Electronic Hall viscosity: hidden indicator for antiferromagnets

Technical Summary: Electronic Hall Viscosity as a Hidden Indicator for Antiferromagnets

Problem Statement
Antiferromagnets are critical for energy-efficient spintronics and topological electronics due to their negligible stray fields and ultrafast (THz) spin dynamics. However, characterizing and controlling the underlying nontrivial Berry curvature in these materials is severely limited. In the linear response regime, the vanishing net magnetization and the resulting zero anomalous Hall conductivity (AHC) in many antiferromagnetic configurations prevent the detection of these quantum geometric properties. Consequently, there is a need for a new physical quantity that can characterize antiferromagnetic ordering even when the linear anomalous Hall response is symmetry-forbidden.

Methodology
The authors establish a theoretical framework connecting the electronic Hall viscosity (EHV) to the quantum geometry of Bloch bands.

1. Formalism: The study begins with the retarded force-force response function to describe the electronic liquid's response to crystal deformation (e.g., acoustic phonons). By utilizing a tetrad field to capture local stretch and rotation, the authors derive the EHV tensor at zero temperature.
2. Multipole Expansion: The EHV is expressed as the integration of the quadrupole moment of the Berry curvature in momentum space ($k$-space). The authors employ a multipole expansion using $k$-space spherical harmonics to decompose the Berry curvature and the weighting factors ($k_\gamma k_\delta$).
3. Symmetry Analysis: The study derives rigorous symmetry conditions for nonzero EHV in generic antiferromagnets by analyzing magnetic point groups (MPGs) and magnetic Laue groups (MLGs). Specifically, it examines how composite symmetries (e.g., $T C_{n\nu}$, $PT C_{n\nu}$) constrain the EHV tensor components, distinguishing them from the constraints on AHC.
4. Quantum Geometry Bounds: Using the Cauchy-Schwarz inequality and the properties of the quantum geometry tensor, the authors derive a fundamental upper bound for the EHV.
5. First-Principles Validation: The theoretical predictions are validated through direct density functional theory (DFT) calculations on two archetypal antiferromagnets: the $d$-wave altermagnet $\text{RuO}_2$ and the noncollinear antiferromagnet $\text{Mn}_3\text{Sn}$.

Key Contributions and Results

• Relation to Quadrupole Berry Curvature: The paper establishes that the EHV is directly proportional to the quadrupole moment of the Berry curvature in $k$-space. Unlike the AHC, which integrates the Berry curvature (monopole), the EHV integrates the Berry curvature weighted by $k_\gamma k_\delta$.
• Quantum Geometry Bound: A fundamental upper bound for the EHV is derived: $\eta_{\gamma\delta\lambda} \leq 2\hbar \sqrt{v_d \cdot \text{vol}^{2nd}_g}$. This bound is determined by the $d$-orbital factor ($v_d$) modulated second moment of the quantum volume ($\text{vol}^{2nd}_g$). This represents a distinct multipole counterpart to the classical inequality between quantum volume and the Chern number.
• Symmetry Conditions for Nonzero EHV: The authors identify that EHV can exist even when AHC is strictly forbidden by symmetry.
  • While $PT$ and $\tau T$ symmetries enforce both EHV and AHC to vanish, specific composite symmetries like $T C_{n\nu}$ allow for nonzero EHV components even when the Berry curvature satisfies conditions that force AHC to zero.
  • For instance, under $T C_{4z}$ symmetry, the AHC vanishes, but the quadrupolar components (e.g., $k_x k_y$) acquire an additional sign change that compensates for the Berry curvature transformation, yielding finite EHV components ($\eta_{xxz}, \eta_{xyz}, \eta_{yyz}$).
• Case Study: $\text{RuO}_2$ (Altermagnet):
  • Calculations show that when the Néel vector lies along the $c$-axis, the AHC vanishes. However, the EHV remains nonzero, with specific components ($\eta_{zzz}$) emerging that are sensitive to the Néel vector orientation.
  • The EHV acts as a "smoking-gun" transport signature for the unique spin-splitting Fermi surface in $d$-wave altermagnets.
• Case Study: $\text{Mn}_3\text{Sn}$ (Noncollinear Antiferromagnet):
  • The study examines two candidate magnetic ground states (Type-III and Type-IV). While both configurations share the same MLG and allow for nonzero AHC, the magnitude and sign of specific EHV tensor components exhibit dramatic differences between the two.
  • This suggests that anisotropic EHV can serve as a decisive probe to identify the true magnetic ground state of $\text{Mn}_3\text{Sn}$, distinguishing between nearly degenerate configurations.

Significance and Claims
The paper claims to reveal a new and fundamental quantum geometry quantity for generic antiferromagnets. By demonstrating that EHV is closely related to the quadrupole Berry curvature and is bounded by the second moment of the quantum volume, the work offers a broadly applicable method to characterize antiferromagnetic ordering. Crucially, it provides a way to detect and design antiferromagnetic spintronic devices via unconventional Hall viscosity in regimes where the linear anomalous Hall response is forbidden. The authors posit that this analysis, based on $k$-space spherical harmonics, may also be applicable to other quantum geometric quantities, such as orbital and heat magnetic moments, in both magnetic and non-magnetic materials.

Experimental Detection
The paper briefly notes potential experimental avenues:

• For metallic systems, nonlocal Hall measurements or local voltage measurements near current-injecting contacts could detect EHV responsible for inhomogeneous electron fields or acoustic phonons.
• For insulating systems, EHV manifests in phonon physics and could be accessed via the acoustic Faraday effect.