Knot-Driven Spin Selectivity in Molecules

• Knot-driven spin selectivity is a phenomenon where trefoil knot topology and curvature-induced SOC generate robust spin polarization, achieving up to 90% efficiency.
• The approach employs a discretized trefoil geometry with site-dependent geometric SOC and asymmetric multi-terminal contacts to differentiate TCISS from conventional CISS.
• This design offers high thermal stability and tunability, paving the way for nonmagnetic spin filters, spin-polarized devices, and efficient molecular electronics.

Searching arXiv for the cited papers and closely related CISS/topological transport work to ground the article. Knot-driven spin selectivity denotes a spin-polarized transport phenomenon in which a topologically nontrivial molecular knot, specifically a trefoil knot, generates robust spin polarization through topological chirality-induced spin selectivity (TCISS). In the formulation of Sun et al., the effect is distinguished from conventional chirality-induced spin selectivity (CISS) in structurally chiral systems such as DNA and helicene by a combination of ultra-high spin polarization, significant conductivity, and robustness against lattice-number reduction, strain regulation, and topological-preserving deformation. The same work reports that topological chirality in trefoil knot molecules has demonstrated spin polarization of nearly 90%, conductivity increased by two orders of magnitude, and high-temperature stability up to $350^{\circ}\mathrm{C}$, while its theoretical single-molecule calculations yield spin polarization exceeding $60\%$ and up to $75\%$ under asymmetric multi-terminal conditions (Sun et al., 27 Jul 2025).

1. Conceptual setting and distinction from conventional CISS

TCISS is the name given to spin selectivity induced by topological chirality rather than solely by structural chirality. In the trefoil-knot setting, the essential claim is that the ultrahigh spin polarization correlates strongly with knot topology: when the nontrivial knot degenerates into a trivial structure and the system transitions from topological chirality to structural chirality, the spin polarization sharply declines (Sun et al., 27 Jul 2025).

This distinction is operationalized by comparing two regimes. In the nontrivial trefoil regime, the spin polarization lies in the $60\%-75\%$ range in the theoretical treatment, and the effect is described as topologically protected. In the trivial regime obtained after untangling, the remaining $10\%-20\%$ polarization is identified as the familiar CISS from structural chirality. Further flattening to a planar circle yields uniform curvature and $P_s \to 0$ (Sun et al., 27 Jul 2025).

A broader transport-theoretic context is provided by helical-junction analyses showing that spin-resolved currents can arise in time-reversal-symmetric molecular transport when spin-flip processes are accompanied by orbital-channel flips. That result does not contradict Bardarson’s theorem because the transmission retains Kramers-degenerate eigenvalue pairs; the key mechanism is that the transmission eigenchannels need not be organized as opposite-spin partners in the same orbital channel (Utsumi et al., 2020). This comparison is relevant because knot-driven selectivity is likewise framed as an interference-and-SOC phenomenon, but in Sun et al. the decisive ingredients are the topologically knotted backbone, curvature-mapped geometric SOC, and inversion-symmetry breaking by asymmetric contacts (Sun et al., 27 Jul 2025).

2. Trefoil-knot geometry and microscopic Hamiltonian

The trefoil is discretized as a closed loop of $N$ sites with equal arc length $S$, with site $i$ parameterized by $\theta_i = 2\pi i/N$ and four shape parameters $60\%$0. The position vector is

$60\%$1

where $60\%$2 are the maximal and minimal radii in top view, $60\%$3 is the vertical amplitude, and $60\%$4 and $60\%$5 represent left- and right-handed knots, respectively (Sun et al., 27 Jul 2025).

The local curvature is obtained from differential geometry through the tangent $60\%$6, the normal $60\%$7, and

$60\%$8

Electrons confined to the curved molecular path experience a noninertial acceleration

$60\%$9

which induces a geometric spin–orbit term in a curved-space expansion of the Dirac Lagrangian,

$75\%$0

In discretized form,

$75\%$1

with $75\%$2 and $75\%$3, where the Pauli matrices are projected onto the local binormal $75\%$4 (Sun et al., 27 Jul 2025).

The full tight-binding Hamiltonian combines molecular, geometric-SOC, electrode-coupling, and dephasing terms:

$75\%$5

In the parameter set used for the calculations, the nearest-neighbor hopping is $75\%$6, the geometric SOC is $75\%$7 $75\%$8, $75\%$9 couples lattice sites to nonmagnetic electrodes via tunneling rates $60\%-75\%$0, and $60\%-75\%$1 models inelastic dephasing through Büttiker virtual leads (Sun et al., 27 Jul 2025).

3. Topological chirality, Berry phases, and inversion-symmetry breaking

The central mechanism is formulated in terms of two inequivalent tunneling paths around the knot. When an electron tunnels between two sites $60\%-75\%$2 along these alternative routes, it acquires spin-dependent Berry phases

$60\%-75\%$3

with $60\%-75\%$4 controlled by the local curvature distribution and by the handedness sign $60\%-75\%$5 (Sun et al., 27 Jul 2025).

In a symmetric device, these phases cancel through $60\%-75\%$6, and spin polarization vanishes. Spin selectivity therefore requires inversion-symmetry breaking. In the trefoil model this is accomplished by asymmetric electrode contacts $60\%-75\%$7, which induce a nonzero Berry-phase difference,

$60\%-75\%$8

The resulting transport is spin dependent because the geometric SOC is site dependent and because the conductance depends on multi-terminal Green’s-function sums (Sun et al., 27 Jul 2025).

Within the terminology of Sun et al., the necessary conditions for knot-driven spin selectivity are therefore not exhausted by chirality alone. The effect rests on the conjunction of a topologically nontrivial knot, curvature-induced SOC, and device asymmetry sufficient to prevent left- and right-path phase cancellation. This suggests that topology enters not merely as a geometric descriptor of the molecular backbone but as a constraint on the curvature pattern and on the associated interference structure.

4. Transport formalism and definition of spin polarization

Spin-resolved conductances are computed using the Landauer–Büttiker formula combined with non-equilibrium Green’s functions and Büttiker probes. In linear response and low temperature,

$60\%-75\%$9

where

$10\%-20\%$0

and the virtual-probe chemical potentials $10\%-20\%$1 are fixed by the zero-net-current condition $10\%-20\%$2 (Sun et al., 27 Jul 2025).

The spin polarization is defined by

$10\%-20\%$3

Typical parameters are $10\%-20\%$4, $10\%-20\%$5, dephasing rate $10\%-20\%$6, and electrode couplings $10\%-20\%$7 (Sun et al., 27 Jul 2025).

Under asymmetric multi-terminal contacts, the trefoil molecule can exhibit $10\%-20\%$8, with up to $10\%-20\%$9 in single-molecule models. Sun et al. further note that in real STM–substrate experiments each tip couples to several lattice sites and thin films contain multiple knot layers, which pushes $P_s \to 0$0 as observed (Sun et al., 27 Jul 2025).

A useful comparative benchmark is provided by a two-orbital helical-junction model, in which the spin polarization is written as

$P_s \to 0$1

and is generated by intra-atomic SOI that flips spin and orbital simultaneously. There, reversing the helix handedness reverses the sign of $P_s \to 0$2, while the transport remains time-reversal symmetric (Utsumi et al., 2020). This provides a formal analogue for the sensitivity of spin selectivity to chirality and interference, although the knot case attributes the robust high-$P_s \to 0$3 regime specifically to nontrivial knot topology (Sun et al., 27 Jul 2025).

5. Robustness with respect to lattice density, strain, and thermal conditions

A defining feature of knot-driven spin selectivity is robustness. Varying the total lattice number $P_s \to 0$4 from $P_s \to 0$5 to $P_s \to 0$6 at fixed molecular length leaves $P_s \to 0$7 essentially unchanged. In the interpretation given by Sun et al., the polarization is dominated by the intrinsic curvature pattern of the knot and by multi-channel asymmetry, rather than by discretization density (Sun et al., 27 Jul 2025).

The same robustness appears under two strain protocols: lateral stretch, defined by $P_s \to 0$8 with $P_s \to 0$9 fixed, and vertical stretch, defined by $N$0 with $N$1 fixed. In both cases, $N$2 and the conductances remain nearly invariant (Sun et al., 27 Jul 2025).

The stated reason is topological protection associated with the trefoil’s nontrivial linking number and $N$3 rotational symmetry, which preserve the essential curvature-induced Berry phases (Sun et al., 27 Jul 2025). This supports the claim that TCISS is not a fine-tuned feature of a single discretization or a narrowly defined geometry. It also provides the proposed explanation for the reported thermal stability up to $N$4, since the spin selectivity is described as insensitive to chain length, lattice density, and moderate mechanical deformation (Sun et al., 27 Jul 2025).

6. Topological-to-trivial transition, relation to helical models, and device implications

The topological character of the effect is made explicit by tuning the hole-size parameter $N$5. When $N$6, the molecule remains a nontrivial trefoil and $N$7. At $N$8, the central triangle collapses to a point and $N$9 drops sharply to approximately $S$0. For $S$1, the loop untangles into a structurally chiral but topologically trivial ring, where $S$2. If the structure is further flattened by $S$3, it becomes a planar circle with uniform curvature and $S$4 (Sun et al., 27 Jul 2025).

This sequence is described as a topological phase transition and is used to distinguish TCISS from conventional CISS. In Sun et al., TCISS in the $S$5 range is presented as a genuinely topological effect, whereas the residual $S$6 polarization in the trivial phase is identified with conventional structural-chirality-driven CISS (Sun et al., 27 Jul 2025).

A common misconception in spin-selective molecular transport is that spin filtering necessarily requires broken time-reversal symmetry or magnetic contacts. The helical two-orbital analysis of Utsumi, Entin-Wohlman, and Aharony shows that two-terminal spin filtering can occur without breaking time-reversal symmetry, provided the SOI flips spin together with orbital channel, and that this remains consistent with Bardarson’s theorem because transmission eigenvalues still occur in Kramers-degenerate pairs (Utsumi et al., 2020). In the trefoil setting, the corresponding emphasis shifts from orbital-channel structure to multi-terminal interference around a topologically nontrivial knot, encoded through lattice-varying geometric SOC and asymmetric contact geometry (Sun et al., 27 Jul 2025).

The design implications stated for knot-driven spin selectivity are specific. Multiterminal contacts that break spatial inversion, such as three or four electrodes placed asymmetrically, maximize $S$7. The required ingredients are nonmagnetic metal contacts, a topologically knotted molecular backbone, and strong curvature-induced SOC with $S$8. Tunability is proposed through higher-knot types such as cinquefoil $S$9, through adjustment of knot tightness $i$0, and through chemical substitution that varies $i$1, thereby engineering $i$2 and optimizing $i$3 relative to conductance (Sun et al., 27 Jul 2025).

The device concepts explicitly listed are nonmagnetic spin filters and detectors based purely on molecular topology, spin-polarized LEDs and photovoltaic devices using trefoil-knot thin films, and spin-dependent catalysis and enantioselective chemistry leveraging the TCISS effect (Sun et al., 27 Jul 2025). A plausible implication is that molecular topology functions here as a transport-design variable, not only as a stereochemical classification, because the reported blueprint ties spin selectivity directly to knot type, curvature distribution, and contact asymmetry.