Chirality-Induced Axiality via Surface Polarization

• Chirality-induced axiality via surface polarization is a phenomenon where an intrinsic polar field couples chiral and axial orders at interfaces, leading to preferential domain stabilization.
• A minimal s–p hybridized tight-binding model with spin–orbit coupling and group-theoretical multipole analysis quantifies the symmetry-allowed interactions driving the interconversion.
• This mechanism enables robust interfacial functionalities such as enantioselective adsorption and controlled domain formation without relying on external fields.

Chirality-induced axiality via surface polarization is a phenomenon in which the presence of a polar field at an interface or surface enables a coupling between chiral order (handedness) and axial order (directionality), leading to preferential stabilization or selection of one over the other. This interaction, and its reciprocal process (axiality-induced chirality), mediates a variety of domain and interfacial functionalities in materials with broken inversion symmetry. The concept is grounded in group-theoretical multipole analysis, tight-binding microscopic modeling, and symmetry arguments, establishing a rigorous framework for the energetic and functional consequences of chirality–axiality–polarity interconversion in condensed matter systems (Hayami et al., 21 Oct 2025).

1. Symmetry-Induced Multipole Coupling at Surfaces

The underlying mechanism relies on the existence and interplay of three fundamental multipoles:

• Electric toroidal monopole (G₀): A rank-0, time-reversal even pseudoscalar encoding chirality.
• Electric toroidal dipole (𝐆): A rank-1, time-reversal even axial vector encoding axiality or handed directionality (not to be confused with a polar vector).
• Electric dipole (𝐐): A rank-1 polar vector capturing the local polarity or the lack of inversion symmetry, generically present at surfaces and interfaces.

Their mutual relations are captured by the symmetry-allowed couplings:

$$G_0 \leftrightarrow \boldsymbol{Q} \cdot \boldsymbol{G}, \quad \boldsymbol{G} \leftrightarrow G_0 \boldsymbol{Q}, \quad \boldsymbol{Q} \leftrightarrow G_0 \boldsymbol{G}$$

This means that the emergence of a nonzero chiral order parameter $G_0$ (e.g., from bulk structure or interface adsorption) in the presence of a local polar field $\boldsymbol{Q}$ naturally induces a nonzero axial moment $\boldsymbol{G}$ at the surface (and reciprocally). The essential coupling term in the Hamiltonian is:

$$H_{\rm couple} \propto G_0\, (\boldsymbol{Q} \cdot \boldsymbol{G}),$$

which directly ties the preferred orientation of axiality to the handedness of chirality, with the surface polar field acting as a conduit.

2. Microscopic Model and Mathematical Formulation

To capture the interconversion phenomena, the authors construct a minimal s–p hybridized tight-binding chain with spin–orbit coupling and site-dependent mean fields. The model Hamiltonian is: $$\mathcal{H} = \sum_{i,j,\alpha,\alpha',\sigma} t_{ij}^{\alpha\alpha'}\, c_{i\alpha\sigma}^\dagger c_{j\alpha'\sigma} + \frac{\lambda}{2} \sum_{i,\tilde\alpha,\tilde\alpha',\sigma,\sigma'} c_{i\tilde\alpha\sigma}^\dagger (\mathbf{l} \cdot \boldsymbol{\sigma})_{\tilde\alpha\tilde\alpha'}^{\sigma\sigma'} c_{i\tilde\alpha'\sigma'} + \sum_{i,\alpha,\alpha',\sigma,\sigma'} h^i_{G_0} c_{i\alpha\sigma}^\dagger [G_0^{(s)}]_{\alpha\alpha'}^{\sigma\sigma'} c_{i\alpha'\sigma'} + \sum_{i,\alpha,\alpha',\sigma,\sigma'} h^i_{G_x} c_{i\alpha\sigma}^\dagger [G_x^{(s)}]_{\alpha\alpha'}^{\sigma\sigma'} c_{i\alpha'\sigma'}$$ with operator definitions:

• $$G_0^{(s)} = \frac{1}{\sqrt{3}}(T_x \sigma_x + T_y \sigma_y + T_z \sigma_z)$$
• $$G_x^{(s)} = \frac{1}{\sqrt{2}} (l_y \sigma_z - l_z \sigma_y)$$

Here, $T_\mu$ (for $\mu=x,y,z$) are atomic-scale magnetic toroidal dipoles associated with imaginary components of s–p hybridization, and $\mathbf{l}$, $\boldsymbol{\sigma}$ are orbital and spin operators, respectively. The $h_{G_0}$ and $h_{G_x}$ fields control the bulk chirality and axiality. The surface polar field $Q_x$ introduces a real on-site s–p hybridization term.

Solving this Hamiltonian for a substrate with fixed $h_{G_0}$ (bulk chirality) and a surface $Q_x$, the model shows that $\langle G_x^{(s)} \rangle$ is induced at the surface, with the sign of $G_0$ and $Q_x$ dictating the favored sign of $G_x$. The energy splitting between opposite domain configurations grows with $h_{G_0}$, $h_{G_x}$, and $|Q_x|$, enabling stabilization of a unique chiral/axial state at the surface.

3. Chirality-Induced Axiality via Surface Polarization (CIAS)

Chirality-induced axiality via surface polarization (CIAS) refers to the process where the presence of bulk handedness (nonzero $G_0$) and a finite polar field at the surface ($\boldsymbol{Q} \neq 0$) leads to the surface stabilization of a particular component of axiality ($\boldsymbol{G}$) via energetic preference. The coupling is such that the term $G_0 (\boldsymbol{Q} \cdot \boldsymbol{G})$ lowers the energy for a surface domain where the direction of $\boldsymbol{G}$ matches the product $G_0 \boldsymbol{Q}$. Explicitly:

• Bulk chirality selects the sign/orientation of the surface axial vector.
• Surface domain walls become energetically biased, stabilizing single-domain axial states compatible with $G_0$.

This mechanism does not require external fields or global symmetry breaking, as the surface polar field is intrinsic to any boundary (due to the loss of inversion symmetry), and is thus a robust route for domain control. Variation of the local surface polar field (e.g., via surface termination or gating) can tune the magnitude of the surface-induced axial moment.

4. Axiality-Induced Chirality via Surface Polarization (AICS)

The process is reciprocal: axiality-induced chirality at the surface (AICS) occurs in an analogous fashion when the roles of chirality and axiality are reversed. In a substrate with nonzero bulk axiality ($\boldsymbol{G}$), interfacing with a chiral molecule or layer at the surface, the polar field $\boldsymbol{Q}$ energetically favors one sign of $G_0$ over the other, fixing the surface chirality to match the energetically preferred orientation. In both CIAS and AICS, the key role of surface polarization is to mediate the unidirectional coupling, enabling energetic control over interfacial chiral/axial order.

5. Stabilization of Single-Domain States and Interfacial Functionalities

As a consequence of the multipole coupling mechanism, stabilization of single-domain states—either chiral or axial—becomes possible in systems where otherwise degenerate domains might coexist. Control, selection, and switching of these domains require only the engineering of local surface polarity (e.g., via atomic-scale structure, gating, or chemical functionalization), not external fields. This approach enables:

• Deterministic selection of handedness (or axial orientation) at surfaces and interfaces.
• Robust stabilization against thermal fluctuations or domain wall migration.
• Template-based control of surface or interface functionalities, such as selective molecular adsorption or catalysis.

This is particularly relevant for designing materials where enantioselectivity, spin selectivity, or other symmetry-derived functionalities are needed at the interface—e.g., enantioselective crystallization, spintronic applications, or chiral separation.

6. Broader Implications and Prospects

The surface-mediated coupling between chirality and axiality provides a new thermodynamic and symmetry-based pathway for interfacial engineering. This framework applies universally where surfaces or interfaces exhibit finite polarity, including the design of functional multilayers, molecular films on substrates, and artificial heterostructures incorporating both chiral and axial components. It also sets the stage for externally tunable chiral/axial phenomena (e.g., via surface field modulation), and offers theoretical guidance for predicting the interfacial behaviors of complex systems such as polar metals, topological insulators with chiral terminations, and hybrid chiral–axial architectures.

Moreover, the analysis provides symmetry-based selection rules for domain nucleation and orientation at surfaces, which is valuable for the prediction and rational design of next-generation materials with cross-correlated chiroptical, spintronic, and catalytic functionalities.

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The discussion presented here reflects the detailed theoretical and symmetry-based analysis of chirality-induced axiality via surface polarization, including explicit Hamiltonian formulations and multipole expansions, and outlines connections to experimental strategies for domain control and interfacial functionality in chiral systems (Hayami et al., 21 Oct 2025).