulators, Weyl Semimetals, and Magnetic Insulators

Topological Insulators

In magnetic TI films, the effective action for the bulk Dirac electrons in the presence of strain encodes the strain in the frame field (vierbein). Integrating the electrons out yields a Nieh–Yan term: $$S_{\text{NY}} = \eta_0 \int dt\, d^3 r\, \Phi(r) \epsilon^{alb} \partial_l \Delta_a^i \partial_t \Delta_b^j \delta_{ij}$$ with $\Phi(r)$ the complex Dirac mass phase, and $\Delta_a^j$ linear in the strain tensor $u_{ij}$. The term is a total derivative in $z$, localizing flux of PHV at the gapped surfaces: $$S_{\text{PHV}} = - \int dt\, d^3r\, (\partial_z \Phi)\, \eta_{ijmn} u_{ij}\, \dot{u}_{mn}$$ where the only symmetry-allowed tensor components for $D_{3d}$ crystals are $\eta_1 \equiv \eta_{xxxy}$, $\eta_2 \equiv \eta_{xzyz}$, $\eta_3 \equiv \eta_{xyyz}/2$ (Chatterjee et al., 19 Jan 2026).

Weyl Semimetals

In strained Weyl semimetal thin films (such as Porphyrin models), strain acts as a chiral pseudo-gauge field $A_5^i \sim u^{(a)} - u^{(b)}$ (difference in sublattice displacement). Integrating out the electrons introduces both a topological (anomaly-induced) Hall viscosity,

$$\eta_{ij}^{\text{anom}} = \frac{|{\bf b}|}{12\pi^2} \epsilon_{ij3}$$

arising from the axionic action, and a conventional Fermi-surface contribution in the presence of $B$-fields (Hiedari et al., 2019).

Magnetic Insulators

For magnetic insulators, TRS breaking (by magnetic order or field) allows a Hall viscosity via lattice symmetry-allowed magnetoelastic couplings and magnon fluctuations. For a square-lattice AFM with spin-orbit coupling and external field $h \hat{z}$, integrating out the magnons produces

$$\eta_{\Gamma\Gamma'}^H({\bf q}_\perp, 0) = \gamma_{\Gamma\Gamma'}\,(q_x^2 + q_y^2 + \delta_\alpha^2)$$

with the detailed form set by the magnetoelastic coupling $\lambda_\Gamma$ and magnon parameters (Ye et al., 2021). Surface Rayleigh-wave modes inherit an effective $2$-dimensional Hall viscosity via profile projection.

3. Calculation and Modeling Methods

Multiple computational paradigms have been established for evaluating surface phonon Hall viscosity:

• Linear Response (Kubo) Formalism: For a gapped electronic Hamiltonian (e.g., the surface Dirac cone of a 3D TI), the PHV is computed through the Kubo formula relating the (antisymmetric) correlation of surface stress operators under infinitesimal strain:

$$\zeta_H = -\frac{2}{L^2}\, \operatorname{Im} \sum_{k,\alpha\in \text{occ},\beta\in \text{unocc}} \frac{\langle \alpha | \hat{\mathcal{T}}_{11} | \beta\rangle \langle \beta | \hat{\mathcal{T}}_{12} | \alpha\rangle}{(E_\beta - E_\alpha)^2}$$

(Shapourian et al., 2015).

• Gauge (minimal-coupling) and Electron–Phonon (gradient) Lattice Deformation Schemes: Lattice models can implement deformations either by momentum shifts or by expansion of the hopping integrals, generating consistent stress operators for Kubo analysis.
• Topological Field Theory Approach: Surface PHV can be identified as boundary terms in the low-energy effective action (e.g., Nieh–Yan), producing explicit forms for the Hall viscosity tensor by symmetry analysis and integrating out bulk fermions or magnons (Chatterjee et al., 19 Jan 2026, Ye et al., 2021).
• Projection Onto Surface Modes: The bulk Hall-viscosity tensor and elastic constants are projected onto the surface-localized mode profiles, generating effective surface parameters (Ye et al., 2021).

4. Effects on Surface Phonon Dynamics

The presence of surface phonon Hall viscosity modifies the dispersion, polarization, and reciprocity of surface (and optical) phonon modes:

• Mode Splitting and Nonreciprocity: For optical phonons in 2D Weyl semimetal films, PHV induces nonreciprocal splitting of degenerate $E$-modes:

$$\Delta \omega_{\rm Raman} = (\eta_H/\rho)\,k^2/\omega_0$$

where $\omega_0$ is the bare optical gap (Hiedari et al., 2019). In magnetic TIs, the surface Hall viscosity term modifies the 2D in-plane acoustic dynamical matrix by off-diagonal components $\sim i\omega \eta_S k^2$ (Chatterjee et al., 19 Jan 2026).

• Chiral and Nonreciprocal Propagation: In TI films with parallel (FM) magnetization alignment of top and bottom surfaces, net PHV is finite and the eigenvectors of surface-acoustic branches acquire angular momentum (chiral phonons), though the spectrum remains reciprocal $\omega(k)=\omega(-k)$. With antiparallel (AFM) alignment, net PHV vanishes in 2D but nonreciprocal propagation (frequency nondegeneracy at $k$ and $-k$) occurs in the full slab geometry (Chatterjee et al., 19 Jan 2026).
• Faraday Rotation and Ellipticity: For surface transverse-acoustic (TA) modes, PHV at the boundary mixes orthogonal polarizations resulting in "acoustic Faraday rotation". The ratio of mixed-mode amplitudes, for an incident TA wave polarized along $x$, is

$$\frac{A_2}{A_1} = \frac{\zeta_H}{\rho v_s^3} \omega^2$$

and the polarization is rotated by angle $\theta \propto \eta_H \omega^2$ (Shapourian et al., 2015).

• Berry Curvature and Thermal Hall Response: The PHV modifies the phonon Berry curvature, yielding a finite phonon thermal Hall conductivity at low $T$:

$\kappa_{xy} \propto \eta_S k_B^3 T^2$

which distinguishes thin-film surface PHV ($T^2$ scaling) from bulk examples ($T^3$) (Chatterjee et al., 19 Jan 2026).

• Magnon–Phonon Polaron Branches: In magnetic TIs, surface magnons couple via PHV-enhanced interactions to create hybrid magnon–phonon polarons with berry-curvature hotspots and enhanced $\kappa_{xy}$ near the anti-crossing (Chatterjee et al., 19 Jan 2026).

5. Symmetry Requirements and Mechanism Classification

A nonzero surface phonon Hall viscosity necessitates the breaking of time-reversal symmetry, either via:

• Magnetization (ferromagnetic or antiferromagnetic order)
• External Fields (applied $B$ field)
• Chiral Anomalies (in Dirac/Weyl systems)
• Spin–Orbit Coupling (required for effective strain-spin coupling in magnetic insulators)

Vertical mirror ($M_v \perp \hat{z}$) symmetries must also be absent, or time-reversal composed with such mirrors must not be a symmetry (Ye et al., 2021). The symmetry of the crystal determines the allowed tensor components of $\eta_H$ and which strain/mode pairs are coupled. Electron systems with topological surface states and gap-inducing perturbations (such as surface magnetic films) are natural platforms, and the configuration of surface magnetization gives experimental "knob" control of PHV magnitude and sign (Chatterjee et al., 19 Jan 2026).

6. Experimental Probes and Signatures

Various experimental approaches can directly or indirectly access surface phonon Hall viscosity:

• Raman and Infrared (ATR) Spectroscopy: Detect nonreciprocal splitting of doubly degenerate optical phonon lines under reversal of in-plane wavevector or incident angle. Field rotation can distinguish topological (anomaly-driven) vs field-induced PHV via their dependencies (Hiedari et al., 2019).
• Surface-Acoustic-Wave (SAW) Measurements: Ellipticity or polarization rotation of SAW on a magnetic TI crystal or in ultrafast pump–probe setups, with rotation angle $\theta \sim \zeta_H \omega^2/(\rho v^3)$. Effect reverses sign with reversal of surface magnetization—an unambiguous signature (Shapourian et al., 2015, Chatterjee et al., 19 Jan 2026).
• Thermal Hall Conductivity (Nanocalorimetry): Measurement of $\kappa_{xy}$ as a function of temperature (expect $\sim 10^{-10}$ W/K at 100 mK for relevant magnetic TIs) (Chatterjee et al., 19 Jan 2026).
• Brillouin Light Scattering: Phase-sensitive detection of chiral phonon modes via scattering polarization, particularly sensitive to angular momentum transport in the surface phonon branches (Chatterjee et al., 19 Jan 2026).
• Pump–Probe and Neutron Scattering: Detection of nonreciprocity in slab delay lines and identification of magnon–polaron hybrid branches with enhanced Berry curvature (Chatterjee et al., 19 Jan 2026).
• Faraday Rotation for Bulk and Surface Acoustic Modes: Observation of rotation per path length at characteristic frequencies, with anticipated scales $\sim 10^{-7}$ rad/m in representative cuprate AFMs (Ye et al., 2021).

7. Material Systems, Parameter Dependence, and Prospects

Surface phonon Hall viscosity is expected in:

• Magnetic topological insulator films (MBE-grown $$\mathrm{(Cr/Bi,Sb)_2Te_3 / (Bi,Sb)_2Te_3 / (V/Bi,Sb)_2Te_3}$$, MnBi$_2$Te$_4$ 4–10 layer slabs): with Dirac mass $m\sim0.1$ eV, surface Fermi velocity $v_f\sim 3\times10^{5}$ m/s, PHV coefficient $\eta_0 \sim 1$–$10$ m$^3$/s$^2$, thickness $\sim 10$–$100$ nm (Chatterjee et al., 19 Jan 2026).
• Thin-film Weyl semimetals (strained porphyrin systems): PHV governed by Weyl node separation $|{\bf b}|$, Fermi velocity, and Grüneisen parameter (Hiedari et al., 2019).
• Magnetic insulators with strong SOC (cuprates, etc.): PHV proportional to magnon gap and spin-lattice coupling (Ye et al., 2021).

Control of PHV via external fields, gating, or engineered strain/stacking is possible. The magnitude and sign of the effect can be tuned by the alignment (parallel/antiparallel) of surface magnetization, and multiple measurement modalities are feasible with current experimental capabilities. Surface phonon Hall viscosity remains a highly active research area, providing a robust link between topology, spintronics, and mesoscale acoustic phenomena.

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Key References:

• "Surface Phonon Hall Viscosity Induced Phonon Chirality and Nonreciprocity in Magnetic Topological Insulator Films" (Chatterjee et al., 19 Jan 2026)
• "Hall viscosity for optical phonons" (Hiedari et al., 2019)
• "Viscoelastic response of topological tight-binding models in two and three dimensions" (Shapourian et al., 2015)
• "Phonon Hall Viscosity in Magnetic Insulators" (Ye et al., 2021)