Volosniev et al. (Phys. Rev. B 104, 024430 (2021)) showed that a one-dimensional quantum carrier subject simultaneously to spin-orbit coupling (SOC) and phenomenological friction reaches a steady state in which opposite spins are spatially separated, and proposed this as an equilibrium mechanism behind chirality-induced spin selectivity (CISS) in chiral molecules, including Shiba-like states (localized electronic states inside the superconducting energy gap) on superconductors. The result rests on a nonlinear dissipative Schrödinger equation with the Albrecht friction potential, assumes spin is conserved on the way to the steady state, and ignores temperature. The open problem, stated by the authors and left unresolved by subsequent literature, is to give this mechanism a microscopic foundation and to test its physical robustness: (i) derive the effective model from a concrete electron-plus-environment Hamiltonian, establishing the quantitative mapping between microscopic electron-bath couplings and the friction coefficient γ\gamma and between molecular chirality and the direction of the effective SOC; (ii) determine whether the steady-state spin separation survives when spin-flip processes are present; (iii) establish the finite-temperature behavior of the polarization, including whether an optimal temperature exists. A 2023 follow-up by part of the same group derived a Caldeira-Leggett-type master equation (the standard effective description of a quantum particle linearly coupled to a bath of harmonic oscillators) for a spin-orbit-coupled impurity in a cold-atom bath, but kept γ\gamma phenomenological and did not address the molecular setting, leaving the core open.

Background

When an electron moves through a chiral (handed) molecule, its transmission can depend on its spin — the chirality-induced spin selectivity (CISS) effect, a long-standing puzzle because organic molecules are made of light elements whose intrinsic spin-orbit coupling (SOC), i.e. the interaction tying the electron spin to its orbital motion, is weak, yet observed spin polarizations can be large. One analytic precursor (Ghazaryan, Paltiel, Lemeshko, J. Phys. Chem. C 124, 11716 (2020)) mapped a chiral molecule onto an anisotropic wire and showed that a helical structure is not necessary for spin selectivity. Volosniev et al. (Phys. Rev. B 104, 024430 (2021)) then proposed a complementary, deliberately minimal mechanism: friction plus SOC. They studied a particle of mass mm confined to a one-dimensional box (half-width aa) obeying a modified Schrödinger equation Hs=p2/2m+αps+V(x)+γWH_s = p^2/2m + \alpha p s + V(x) + \gamma W with s=±1s = \pm 1 the conserved spin projection, αps\alpha p s a 1D SOC term, and γW\gamma W a friction potential in the tradition of nonlinear dissipative Schrödinger equations (Caldirola-Kanai, Kostin, Süssmann, Albrecht), which are constructed phenomenologically so that quantum expectation values obey the classical equations of motion with Stokes drag −γp-\gamma p. For Albrecht's choice W=⟨p⟩s(x−⟨x⟩s)W = \langle p \rangle_s (x - \langle x \rangle_s) (a nonlinear, state-dependent potential; ⟨⋅⟩s\langle \cdot \rangle_s denotes the expectation value in the spin-ss state), a gauge transformation folds the SOC term into a renormalized dissipation strength β=−γαs/2\beta = -\gamma\alpha s/2, and the steady state is an Airy-function state in which particles of opposite spin settle at opposite ends of the box: ⟨x⟩+1=−⟨x⟩−1\langle x \rangle_{+1} = -\langle x \rangle_{-1}, with separation controlled by the product γα\gamma\alpha, so strong friction compensates weak SOC. With parameters estimated for heptahelicene (a chiral, helix-shaped polycyclic molecule; ℏα≈5.2\hbar\alpha \approx 5.2 meV·Å, γ≈4.19×1014\gamma \approx 4.19\times10^{14} s−1^{-1} from mobility data), the model predicts equilibrium spin accumulation at the molecular edges — implying that a chiral molecule acts as a magnet in equilibrium, consistent with Shiba-like in-gap states (localized states inside the superconducting energy gap) observed when chiral molecules sit on an ss-wave superconductor (a superconductor with isotropic, momentum-independent pairing). However, three pillars of the construction are unproven. First, the friction term is phenomenological: no microscopic electron-plus-environment Hamiltonian is known from which the Albrecht (or related) form of WW emerges with a quantitative mapping between the bath couplings and γ\gamma, or between molecular chirality and the sign of α\alpha; this matters because 1D SOC can be gauged away, so a faithful microscopic parent must contain genuinely beyond-1D ingredients. Deriving nonlinear friction Schrödinger equations from first principles is a recognized open difficulty even without SOC (K. Albrecht, Phys. Lett. B 56, 127 (1975); see Mousavi and Miret-Artés, Eur. Phys. J. Plus 134, 431 (2019) for a survey of this equation class). Second, the analysis assumes spin is conserved on the timescale for reaching the steady state (the spin-coherence length exceeds the mean free path), so the fate of the spin separation when spin flips are present is unknown. Third, the model is at zero temperature; how thermal fluctuations and the temperature dependence of spin coherence and bath coupling modify the polarization — in particular whether an optimal temperature exists, as the authors speculate for biological relevance — is undetermined.

Problem Statement

Decide whether friction-induced spin polarization is microscopically real and physically robust, by completing the three linked tasks stated as future work in Volosniev et al. (Phys. Rev. B 104, 024430 (2021)):

(1) Microscopic foundation. Starting from a concrete microscopic Hamiltonian for a charge carrier coupled to an environment, derive the effective one-dimensional dissipative SOC model

Hs=p22m+αps+V(x)+γW,H_s = \frac{p^2}{2m} + \alpha p s + V(x) + \gamma W,

with s=±1s = \pm 1 and WW of Albrecht's (or a related Kostin/Süssmann) nonlinear form. The derivation must state its approximations explicitly, establish the quantitative mapping between the microscopic electron-bath couplings and the phenomenological friction coefficient γ\gamma, and establish how molecular chirality fixes the sign and direction of the effective SOC α\alpha (noting that a faithful microscopic parent must contain genuinely beyond-1D ingredients, since 1D SOC is gauge-removable).

(2) Robustness to spin flips. Within the microscopic description (or an equally justified open-system treatment), determine the steady state of the confined carrier when spin-flip processes are no longer negligible relative to spin-preserving relaxation: does the spatial separation of opposite spins survive, and with what magnitude as a function of the spin-flip rate?

(3) Finite temperature. Determine how thermal fluctuations, the temperature dependence of spin coherence, and the temperature dependence of the bath coupling modify the steady-state spin polarization, and establish whether an optimal temperature exists as the authors speculated.

These three tasks jointly serve one objective: to convert the phenomenological demonstration that friction and SOC act in unison to spatially separate spins into a microscopically grounded, physically robust (or refuted) mechanism for equilibrium spin polarization in chiral molecules.

References

  1. A. G. Volosniev, H. Alpern, Y. Paltiel, O. Millo, M. Lemeshko, A. Ghazaryan, 'Interplay between friction and spin-orbit coupling as a source of spin polarization', Physical Review B 104, 024430 (2021). DOI: 10.1103/PhysRevB.104.024430, https://doi.org/10.1103/PhysRevB.104.024430 (arXiv:2101.05173)
  2. K. Albrecht, 'A new class of Schrödinger operators for quantized friction', Physics Letters B 56, 127-129 (1975). DOI: 10.1016/0370-2693(75)90283-X, https://doi.org/10.1016/0370-2693(75)90283-X
  3. A. Ghazaryan, A. Cappellaro, M. Lemeshko, A. G. Volosniev, 'Dissipative dynamics of an impurity with spin-orbit coupling', Physical Review Research 5, 013029 (2023). DOI: 10.1103/PhysRevResearch.5.013029, https://doi.org/10.1103/PhysRevResearch.5.013029 (arXiv:2210.01829)
  4. A. Varshney, A. Ghazaryan, A. Volosniev, 'Classical Spin Filtering with Two Degrees of Freedom and Dissipation', Few-Body Systems 65, 12 (2024). DOI: 10.1007/s00601-024-01880-x, https://doi.org/10.1007/s00601-024-01880-x (arXiv:2401.08454)
  5. A. Ghazaryan, Y. Paltiel, M. Lemeshko, 'An analytic model of chiral-induced spin selectivity', The Journal of Physical Chemistry C 124, 11716-11721 (2020). DOI: 10.1021/acs.jpcc.0c02584, https://doi.org/10.1021/acs.jpcc.0c02584 (arXiv:2002.09938)
  6. S. V. Mousavi, S. Miret-Artés, 'On non-linear Schrödinger equations for open quantum systems', The European Physical Journal Plus 134, 431 (2019). DOI: 10.1140/epjp/i2019-12965-6, https://doi.org/10.1140/epjp/i2019-12965-6 (arXiv:1902.04410)
  7. T. K. Das, F. Tassinari, R. Naaman, J. Fransson, 'Temperature-Dependent Chiral-Induced Spin Selectivity Effect: Experiments and Theory', The Journal of Physical Chemistry C 126, 3257-3264 (2022). DOI: 10.1021/acs.jpcc.1c10550, https://doi.org/10.1021/acs.jpcc.1c10550
  8. J. Fransson, 'Chiral Induced Spin Polarized Electron Current: Origin of the Chiral Induced Spin Selectivity Effect', The Journal of Physical Chemistry Letters 16, 4346-4353 (2025). DOI: 10.1021/acs.jpclett.5c00104, https://doi.org/10.1021/acs.jpclett.5c00104
  9. P. M. Theiler, M. C. Beard, 'Equilibrium spin polarization arising from chirality', Newton 2, 100480 (2026). DOI: 10.1016/j.newton.2026.100480, https://doi.org/10.1016/j.newton.2026.100480