Anomalous Hall viscosity of altermagnets

Abstract: We show that the phonon Hall viscosity at zero magnetic field is a natural probe of altermagnetism. First, we demonstrate that the finite elements of the Hall viscosity tensor unambiguously distinguish altermagnets from ferromagnets and conventional antiferromagnets. We then microscopically compute the Hall viscosity in models for d-wave and g-wave altermagnets, and find a strong sensitivity to electronic spectrum features such as gapped Dirac points and Lifshitz transitions. This sensitivity reflects a strain-space Berry curvature monopole, which contrast to the multipolar character of the standard momentum-space Berry curvature in altermagnets. Since the Hall viscosity can be probed experimentally through magneto-acoustic measurements, it provides a compelling method to probe the broken symmetries and topology of insulating altermagnets.

• The paper establishes that altermagnets can exhibit finite zero-field Hall viscosity, whose tensor structure distinguishes pure altermagnetic, ferromagnetic, and conventional antiferromagnetic order.
• The authors show that strain-space Berry curvature forms monopoles near SOC-gapped Dirac points and changes sharply at Lifshitz transitions, producing viscosities of roughly 8.15 μPa·s in representative models.
• The paper proposes acoustic birefringence and polarization-rotation measurements as practical ways to detect altermagnetic order in insulating materials where anomalous Hall conductivity is unavailable.

The Hall viscosity — the antisymmetric, non-dissipative part of the viscosity tensor relating time-dependent strain to transverse stress — has historically been studied in quantum Hall systems and magnetic insulators under an applied magnetic field. In "Anomalous Hall viscosity of altermagnets" (2606.26239), Jang, Aquino, Schmalian, and Fernandes establish that this response is finite at zero field in altermagnets, that its tensor structure unambiguously distinguishes altermagnetic from ferromagnetic and antiferromagnetic order, and that it is governed by a strain-space Berry curvature monopole whose magnitude is dictated by band-structure singularities such as SOC-gapped Dirac points and Lifshitz transitions. Because it is a lattice response measurable via acoustic probes, it applies to insulating altermagnets, where the anomalous Hall conductivity is unavailable.

Symmetry classification

The authors decompose the 15-dimensional representation of non-zero Hall viscosity tensor elements into irreps of the paramagnetic point group using Jahn symbols, $a\{[V^2][V^2]\}$. For tetragonal $D_{4h}$, this yields $$A_{1g}^{-}\oplus2A_{2g}^{-}\oplus2B_{1g}^{-}\oplus2B_{2g}^{-}\oplus4E_{g}^{-}$$, all TRS-odd irreps corresponding to distinct even-parity $\boldsymbol{Q}=0$ magnetic orders with SOC. The key result is that each magnetic order parameter maps to a unique combination of non-zero $\eta^{H}_{ijkl}$ elements: $A_{1g}^{-}$, $B_{1g}^{-}$, and $B_{2g}^{-}$ describe pure $g$-, $d_{xy}$-, and $D_{4h}$0-wave altermagnets, while $D_{4h}$1 and $D_{4h}$2 describe ferromagnetic or mixed altermagnetic order.

A particularly diagnostic case contrasts a $D_{4h}$3 altermagnet ($D_{4h}$4) with an out-of-plane ferromagnet ($D_{4h}$5): both share the same non-zero elements, but with opposite relative signs, $D_{4h}$6 versus $D_{4h}$7. Consequently, the same dynamic shear strain generates symmetry-preserving stress ($D_{4h}$8) in the altermagnet but tetragonal-symmetry-breaking stress ($D_{4h}$9) in the ferromagnet. Conventional antiferromagnets, by contrast, are forced to zero by combined time-reversal and translational (or inversion) symmetries. The End Matter tabulates the allowed tensors for $$A_{1g}^{-}\oplus2A_{2g}^{-}\oplus2B_{1g}^{-}\oplus2B_{2g}^{-}\oplus4E_{g}^{-}$$0, $$A_{1g}^{-}\oplus2A_{2g}^{-}\oplus2B_{1g}^{-}\oplus2B_{2g}^{-}\oplus4E_{g}^{-}$$1, and $$A_{1g}^{-}\oplus2A_{2g}^{-}\oplus2B_{1g}^{-}\oplus2B_{2g}^{-}\oplus4E_{g}^{-}$$2, noting one exception: the cubic $$A_{1g}^{-}\oplus2A_{2g}^{-}\oplus2B_{1g}^{-}\oplus2B_{2g}^{-}\oplus4E_{g}^{-}$$3-wave ($$A_{1g}^{-}\oplus2A_{2g}^{-}\oplus2B_{1g}^{-}\oplus2B_{2g}^{-}\oplus4E_{g}^{-}$$4) altermagnet has no non-zero Hall viscosity element.

Microscopic mechanism

Using a quasi-adiabatic expansion (with a derivation free of this approximation given in the Supplemental Material), the Hall viscosity is expressed as a Brillouin-zone sum over occupied states of a strain-space Berry curvature,

$$A_{1g}^{-}\oplus2A_{2g}^{-}\oplus2B_{1g}^{-}\oplus2B_{2g}^{-}\oplus4E_{g}^{-}$$5

where $$A_{1g}^{-}\oplus2A_{2g}^{-}\oplus2B_{1g}^{-}\oplus2B_{2g}^{-}\oplus4E_{g}^{-}$$6 involves the electron–strain coupling matrices $$A_{1g}^{-}\oplus2A_{2g}^{-}\oplus2B_{1g}^{-}\oplus2B_{2g}^{-}\oplus4E_{g}^{-}$$7. Finiteness requires that the strain Hamiltonian not commute with the strain-free Hamiltonian, and SOC must be present to generate local Berry curvature. Notably, the paper emphasizes that weak SOC does not necessarily imply a small Hall viscosity.

Tetragonal $$A_{1g}^{-}\oplus2A_{2g}^{-}\oplus2B_{1g}^{-}\oplus2B_{2g}^{-}\oplus4E_{g}^{-}$$8-wave altermagnets

For the Lieb-lattice model relevant to $$A_{1g}^{-}\oplus2A_{2g}^{-}\oplus2B_{1g}^{-}\oplus2B_{2g}^{-}\oplus4E_{g}^{-}$$9Mn$\boldsymbol{Q}=0$0Se$\boldsymbol{Q}=0$1O$\boldsymbol{Q}=0$2, $\boldsymbol{Q}=0$3Te$\boldsymbol{Q}=0$4O, and Fe$\boldsymbol{Q}=0$5O families, the spectrum hosts four spin-polarized Dirac points for $\boldsymbol{Q}=0$6, gapped by SOC into masses $\boldsymbol{Q}=0$7. The computed strain-space Berry curvature is sharply peaked at these gapped Dirac points; the symmetric combination $\boldsymbol{Q}=0$8 carries the same sign at all four points — a Berry curvature monopole — whereas the antisymmetric combination forms a quadrupole mirroring the multipolar momentum-space Berry curvature. This contrast is central to the paper's argument: the strain-space curvature disentangles intrinsic multipolar structure that the momentum-space curvature cannot.

Expanding around each Dirac valley, dynamic strain enters as spin-dependent emergent electromagnetic gauge fields, analogous to static strain gauge fields in graphene but acting as emergent electric fields because the strain is dynamic. The resulting closed-form result,

$\boldsymbol{Q}=0$9

identifies the Hall viscosity with the Hall conductivity of a single Dirac point (of unit charge), equal to $\eta^{H}_{ijkl}$0 when the chemical potential lies in the gap and $\eta^{H}_{ijkl}$1 otherwise, with $\eta^{H}_{ijkl}$2. This expression agrees quantitatively with the full tight-binding calculation for small SOC gaps. Summing over all valleys cancels the net Hall conductivity while preserving the viscosity; the effective action reduces to two Chern-Simons terms of opposite sign for the two spin sectors. Two further results follow directly: $\eta^{H}_{ijkl}$3 is proportional to the altermagnetic order parameter $\eta^{H}_{ijkl}$4, and it is largest when the chemical potential sits in the insulating phase — precisely the regime inaccessible to anomalous Hall conductivity measurements. One caveat noted by the authors: within the Dirac gap the viscosity is independent of both $\eta^{H}_{ijkl}$5 and $\eta^{H}_{ijkl}$6, but the limit $\eta^{H}_{ijkl}$7 cannot be taken continuously since well-separated Dirac points require finite $\eta^{H}_{ijkl}$8.

Hexagonal $\eta^{H}_{ijkl}$9-wave altermagnets

For the intrinsically 3D hexagonal model (relevant to CrSb, MnTe, Co$A_{1g}^{-}$0NbSe$A_{1g}^{-}$1), with moments along $A_{1g}^{-}$2 transforming as $A_{1g}^{-}$3, group theory predicts $A_{1g}^{-}$4. Here the dominant contribution arises from the strain-induced modification of the SOC itself: without a strain-dependent SOC term in $A_{1g}^{-}$5, the Hall viscosity vanishes identically. The in-plane-integrated Berry curvature changes abruptly near $A_{1g}^{-}$6 due to a Lifshitz transition of the Fermi surface, producing correspondingly sharp features in $A_{1g}^{-}$7 as a function of chemical potential. As in the $A_{1g}^{-}$8-wave case, $A_{1g}^{-}$9 scales linearly with the order parameter $B_{1g}^{-}$0.

Magnitude and experimental access

The calculated values reach roughly $B_{1g}^{-}$1, which for $B_{1g}^{-}$2\,\AA{} corresponds to $B_{1g}^{-}$3 — comparable to the phonon Hall viscosity recently measured in $B_{1g}^{-}$4-RuCl$B_{1g}^{-}$5 under external field via the acoustic Faraday effect. The paper analyzes how a modified acoustic Faraday geometry could detect the zero-field signal: in a hexagonal $B_{1g}^{-}$6-wave altermagnet, the component $B_{1g}^{-}$7 does not mix strictly $B_{1g}^{-}$8-axis transverse modes, but propagation tilted slightly from the $B_{1g}^{-}$9-axis yields near-degenerate birefringent modes whose polarization rotation acquires a sizable altermagnetic contribution. A detailed magneto-acoustic protocol is deferred to separate work, so the experimental feasibility rests on this qualitative analysis rather than a full calculation presented here.

Limitations and open questions

Several assumptions bound the results. The microscopic calculations use minimal tight-binding models with coupling constants assumed comparable to hopping parameters, taken from prior work rather than derived from first principles for specific materials. The quantitative agreement between the Dirac formula and the full lattice calculation holds only for small SOC-induced gaps. The cubic $B_{2g}^{-}$0-wave altermagnet is a symmetry-enforced exception with vanishing Hall viscosity, so the probe is not universal across all altermagnetic classes. Finally, the connection to concrete measurement geometries remains incomplete pending the dedicated analysis of acoustic setups referenced but not contained in this paper.

Conclusion

This work establishes the zero-field phonon Hall viscosity as a bulk geometric response unique to time-reversal-breaking compensated magnets, with a tensor signature that separates pure altermagnets from ferromagnets and conventional antiferromagnets. Its microscopic origin — a strain-space Berry curvature monopole concentrated at SOC-gapped Dirac points and sensitive to Lifshitz transitions — makes it simultaneously a symmetry diagnostic and a probe of band topology, applicable to insulators that constitute most altermagnetic candidates. The proportionality to the altermagnetic order parameter and the predicted magnitude comparable to existing acoustic Faraday measurements make this a concrete proposal for experimental detection of altermagnetic order and its underlying topology.