Acoustic Faraday Effect

• The acoustic Faraday effect is the rotation of polarization in transverse acoustic waves under a magnetic field via circular birefringence, characterized by differences in phase velocities and quantified by the magneto-acoustic Verdet constant.
• It is observed in diverse systems such as superfluid Helium-3 and paramagnetic crystals like Tb3Ga5O12, where mechanisms including Zeeman splitting and magnetoelastic coupling drive the birefringence.
• The phenomenon extends to inverse effects, where circularly polarized acoustic waves generate measurable DC responses, offering insights into Hall viscosity and topologically induced magnetization.

Searching arXiv for recent and foundational papers on the acoustic Faraday effect and closely related inverse acoustic Faraday phenomena. The acoustic Faraday effect is the rotation of the polarization of a linearly polarized transverse acoustic wave as it propagates through a medium in a magnetic field aligned with the propagation direction. Its immediate kinematic origin is circular birefringence: the left- and right-circularly polarized transverse modes acquire different phase velocities, so their relative phase accumulates during propagation and rotates the linear superposition. In canonical form, the rotation angle may be written as $\theta = (k_+ - k_-)L/2$, or, per unit length, $d\theta/dz = V_s B$ when a magneto-acoustic Verdet constant $V_s$ is introduced. Across the systems treated in the literature, the microscopic source of the birefringence differs substantially: Zeeman-split collective modes in superfluid ${}^3$He-B, magnetoelastic coupling between strain and $4f$ quadrupoles in Tb$_3$Ga$_5$O$_{12}$, and, in related inverse phenomena, nonlinear coupling of circularly polarized sound to dc flow or magnetization in quantum fluids and topological insulators (Collett et al., 2012, Sytcheva et al., 2010, Sharma, 2024, Su et al., 2024).

1. Definition, geometry, and kinematic description

In Faraday geometry, a transverse acoustic wave propagates parallel to an applied magnetic field, and the two transverse polarizations are most naturally represented as right- and left-circularly polarized components. At zero field, these components are degenerate when symmetry permits doubly degenerate transverse modes. A magnetic field breaks time-reversal symmetry and lifts the degeneracy, so the circular components propagate with different wave numbers $k_\pm = \omega/v_\pm$. The accumulated phase difference rotates the polarization vector of the transverse wave as it travels (Sytcheva et al., 2010).

For a wave of length $L$, the standard relation is

$d\theta/dz = V_s B$0

with $d\theta/dz = V_s B$1. In a cavity geometry this becomes $d\theta/dz = V_s B$2, where $d\theta/dz = V_s B$3 and $d\theta/dz = V_s B$4 is the cavity spacing (Collett et al., 2012). In the conventional phenomenological form, the rotation per unit length is summarized as

$d\theta/dz = V_s B$5

which is the acoustic analogue of optical Faraday rotation (Sharma, 2024).

The basic observable is therefore not merely a velocity anomaly but a differential phase advance between circular polarizations. In experiments, this commonly appears as an oscillation in the received acoustic amplitude as a function of magnetic field, with maxima and minima corresponding to specific polarization angles. In Tb$d\theta/dz = V_s B$6Ga$d\theta/dz = V_s B$7O$d\theta/dz = V_s B$8, the detected amplitude shows maxima at $d\theta/dz = V_s B$9 and minima at $V_s$0 once magneto-acoustic Faraday oscillations develop (Sytcheva et al., 2010). In superfluid $V_s$1He-B, envelope minima of the cavity signal correspond to $V_s$2, $V_s$3, providing a direct polarization calibration even for very large rotations (Collett et al., 2012).

2. Symmetry, constitutive structure, and Hall-viscosity formulation

The symmetry requirements for the acoustic Faraday effect are explicit. One needs transverse waves, an axis selecting circular polarizations, and broken time-reversal symmetry, typically via an applied magnetic field. In a three-dimensional viscoelastic medium supporting transverse waves, these conditions permit a nonzero Hall viscosity, a dissipationless component of the stress response to strain rate that is antisymmetric under exchange of spatial indices and odd under time reversal (Tuegel et al., 2017).

The linearized constitutive structure is written in terms of momentum conservation and the stress response,

$V_s$4

with

$V_s$5

For a plane transverse wave with wavevector $V_s$6, the transverse-sector equation becomes

$V_s$7

In an isotropic transverse plane, the odd part of the viscosity is parameterized by $V_s$8, yielding circularly polarized eigenmodes with dispersions

$V_s$9

where ${}^3$0 is the transverse shear modulus. The resulting rotation rate is

${}^3$1

with ${}^3$2 (Tuegel et al., 2017).

This formulation makes clear that circular birefringence can be viewed as the propagation signature of a nonzero odd viscosity in the transverse sector. In superfluid ${}^3$3He-B, the same paper shows that the Zeeman effect on the collective modes generates such a Hall viscosity coefficient, linking the observed acoustic Faraday effect directly to the antisymmetric momentum-stress response rather than treating rotation as an isolated spectroscopic anomaly (Tuegel et al., 2017). A plausible implication is that the Hall-viscosity language supplies a unifying description across otherwise distinct microscopic platforms whenever transverse waves and time-reversal breaking coexist.

3. Superfluid ${}^3$4He-B: collective-mode birefringence and giant rotations

Superfluid ${}^3$5He-B is a central realization because it supports transverse zero sound and because this transverse sound couples strongly to the order-parameter collective mode known as the imaginary squashing mode (ISQ), a ${}^3$6 mode with five Zeeman sub-states. Transverse sound couples specifically to the ${}^3$7 sublevels. A magnetic field splits these sublevels, so right- and left-circularly polarized transverse sound acquire different wave numbers and the linear polarization rotates. This effect both proves the existence of transverse zero sound in the superfluid and provides a sensitive probe of the magnetic-field dependence of the ISQ (Collett et al., 2012).

The standard near-mode dispersion is

${}^3$8

with

${}^3$9

To model the magnetic-field dependence of the birefringence, the denominator is replaced by

$4f$0

where $4f$1, $4f$2, and $4f$3 parameterize linear, quadratic, and cubic field dependence, respectively (Collett et al., 2012).

Experimentally, transverse acoustics studies in superfluid $4f$4He-B at fields up to $4f$5 T and frequency $4f$6 MHz observed Faraday rotations as large as $4f$7, nearly five complete rotations. The experiment employed acoustic cavity interferometry with cavity spacing $4f$8, at temperatures near $4f$9K, with pressure swept from approximately $_3$0 bar down to $_3$1 bar (Collett et al., 2012). The measured transducer signal obeys

$_3$2

so both $_3$3 and $_3$4 can be extracted from the same trace (Collett et al., 2012).

A central result is that the field dependence is not purely linear. Fits show that the transverse sound velocity $_3$5 is nearly field independent in the explored range, giving only an upper bound on the quadratic term $_3$6, whereas the polarization rotation requires both linear and cubic terms for an accurate description. Extrapolating the linear coefficient $_3$7 to $_3$8 gives the ISQ Landé $_3$9-factor through

$_5$0

The paper reporting nonlinear field dependence and $_5$1-wave interactions further states that fitting $_5$2 near the mode yields $_5$3, giving $_5$4 at $_5$5 bar, and uses this to infer that the pairing interaction in the $_5$6-wave channel is attractive at that pressure (Collett et al., 2012).

Within the Hall-viscosity interpretation, the same birefringence can be re-expressed as arising from a Zeeman-induced odd viscosity,

$_5$7

which leads to

$_5$8

This reformulation does not replace the collective-mode picture; rather, it recasts the same physics as an odd-viscous stress response (Tuegel et al., 2017).

4. Paramagnetic garnets and the Tb$_5$9Ga$_{12}$0O$_{12}$1 mechanism

Tb$_{12}$2Ga$_{12}$3O$_{12}$4 provides a distinct realization of the acoustic Faraday effect in a paramagnetic crystal. Along a fourfold cubic axis such as $_{12}$5, the transverse acoustic $_{12}$6 mode is doubly degenerate at zero field. A magnetic field applied parallel to the propagation direction lifts this degeneracy by coupling the transverse strain to quadrupolar operators of the Tb$_{12}$7 $_{12}$8 ions. The linearly polarized transverse wave is decomposed into left- and right-circularly polarized phonons, which in field acquire different wave numbers $_{12}$9, producing the rotation (Sytcheva et al., 2010).

The magnetoelastic interaction is written as

$k_\pm = \omega/v_\pm$0

and the corresponding elastic constant renormalization is

$k_\pm = \omega/v_\pm$1

For the $k_\pm = \omega/v_\pm$2 shear mode, the dominant symmetry channel is $k_\pm = \omega/v_\pm$3, and the analysis uses the composite quadrupolar operator

$k_\pm = \omega/v_\pm$4

In the field-dependent fit of the $k_\pm = \omega/v_\pm$5 velocity, the extracted magnetoelastic coupling is $k_\pm = \omega/v_\pm$6 K, compared with $k_\pm = \omega/v_\pm$7 K from the temperature dependence of $k_\pm = \omega/v_\pm$8 (Sytcheva et al., 2010).

The experiments were performed on a TGG single crystal with wavevector $k_\pm = \omega/v_\pm$9, displacement $L$0, magnetic field $L$1, and sample length $L$2 mm, using LiNbO$L$3 transducers. Static-field measurements up to $L$4 T were made at $L$5, $L$6, $L$7, $L$8, and $L$9 MHz and $d\theta/dz = V_s B$00 K; pulsed-field measurements extended to $d\theta/dz = V_s B$01 T at $d\theta/dz = V_s B$02 MHz (Sytcheva et al., 2010). In static fields, clear magneto-acoustic Faraday oscillations appear above about $d\theta/dz = V_s B$03 T. The $d\theta/dz = V_s B$04 velocity shows a pronounced minimum at about $d\theta/dz = V_s B$05 T with maximum reduction $d\theta/dz = V_s B$06, while in pulsed fields the velocity anomaly is roughly $d\theta/dz = V_s B$07 stronger, with hysteresis attributed to the magnetocaloric effect (Sytcheva et al., 2010).

A major theoretical issue in TGG is the frequency dependence. Earlier theories based only on direct acoustic coupling to magnetic excitations predicted $d\theta/dz = V_s B$08. The TGG experiments, however, found a linear dependence of the rotation per unit length on frequency, with $d\theta/dz = V_s B$09 collapsing onto a single curve over $d\theta/dz = V_s B$10–$d\theta/dz = V_s B$11 MHz at $d\theta/dz = V_s B$12 K (Sytcheva et al., 2010). The resolution proposed in the companion theory is that, in non-Bravais lattices such as TGG, long-wavelength optical $d\theta/dz = V_s B$13-symmetry phonons can directly influence the acoustic shear waves via symmetry-allowed acoustic-optical coupling. Magnetoelastic coupling splits the doubly degenerate optical circular phonons even as $d\theta/dz = V_s B$14, and this optical splitting induces an indirect splitting of the acoustic left and right modes that remains finite in the long-wavelength limit, yielding $d\theta/dz = V_s B$15 (Thalmeier, 2010).

In that theory, the indirect contribution to the Faraday rotation per unit length is written as

$d\theta/dz = V_s B$16

with split optical frequencies

$d\theta/dz = V_s B$17

and effective acoustic-optical coupling

$d\theta/dz = V_s B$18

Because the optical splitting remains finite for $d\theta/dz = V_s B$19, the resulting rotation is linear in frequency rather than quadratic (Sytcheva et al., 2010).

The resonance in the field range $d\theta/dz = V_s B$20–$d\theta/dz = V_s B$21 T is then tied to the crystal-electric-field structure. The simplified cubic model uses a ground-state doublet and an excited quasi-triplet at $d\theta/dz = V_s B$22 K, with a level crossing in the $d\theta/dz = V_s B$23–$d\theta/dz = V_s B$24 T range. The quadrupolar susceptibilities entering the optical splitting are dominated by transitions between the ground-state doublet and the field-split triplet components, naturally explaining the resonance-like features in rotation and the strong damping near $d\theta/dz = V_s B$25–$d\theta/dz = V_s B$26 T (Sytcheva et al., 2010). This directly addresses the longstanding discrepancy between experiments showing $d\theta/dz = V_s B$27 and acoustic-only theories requiring $d\theta/dz = V_s B$28 (Thalmeier, 2010).

5. Inverse variants and acoustic angular momentum

The inverse acoustic Faraday effect is distinct from the conventional acoustic Faraday effect. In the direct effect, a magnetic field modifies the propagation of a transverse acoustic wave and rotates its polarization. In the inverse effect, a circularly polarized acoustic wave generates a dc quantity, such as a static magnetization or a dc circulating flow, without requiring polarization rotation as the primary observable (Sharma, 2024).

In the proposal for normal liquid $d\theta/dz = V_s B$29He in aerogel, a circularly polarized transverse shear wave

$d\theta/dz = V_s B$30

is predicted to induce a static circulating current when the wave propagates along $d\theta/dz = V_s B$31. The central scaling relations are

$d\theta/dz = V_s B$32

with the sign reversing when the handedness of the circular polarization is reversed. The author emphasizes that aerogel is essential because it supports a low-attenuation transverse sound mode in the coupled $d\theta/dz = V_s B$33He-aerogel system, whereas in pure liquid $d\theta/dz = V_s B$34He the strong attenuation would suppress the propagating wave amplitude and therefore the induced dc current. The temperature range identified as most suitable for detection is approximately $d\theta/dz = V_s B$35–$d\theta/dz = V_s B$36 mK, and a mass flow of $d\theta/dz = V_s B$37 kg/sec is stated to be measurable in the appropriate context (Sharma, 2024).

A topological inverse acoustic Faraday effect is proposed in inversion-broken, time-reversal-invariant Dirac insulators such as hBN and transition-metal dichalcogenide monolayers. In that setting, a circularly polarized acoustic wave produces a static magnetization through valley-contrasting band topology rather than through conventional magnetic order or an external field. The induced magnetization is written as

$d\theta/dz = V_s B$38

so that

$d\theta/dz = V_s B$39

with the sign set by acoustic helicity and the susceptibility proportional to the valley Chern number $d\theta/dz = V_s B$40 (Su et al., 2024).

These inverse results sharpen the role of acoustic angular momentum. In the $d\theta/dz = V_s B$41He-aerogel proposal, the dc response is traced to nonlinear coupling of the circularly polarized shear wave to the Fermi liquid, with angular momentum density proportional to $d\theta/dz = V_s B$42 (Sharma, 2024). In the Dirac-insulator setting, the same circular structure enters through $d\theta/dz = V_s B$43, but the response is framed in terms of a quantized cross Chern-Simons action and strain-induced pseudogauge fields (Su et al., 2024). This suggests a useful conceptual distinction: conventional AFE is a propagation effect caused by circular birefringence, whereas inverse AFE is a nonlinear generation effect caused by the angular momentum carried by circularly polarized sound.

6. Conditions, diagnostics, and recurrent points of confusion

Observation of a robust acoustic Faraday effect is system-dependent, but several conditions recur. One needs a medium that supports transverse waves or transverse zero sound, a mechanism that differentiates right- and left-circular polarizations, and a geometry with field aligned to the propagation direction. In cubic garnets such as TGG, this further requires propagation along a fourfold axis so that doubly degenerate transverse $d\theta/dz = V_s B$44-type modes exist, together with strong magnetoelastic coupling and, for the linear-in-frequency mechanism, a non-Bravais lattice that allows symmetry-allowed coupling between acoustic and optical modes (Sytcheva et al., 2010). In superfluid $d\theta/dz = V_s B$45He-B, the optimal regime is low temperature and frequencies near the ISQ, where the birefringence is resonantly enhanced (Collett et al., 2012).

One common misconception is that the acoustic Faraday effect requires magnetic order. The TGG case is explicitly paramagnetic: the effect arises from magnetoelastic coupling between strain and localized $d\theta/dz = V_s B$46 quadrupoles in an applied magnetic field, not from ferromagnetic order (Thalmeier, 2010). Another recurring misconception is that the frequency dependence must generically be quadratic. Acoustic-only theories indeed predict $d\theta/dz = V_s B$47 in several settings, because the splitting must vanish as $d\theta/dz = V_s B$48; however, TGG shows that optical-phonon-mediated coupling in a non-Bravais lattice can yield $d\theta/dz = V_s B$49 and dominate experimentally (Sytcheva et al., 2010).

A further point of confusion concerns attenuation and dichroism. In the superfluid $d\theta/dz = V_s B$50He-B measurements at fields of order $d\theta/dz = V_s B$51 T, circular dichroism is described as negligible, so birefringence dominates the rotation signal (Collett et al., 2012). More generally, differences in attenuation of the two circular modes can also affect polarization, and the Hall-viscosity analysis therefore emphasizes low-attenuation transverse-wave regimes and the simultaneous measurement of both rotation and amplitude (Tuegel et al., 2017). In the normal-state $d\theta/dz = V_s B$52He-aerogel inverse proposal, the requirement $d\theta/dz = V_s B$53 is correspondingly crucial, since $d\theta/dz = V_s B$54 places the medium in a hydrodynamic regime with high shear attenuation and a strongly suppressed signal (Sharma, 2024).

As an experimental diagnostic, two controls recur across direct and inverse phenomena. The first is sign reversal under reversal of the circular basis: in the direct effect, $d\theta/dz = V_s B$55 changes sign under $d\theta/dz = V_s B$56 because the birefringence is odd in field; in inverse effects, the dc response reverses with acoustic helicity (Tuegel et al., 2017, Sharma, 2024). The second is scaling: linear response in the conventional AFE is associated with field-induced circular birefringence, whereas inverse effects in $d\theta/dz = V_s B$57He-aerogel are explicitly quadratic in the drive amplitude, $d\theta/dz = V_s B$58, which distinguishes them from the oscillatory flow linear in $d\theta/dz = V_s B$59 (Sharma, 2024).

Taken together, these results establish the acoustic Faraday effect as a family of magneto-acoustic birefringence phenomena rather than a single mechanism. In superfluid $d\theta/dz = V_s B$60He-B it is a probe of Zeeman-split order-parameter dynamics and, equivalently, of Hall viscosity (Collett et al., 2012, Tuegel et al., 2017). In Tb$d\theta/dz = V_s B$61Ga$d\theta/dz = V_s B$62O$d\theta/dz = V_s B$63 it is governed by magnetoelastic coupling to $d\theta/dz = V_s B$64 quadrupoles and, crucially, by optical-phonon-mediated splitting in a non-Bravais lattice (Sytcheva et al., 2010, Thalmeier, 2010). In inverse variants, circularly polarized sound acts not as a probe but as a generator of dc flow or magnetization, extending the conceptual scope of Faraday-type acoustics into nonlinear and topological response (Sharma, 2024, Su et al., 2024).