Reservoir-Locked Spin Configurations and the Onsager Objection to CISS (Fransson, 2025)

In a September 2025 preprint (arXiv:2509.17817, dated 23 September 2025), Jonas Fransson of the Department of Physics and Astronomy at Uppsala University offers a theoretical answer to an objection raised against Chirality-Induced Spin Selectivity (CISS) (CISS) transport experiments. The paper states the problem this way: Onsager reciprocity requires time-reversal symmetry to remain intact at equilibrium, yet the measured CISS signal changes only weakly with bias, which conflicts with that requirement. Fransson's answer is that a chiral molecule attached to an electron reservoir settles into a fixed, chirality-determined spin arrangement. Its charge distribution then shifts linearly with the reservoir's magnetisation, so there is no linear-response regime in which the reciprocity theorem would apply. The paper is purely theoretical: it contains a model, analytic estimates and numerical examples, and no new measurements.

Which experiments and earlier work the paper leans on

For the weak, same-sign bias dependence, the paper points to six transport studies: Xie and co-workers on DNA oligomers (Nano Letters, 2011); Mishra and co-workers on oligopeptides and DNA (Journal of Physical Chemistry C, 2020); Hossain and co-workers on two-terminal chiral devices (Nanoscale Horizons, 2023); Safari and co-workers on single chiral molecules (Small, 2024); Singh Bisht and co-workers on self-assembled nanofibers (Journal of Physical Chemistry Letters, 2024); and the 2025 Albro study discussed below. The paper cites these as background and does not reanalyse their data.

On the theory side, it places itself in an existing debate over time-reversal symmetry and Onsager reciprocity in CISS, citing among others Dalum and Hedegård (2019) and the Tirion–van Wees paper (2024). It names two precedents for a spontaneously time-reversal-broken state: a 2023 study by Silveirinha, Terças and Antezza on chiral atomic systems, and Fransson's own 2023 work with Shiranzaei and Kalhöfer on magnetism emerging from interactions plus a reservoir.

The argument in outline

The reasoning can be grouped around a few questions a reader might ask:

• Why does an isolated molecule show no spin? Fransson treats the paired electrons of a stable molecule as mutually dependent, so that what one electron's spin does is tied to the rest. On this view the zero average arises because the molecule keeps switching among several equally likely spin arrangements, not because every site is spin-free.
• What links the molecule's shape to spin? In the model, spin-orbit coupling is the sole link between structure and spin. Fransson adds that this coupling alone cannot pick out any one arrangement.
• What changes when an electrode is attached? Contact with a metal or semiconductor lets electrons pass between molecule and electrode, so the molecular states no longer persist indefinitely and energy drains away into the electrode. The paper argues that this broadening of the energy levels, combined with spin-orbit coupling, acts like a local Zeeman-like splitting that settles the molecule into one arrangement set by its chirality.
• Why would reciprocity then fail? Fransson argues that the molecule's electronic structure ends up depending linearly on the electrode's magnetisation, which leaves no linear-response regime.

How the model is built

The molecule is represented as a chain of sites with nearest-neighbour and next-nearest-neighbour hopping. Each hopping process has an elastic version and a vibrationally assisted version. The spin-orbit term depends on a curvature vector defined from three consecutive sites. In a straight chain this vector vanishes. In a flat zig-zag it points only out of the molecular plane. Only a helix, or another chiral shape, also gives it a component along the molecular axis. When the vibrations are integrated out, they leave effective electron-electron interactions. The paper says its conclusions, apart from their temperature dependence, would be the same with direct Coulomb interactions instead.

A mean-field treatment of a four-site chain illustrates the mechanism. In the isolated closed shell, the relevant off-diagonal spin terms (coherences) are real, and the interactions barely change the spectrum. With the chain coupled to reservoirs, Fransson's Green-function estimates make these coherences non-zero. A definite spin texture then lowers the total energy, and the stabilising gap grows with the reservoir coupling strength. A lower vibrational energy is said to make the texture more favourable. The energy depends on whether curvature exists, not on which way it twists. The two enantiomers are therefore degenerate in energy but carry mirror-image spin configurations, and the paper describes them as forming a Kramers doublet. In this picture the reservoir-coupled system sits in a spontaneously time-reversal-broken state.

Why Onsager is said not to apply

The key analytic result is an expression for the charge on the edge sites. It contains a term proportional to the dot product of the reservoir's spin polarisation and the local curvature vector. The molecule's charge therefore rises or falls depending on whether the electrode is magnetised parallel or antiparallel to the chirality-defined direction. Because current through a junction depends on that charge density, Fransson argues that the current varies linearly with the reservoir's polarisation. From the time-reversal-broken state and this absence of a linear-response regime, he concludes that "the Onsager reciprocity theorem does not apply" to a chiral molecule interfaced with a reservoir.

In the uncorrelated version of the model this asymmetry disappears. The total charge does not depend on the external magnetisation, and the ferromagnet's influence barely penetrates the molecule.

Numerical examples

The calculations use a helical molecule of 3 × 6 sites, with the left electrode spin-polarised at p_L = ±0.2 along the axis and the right electrode non-magnetic. The parameters are given in units of t₀ = 0.1 eV, with temperature 300 K. The paper reports:

• With non-magnetic electrodes, both enantiomers carry equal charge distributions and mirror-image spin distributions.
• With a magnetic electrode, one magnetisation direction concentrates charge in the molecule more than the other. Fransson reads this stronger localisation as a higher barrier to current.
• Calculated currents stay small for |V| ≲ 0.75 V. Beyond that range, the current for p_L = −0.2 grows faster. The normalised magneto-current is negative, changes only weakly with voltage and keeps its sign for both bias polarities. The paper says both features are often seen in experiments.
• Moving the magnetic electrode from the left side to the right reverses the sign of the magneto-current. Fransson describes this as in "excellent agreement" with a 2025 direction-dependent magnetic conductive-AFM study by Albro, Bloom, Waldeck and co-workers (ACS Nano 19, 17941).
• The current itself is spin-polarised. The paper links this to photoemission results but notes that spin polarisation of the transport current has not yet been measured.
• A non-helical chiral geometry, a zig-zag chain with one site sticking out of the plane, also responds asymmetrically. Its normalised magneto-current is non-zero but below 1 %.

What the paper does not establish

The paper shows that a particular model, under mean-field and lowest-order approximations, produces a reservoir-induced spin texture and a magnetisation-dependent charge distribution. Its parameters are given in model units of t₀ = 0.1 eV and are not tied to a specific molecule or to measured data, and the paper presents no quantitative fits to measured current-voltage data. The Sarkar–Dubi perspective said Fransson's earlier chiral-phonon theory likewise lacked quantitative fits to raw data. The paper reports no quantitative comparison between the size of the computed effect and measured values, and its non-helical example gives a normalised magneto-current below 1 %. Comparisons with experiment are qualitative: weak voltage dependence, consistent sign, and reversal when the magnet changes sides. In the acknowledgements, Fransson thanks Per Hedegård and Jan van Ruitenbeek for critical questions and Ron Naaman for encouragement. He does not report what those critics concluded.

Place in the dossier

The paper addresses the Onsager objection discussed in the Evers–Kronik review, which records that time-reversal symmetry rules out spin filtering in the low-bias regime of a two-terminal device. It shares one idea with the Tirion–van Wees electrostatic mechanism: the equilibrium state itself depends on magnetisation direction. Tirion and van Wees place that dependence in an interface contact potential without spin-dependent transport, while Fransson places it in a spin configuration locked inside the molecule. The spinterface model also uses dissipation to stabilise moments, but those moments sit on the electrode surface. The Sarkar–Dubi perspective admits that the spinterface model assumes, rather than derives, dissipation-driven time-reversal breaking. Fransson's paper tries to derive that step within its own model. For the symmetry background, see Nonreciprocity.

References

• Fransson, J. "Breaking of Time-Reversal Symmetry and Onsager Reciprocity in Chiral Molecule Interfaced with an Environment." arXiv:2509.17817v1 (cond-mat.mes-hall), September 2025.
• Albro, J. A. et al. "Measurement platform to probe the mechanism of chiral-induced spin selectivity through direction-dependent magnetic conductive atomic force microscopy." ACS Nano 19, 17941 (2025).
• Onsager, L. "Reciprocal relations in irreversible processes. I." Physical Review 37, 405 (1931).
• Fransson, J. "Vibrational origin of exchange splitting and chiral-induced spin selectivity." Physical Review B 102, 235416 (2020).