Modern Theory of Chiralization

• Modern Theory of Chiralization is a multidisciplinary framework that extends traditional molecular chirality to bulk systems by defining measurable chiral order parameters analogous to polarization in dielectrics.
• It integrates quantum, statistical, and field-theoretical methods to analyze symmetry breaking, topological invariants, and chiral-induced phenomena such as spin selectivity.
• The framework supports advances in materials science and prebiotic chemistry by providing computational tools to design and predict chiral phase transitions and electronic magneto-chiral effects.

The modern theory of chiralization encompasses a rapidly developing, multidisciplinary framework for understanding the emergence, quantification, and control of chirality in physical, chemical, and biological systems. Chiralization extends beyond the recognition or classification of objects as chiral, aiming to rigorously define, measure, and engineer handedness as a bulk or collective property—analogous to the development of modern polarization theory for dielectrics. Recent advances integrate quantum, statistical, and field-theoretical perspectives, addressing bulk order parameters, topological invariants, many-body effects, electronic correlations, nonequilibrium mechanisms, and multipole expansions. The concept unifies the spontaneous symmetry breaking driving homochirality in prebiotic chemistry, chiral-induced spin selectivity in electronic transport, chiral order in soft and condensed matter, and the precise algebraic treatment of chirality in quantum field theory and algebraic structures.

1. Bulk Chirality: Definitions, Order Parameters, and Operator Approach

Bulk chirality—termed "chiralization"—demands an intensive, translationally invariant quantity that meaningfully extends the notion of molecular handedness to periodic solids, fields, and many-body systems. This is conceptually parallel to the transition from molecular dipole moments to bulk polarization in insulators.

Operator Construction and Multipole Frameworks

The electric toroidal multipole expansion rigorously classifies quantities according to time-reversal (T) and parity (P) symmetry:

• Electric-toroidal monopole $G_0$ (P-odd, T-even): The canonical bulk chirality order parameter, directly mapping to the helicity operator $\mathbf{p}\cdot\boldsymbol{\sigma}$ for electrons and $\psi^\dagger\gamma^5\psi$ in Dirac theory (Kusunose et al., 2024, Spaldin, 22 Jan 2026).
• Extensions to higher toroidal multipoles (e.g., $G_2$, $G_4$): Achieve sensitivity to more subtle chiral arrangements and avoid false zeros present in $G_0$ during continuous deformations between enantiomers (Spaldin, 22 Jan 2026).

For phonon systems and collective vibrations, bulk chirality can be constructed as a mode chirality, for example via the difference of right- and left-circular polarization projections of phonon eigenvectors (Spaldin, 22 Jan 2026).

Topological and Berry-Phase Representation

A fully modern theory aspires to represent bulk chirality as a $k$-space Berry phase or gauge-invariant integral: $$\chi = \sum_{n \in \text{occ}} \int_{\text{BZ}} d^3k\, \mathcal{F}_{\text{chir}}[u_{n\mathbf{k}}]$$ where $\mathcal{F}_{\text{chir}}$ is the Berry curvature associated with the appropriate chiral operator (Spaldin, 22 Jan 2026).

Chiral Lattice and Quantum of Chirality

In analogy to the polarization lattice, chiralization in periodic systems is only defined modulo a quantum. This multi-valuedness must be managed when computing physical observables or comparing different domain terminations (Spaldin, 22 Jan 2026).

2. Chiral Symmetry Breaking: Mechanisms, Phase Transitions, and Emergence

Spontaneous chiralization arises as a symmetry-breaking transition in diverse contexts, classically and quantum-mechanically. Modern statistical and dynamical models demonstrate generic routes to robust homochirality or chiral domains.

Non-Equilibrium Reaction Networks and Prebiotic Homochirality

Extending Frank–Sandars autocatalytic models, large non-equilibrium networks of autocatalytic reactions with many chiral species exhibit phase transitions to global homochirality as a consequence of combinatorial explosion in chiral molecular types (Laurent et al., 2021). The order parameter is the enantiomeric excess, whose stability and critical behavior can be analyzed via random-matrix approaches: $\frac{dc}{dt} = F(c) + \text{inflow/outflow}$ where $F$ encodes the network, and instability of the racemic fixed point yields chiral order (Laurent et al., 2021).

Environmental and Punctuated Chiralization

In prebiotic or planetary environments, strong, stochastic perturbations (e.g. impacts, UV events) act as agents of symmetry breaking. The "punctuated chirality" paradigm describes how environmental noise above a critical threshold both drives systems into, and can reverse, global chiral states. The dynamics are captured by reaction-diffusion–Langevin systems: $$\partial_t \mathcal{A} - k \nabla^2 \mathcal{A} = \mathcal{S}\,\mathcal{A} \left(\frac{2f}{\mathcal{S}^2}+\mathcal{A}^2-1\right) + w(t,\mathbf{x})$$ where $\mathcal{A}$ is the local chiral order parameter, $w$ is stochastic forcing, and chiralization occurs via saddle-node bifurcations as noise intensity crosses the Ising-type boundary (0802.1446).

3. Chiralization in Electronic, Spin, and Field Systems

CISS and Non-Hermitian Quantum Theory

The chiral-induced spin selectivity (CISS) effect, in which transmission through a chiral system converts unpolarized electrons into spin-polarized currents, is captured by models incorporating non-Hermitian (pseudo-Hermitian) chiral Hamiltonians: $H = \frac{p^2}{2m} + i\alpha\,\sigma\cdot p + V(x)$ The term $i\alpha\,\sigma\cdot p$ arises from structural chirality—specifically, kinematically allowed "twin-pair" exchanges in a four-electron (tetrahedral) motif—breaking $\mathcal{P}$ and $\mathcal{T}$ individually but preserving combined $\mathcal{PT}$-symmetry. This leads to:

• Spin-momentum locking and non-Hermitian skin effects.
• Equilibrium spin-displacement order $\langle \sigma \cdot x \rangle$ ("cismagnetism").
• Quantitative predictions for interface spin and charge accumulation, and Onsager-Casimir relations modified in the $\eta$-metric (Theiler et al., 9 May 2025, Theiler et al., 28 Oct 2025).

Dynamical Theory and Lindblad Formalism

Chiral-induced spin selectivity in electron transfer is further elucidated by Lindblad-type master equations for open quantum systems. Spin–orbit–coupled transport through a chiral bridge combined with dephasing produces steady-state spin polarization, with maximum polarization achieved when the dephasing time matches the separation of spin transit times (Zhang et al., 4 Sep 2025).

Electronic Chiralization and Magnetotransport

The concept of electronic chiralization as a bulk pseudovector indicator—constructed from gradients of charge and magnetic density—is shown to correlate with the anomalous Hall effect, even in compensated antiferromagnets. The quantity

$$\boldsymbol\chi_1 = \frac{1}{V} \int d^3r\, [\nabla\phi(\mathbf{r})][\nabla\cdot \mathbf{m}(\mathbf{r})]$$

serves as a symmetry-appropriate order parameter, directly measurable via advanced scattering techniques (Chen, 2022).

4. Chiralization Beyond Structure: Dynamic, Rotational, and Field Paradigms

Rotationally Induced Chirality (RIC)

Chirality can be dynamically induced even in equilibrium-achiral molecules by driving them to extreme rotational excitation. An optical centrifuge combined with a weak static field lifts the degeneracy of chiral cluster states and produces macroscopically chiral ensembles, with order parameters expressed as triple products of angular momentum or time-dependent dipole expectation values (Owens et al., 2018).

Chirality Polarization and Multipole Expansion in Fields

The notion of "polarization of chirality" extends chiralization to classical and quantum electromagnetic fields. Chirality density $C(\mathbf{r},t)$ and higher multipoles (dipole, quadrupole, etc.) can be systematically constructed, with conservation laws and constitutive relations in chiral media: $$C(\mathbf{r},t) = \frac{\varepsilon_0}{2} \mathbf{E}\cdot(\nabla\times\mathbf{E}) + \frac{1}{2\mu_0} \mathbf{B}\cdot(\nabla\times\mathbf{B})$$ This enables engineering of racemic but chirality-polarized light fields that drive maximal enantioselective responses in matter, including optical dichroism exceeding 200% (Ayuso et al., 2020).

5. Cooperative Chiral Ordering, Soft Materials, and Multipole Interconversion

Soft Matter and Liquid Crystals

In nematic and cholesteric liquid crystals, cooperative ordering between rapidly interconverting axial-chiral molecular states can give rise to spontaneous deracemization and chiral amplification. Mean-field Maier–Saupe–Ising models predict:

• Continuous chiral transitions and diverging susceptibility,
• Phase diagrams with deracemized cholesteric and racemic nematic regions,
• Pre-transitional enhancement of helical twisting power (Deutsch et al., 5 Jun 2025).

Monte Carlo simulations of lattice models validate these predictions and reveal domain coarsening kinetics.

Multipole Interconversion

Modern theory recognizes that achiral multipoles (electric, magnetic, toroidal) can, through interconversion, yield true chiral order parameters ($G_0$) (Kusunose et al., 2024). For instance, the combination of ET-dipole and E-dipole or the dot product of current and magnetic field (e.g., $\mathbf{J} \cdot \mathbf{B}$) produces an emergent $G_0$, manifesting in electrical magneto-chiral anisotropy and other observables.

6. Open Challenges and Perspectives

Despite substantial progress, several central challenges remain:

• Identification of the universal local chiral operator—potentially a combination of $G_0$ and higher toroidal multipoles—that yields an unambiguous, gauge-invariant bulk chiral order parameter under periodic boundary conditions (Spaldin, 22 Jan 2026).
• Construction of a closed-form Berry-phase–style bulk chiralization formula.
• Implementation of efficient and robust computational algorithms for chiralization in first-principles electronic structure codes.
• Systematic classification and control of chiral phase transitions, including the interplay of spin, lattice, and topological chiral orders.
• Elucidation of the full impact of electron correlations, vibrational exchange, and environmental influences on equilibrium and dynamical chiral phenomena.

Current research bridges foundational quantum mechanical/statistical models, condensed-matter multipole analysis, and nonequilibrium and environmental mechanisms, establishing a comprehensive framework for the quantification, prediction, and control of chiralization across molecular, mesoscale, and bulk regimes (Spaldin, 22 Jan 2026, Kusunose et al., 2024, Laurent et al., 2021, 0802.1446, Theiler et al., 9 May 2025, Theiler et al., 28 Oct 2025, Chen, 2022, Ayuso et al., 2020, Deutsch et al., 5 Jun 2025).