Equilibrium Spin Polarization Arising From Chirality

TL;DR

This work resolves the tension between equilibrium spin polarization and thermodynamic reversibility by formulating a pseudo-Hermitian, $\mathcal{PT}$-symmetric framework in which chirality and electronic correlations generate a spin–displacement order $\langle \sigma\cdot x \rangle$ without violating detailed balance. Through a Dyson map, the non-Hermitian spin–chirality term $i\alpha\,\sigma\cdot p$ is mapped to a Hermitian image with a real spectrum and a nonlocal $\eta$-metric that entangles spin and space, producing a cismagnetic phase. Generalized Onsager relations are derived in the $\eta$-metric, yielding antisymmetric cross-couplings $L_{sc}$ and $L_{cs}$ while preserving diagonal responses, and extending Bardarson’s theorem to chiral systems via a composite antiunitary symmetry $\Theta$. The theory provides experimentally testable predictions for equilibrium spin polarization, spin-to-charge transduction, and energy scales set by $\alpha$, offering a thermodynamically consistent route to link chemical chirality with measurable spin phenomena in light-element systems and guiding future ab initio implementations. The framework thus positions CISS as a finite-system, nonlocal correlation phenomenon beyond conventional Hermitian single-particle pictures, with potential implications for quantum transport, biology, and ultrafast spin dynamics.

Abstract

Chirality-induced spin selectivity (CISS) describes how chiral molecules and materials generate spin polarization even at thermal equilibrium. This observation has challenged established principles of microscopic reversibility and Onsager reciprocity. We resolve this paradox by formulating a pseudo-Hermitian quantum framework in which structural chirality and electron correlations are sufficient to produce CISS observables. Chirality enters through a non-local metric that couples spin and spatial motion, leading to real spectra, unitary evolution, and thermodynamic consistency. The framework predicts a chirality-induced spin magnetic ordering characterized by a spin--displacement order $\langle σ\cdot x \rangle$, which reconciles equilibrium spin polarization with detailed balance and explains the persistence of CISS in materials composed of light elements. We also derive generalized Onsager-Casimir relations that respect the observed parity ($\mathcal{P}$) and time-reversal ($\mathcal{T}$) breaking, while preserving combined $\mathcal{PT}$-symmetry. This approach establishes a coherent foundation for equilibrium CISS and provides a route to link chemical chirality with measurable spin-to-charge conversion effects.

Figures (3)

• Figure 1: Scattering from a chiral object. A scattering process involving a chiral material. A particle traveling form $1 \rightarrow 2$ (top left) scatters from a chiral object (here denoted as a thetrahedron whose four vertices are distinct). Microscopically reversing that process, (bottom left), i.e., applying the time-reversal operator $\mathcal{T}$ produces a distinct sequence of events. To fully reverse the process requires a combined parity, $\mathcal{P}$ and $\mathcal{T}$, or $\mathcal{PT}$ operation. Only in the $\mathcal{PT}$ case the sequence of events (scattering off from the green before blue) is reversed correctly. However, the in chiral systems the $\mathcal{P}$ operation transforms the system from one enatiomer to the other. This paradox illustrates the complexity of applying microscopic reversibility in a chiral system. Arrows show the direction of movement of the point-like particle.
• Figure 2: Dyson mapping for structurally chiral Hamiltonians. (a) The physical space of all Hamiltonians $H$ includes pseudo-Hermitian cases ($H_I$, $H_T$, $H_P$, $H_{PT}$), with Hermitian Hamiltonians ($\alpha=0)$ forming a subset thereof. (b) In the mapped space, Hamiltonians are categorized according to their eigenenergies. Real eigenenergies correspond to the Hermitian and pseudo-Hermitian Hamiltonians and are represented by the shaded region. All chiral pseudo-Hermitian Hamiltonians can be parameterized by the order parameter $\alpha$, where $\alpha = 0$ corresponds to physical Hermitian systems. (c) For each eigenvalue, the corresponding left- and right-sided eigenvectors of the Dyson-mapped Hermitian Hamiltonians $h$ are obtained using the same unitary and antiunitary operations for both enantiomers. Green shading corresponds to the $\eta$-metric, blue shading to $\eta^{-1}$ metric sectors with equivalent mapping. (d) The physical Hilbert-space of the inverse Dyson mapping yields the physical eigenvectors of the pseudo-Hermitian Hamiltonians $H_I$ and $H_{PT}$ for green shading, which describe distinct physical systems but share an equivalent mapping structure. Blue shading would correspond to $H_{P}$ and $H_{T}$. The Dyson mapped unitary and antiunitary operators are defined within each sector (i.e., they do not allow crossing to another sector) and allow for defining the microscopic reversibility symmetry and detailed balance within each sector, resolving the paradox presented in Fig. \ref{['Fig:PT-scattering']}.
• Figure 3: Topological separation of the state space in a chiral system into four dynamically disconnected ergodic components: (a) Symmetry relations of the metrics $\eta$ and $\eta^{-1}$ under parity $\mathcal{P}$ and time-reversal $\mathcal{T}$-operation. States with the same metric have degenerate energies but mirror image wavefunctions. (b) Illustration of Dyson mapping with the same symmetries applied to a triangular potential well problem. The probability distributions is plotted in normalized coordinates $\lambda z$ and staggered according to their normalized energiy $E_n/E_0$. The energy $E_0$ corresponds to the eigenenergy of the ground state of the non-chiral problem $(\alpha=0)$, these eigenstate are plotted as faint white contours. The effect of $\alpha$ is twofold: (1) the eigen energies are shifted by $\alpha^2m/2$ depending upon the chiral sector, (2) spins $\leftarrow$ and $\rightarrow$ are pushed apart to a finite spin-displacement $z\sigma_z$.