Effective phonon models based on symmetry-adapted multipole basis -- Hidden chiral phonon angular momentum splitting in ferroaxial systems

Abstract: We propose a symmetry-based framework for constructing effective harmonic phonon models using a symmetry-adapted multipole basis. By decomposing the force-constant matrix into bond-centered electric multipoles, we identify the minimal microscopic ingredients responsible for phonon angular-momentum splitting. Applying this framework to a minimal zigzag-chain model, we show that ferroaxial order gives rise to a hidden sublattice-resolved chiral phonons, while an additional polar contribution leads to finite global chirality. Our results provide a unified symmetry-based description of hidden and emergent phonon phenomena and suggest a route to control phonon properties via electronic orderings and external fields.

• The paper proposes a symmetry-adapted multipole formalism that decomposes force-constant matrices into electric multipole components to model phonon behavior.
• It demonstrates that ferroaxial order induces hidden chiral splitting in phonon angular momentum between sublattices using a minimal zigzag chain model.
• The study shows that combining ferroaxial and polar orders lifts symmetry-enforced cancellation, enabling a global chiral phonon response.

Symmetry-Adapted Multipole-Based Phonon Modeling and Chiral Angular Momentum in Ferroaxial Systems

Background and Motivation

The study addresses the microscopic origin and symmetry-driven control of phonon angular momentum (AM) splitting, particularly in ferroaxial systems. Chiral phonons, characterized by AM along propagation ($\bm{q}$), have been identified in structurally chiral lattices, leading to symmetry-dependent splitting phenomena observed via Raman and X-ray spectroscopies. Ferroaxial crystals, distinguished by axial-vector order preserving inversion symmetry ($\mathcal{P}$), have recently been observed to show chiral responses despite being nonchiral. However, the mechanisms underlying phonon AM splitting and the conditions for emergent global chirality in these systems, as well as the role of polar contributions, remain insufficiently clarified.

Symmetry-Adapted Multipole Basis (SAMB) Formalism

The paper proposes a symmetry-driven framework for constructing effective harmonic phonon models using the symmetry-adapted multipole basis (SAMB). The underlying principle is a complete decomposition of the force-constant matrix into electric multipole components (monopole and quadrupole), with explicit inclusion of spatial and sublattice symmetries. This approach is rooted in the formalism detailed in [HayamiEtAl2018PRB], [KusunoseOiwaHayami2023PRB], and recent software developments such as MultiPie.

Within this framework, the harmonic potential energy is expressed as a sum over bond-centered multipole contributions, enabling a systematic expansion:

$$\hat{\Phi}^{\rm (on)} = \sum_j z_j^{\rm (on)} \hat{\mathbb{Z}}_j^{\rm (on)}, \quad \hat{\Phi}^{\rm (inter)} = \sum_j z_j^{\rm (inter)} \hat{\mathbb{Z}}_j^{\rm (inter)},$$

where $\hat{\mathbb{Z}}_j$ are symmetry-driven multipole bases, and $z_j$ are model parameters constrained by the acoustic sum rule.

Technical constraints (Hessian symmetry, acoustic sum rule, positive definiteness) ensure the physicality and completeness of the constructed effective Hamiltonians. The irreducible representations (irreps) of the point group symmetry are leveraged such that only symmetry-allowed SAMBs contribute, and symmetry-breaking terms encode effects of structural or electronic orderings.

Minimal Zigzag Chain Model and Ferroaxiality

The theoretical framework is instantiated using a minimal zigzag chain model with two sublattices (A, B), corresponding to space group No. 51 ($D_{2h}$). Simple atomic displacements and bond orientations are considered, with parametrization for both isotropic and bond-oriented anisotropic force constants. This allows transparent examination of how symmetry-adapted multipoles induce distinctive phonon AM phenomena.

A ferroaxial SAMB is constructed as an axial vector ($z$ direction), breaking specific mirrors (e.g., $\sigma_{yz}$, $\sigma_{zx}$) but preserving inversion. The crucial result is that ferroaxial order induces a hidden, sublattice-resolved chiral phonon AM: i.e., $L_x^{\rm (ph),(A,B)}(q_x)$ is antisymmetric in $\mathcal{P}$0 with opposing signs for A and B sublattices, yielding zero net AM. This splitting is an analog to hidden spin polarization observed previously [BhowalSpaldin2024PRR].

Quantitatively, the hidden AM emerges with only nearest-neighbor isotropic force constants, indicating robust minimal ingredients for ferroaxial-driven effects.

Polar Contributions and Emergent Chirality

A polar SAMB induces explicit electric polarization along $\mathcal{P}$1, breaking inversion and additional mirrors (e.g., $\mathcal{P}$2). When solely polar order is present, Rashba-type antisymmetric splitting of the phonon AM appears, requiring both isotropic and anisotropic nearest-neighbor force constants.

Combining ferroaxial and polar SAMBs removes all mirror and inversion symmetries, yielding a truly chiral system. The cancellation of local chirality, enforced by symmetry in the pure ferroaxial case, is lifted, resulting in finite global chiral phonon AM. Notably, global chirality appears with minimal model parameters, indicating that symmetry interplay, rather than force-constant complexity, dictates observable chiral phonon phenomena.

Unified Microscopic Description and Implications

The symmetry-adapted multipole expansion provides a unified framework for classifying phonon AM splitting across ferroaxial, polar, and intrinsically chiral crystals. The identification of minimal ingredients responsible for hidden and emergent AM phenomena has significant theoretical implications:

• Microscopic Origin Clarification: The work demonstrates that bond-centered electric quadrupole and toroidal multipole components are essential for phonon AM splitting, with ferroaxial order generating hidden sublattice chirality and polarity unlocking global chiral responses.
• Symmetry-Driven Materials Engineering: The symmetry-adapted approach allows systematic design of phononic properties through targeted symmetry breaking (e.g., inducing polar order via external electric fields in ferroaxial compounds) to realize desired chiral response in nonchiral lattices.
• Connection to Electronic Orderings: The formalism facilitates parallel exploration of spin, charge, and multipolar degrees of freedom, potentially clarifying phenomena such as hidden spin polarization seen in ferroaxial systems [BhowalSpaldin2024PRR], and provides routes for cross-correlation between electronic and phononic angular momentum.
• Experimental Consequences: The conditions for emergent global chirality derived here inform design of experiments utilizing Raman and X-ray spectroscopies for detecting chiral phonon AM, and suggest practical strategies for controlling phononic and optical activity via symmetry-breaking fields.

Speculation on Future Developments

The symmetry-adapted multipole framework is anticipated to drive further advances in:

• Nonlinear Phononic and Optical Response: Controlled interplay of ferroaxiality and polarity may enable new regimes for phonon-induced transport, phonon-driven electron spin polarization, and selective excitation protocols.
• Topological and Quantum Phenomena: Extension to systems with strong spin-orbit coupling and magnetic order may unveil topological phononic states with robust chiral AM features.
• Materials Informatics and Automated Design: MultiPie and similar libraries could automate search for compounds where symmetry constraints enable emergent phononic chirality, leveraging the minimal ingredient analysis for targeted materials discovery.

Conclusion

This study delivers a rigorous symmetry-driven methodology for effective phonon modeling and elucidates the microscopic conditions for hidden and emergent chiral phonon angular momentum splitting in ferroaxial systems. The symmetry-adapted multipole expansion serves as a unified theoretical foundation, enabling systematic analysis and design of phononic responses via electronic orderings and external fields. The results sharpen understanding of chirality in phononic systems and lay the groundwork for engineering advanced materials with tunable chiral phonon and optical activity (2604.23109).