Topological Chirality-Induced Spin Selectivity

• TCISS is a multifaceted phenomenon where chirality couples to spin via global symmetries, non-Hermitian dynamics, and topological band structures.
• It encompasses diverse models such as chiral junctions, helical edge states, and knot-based molecular systems that achieve spin-polarized transport without magnetic fields.
• Recent research integrates geometric chirality, spin–orbit coupling, and dephasing to enhance spin polarization, yet a unified microscopic theory remains an open challenge.

Topological chirality-induced spin selectivity (TCISS) denotes a family of frameworks that relate chirality-induced spin selectivity (CISS) to topology, topology-like robustness, or symmetry constraints that exceed the usual molecule-only description of chiral spin filtering. In the contemporary literature, the term is not used in a single uniform sense. It can refer to global structural chirality of an entire junction, to a $PT$-symmetric non-Hermitian exchange mechanism in structurally chiral many-electron systems, to spin and orbital polarization in chiral crystals that also host topological electronic structures, to engineered topological quantum wells with helical edge states, or to molecular knots whose chirality is topological rather than merely structural (Dednam et al., 2022, Theiler et al., 9 May 2025, Yang et al., 2023, Liu et al., 23 Mar 2026, Sun et al., 27 Jul 2025).

1. Terminological scope and principal usages

CISS, in the shared baseline sense across these works, is spin-dependent transmission through a non-magnetic chiral medium, often described as the generation of spin polarization from a spin-unpolarized source without magnetic fields or ferromagnetic elements. What distinguishes TCISS from generic CISS is not a single microscopic ingredient, but the claim that chirality couples to spin through a robust organizing principle such as global symmetry, non-Hermitian topology, helical edge topology, nonsymmorphic symmetry, or knot topology.

Usage of “TCISS”	Core organizing principle	Representative work
Global junction chirality	Point-group symmetry of the whole device	(Dednam et al., 2022)
Non-Hermitian TCISS	$PT$-symmetric anti-Hermitian exchange and skin effect	(Theiler et al., 9 May 2025)
Chiral-crystal TCISS	Intrinsic SOC in homochiral solids with topological band structures	(Yang et al., 2023)
Topological-quantum-well TCISS	Helical edge states plus engineered chirality and dephasing	(Liu et al., 23 Mar 2026)
Knot-driven TCISS	Topological chirality of a nontrivial molecular knot	(Sun et al., 27 Jul 2025)

This diversity matters because the protected object differs from one formulation to another. In some papers, robustness is a symmetry selection rule; in others, it is a non-Hermitian skin effect, a helical-edge transport property, or the persistence of spin filtering under knot-preserving deformations. A recurrent source of confusion is that “topological” sometimes means band topology, sometimes global structural chirality, and sometimes topology in the knot-theoretic sense.

2. Symmetry-based formulations in junctions, helices, and chiral interfaces

A major line of work formulates TCISS as a consequence of the symmetry of the entire transport setup rather than of molecular chirality in isolation. In the scattering approach for non-magnetic molecular junctions, the decisive criterion is the absence of improper spatial symmetries of the full junction that do not permute the electrodes and act along the transport axis. The spin polarization along the transport direction is written as

$$P_z(E)=\frac{T_{\uparrow\uparrow}(E)+T_{\uparrow\downarrow}(E)-T_{\downarrow\uparrow}(E)-T_{\downarrow\downarrow}(E)}{T_{\uparrow\uparrow}(E)+T_{\uparrow\downarrow}(E)+T_{\downarrow\uparrow}(E)+T_{\downarrow\downarrow}(E)}.$$

Within this framework, a longitudinal mirror plane $\sigma_l$ enforces $P_z=0$, and point groups such as $C_{nv}$, $D_{nh}$, and $D_{nd}$ suppress all spin-polarization components in two-terminal collinear geometries. By contrast, inversion symmetry alone does not force $P_z=0$, so breaking inversion is not a necessary condition for spin-polarized transport. The same analysis predicts that even achiral molecules, and even standalone metallic nanocontacts, can exhibit spin polarization if the full junction becomes globally chiral through relative rotations or contact geometry (Dednam et al., 2022).

A distinct time-reversal-symmetric route appears in the two-orbital helical-junction model. There, spin-resolved currents arise when spin-flip processes are accompanied by flips between orbital channels, so Bardarson’s theorem is not violated even though spin filtering occurs in a two-terminal device. The helix induces a reciprocal single-turn momentum shift $\Delta k=\pm G$ with $PT$0, and the sign depends on handedness. As a result, spin-up and spin-down states propagate in opposite directions near the relevant avoided crossings without breaking time-reversal symmetry. This formulation treats TCISS as an emergent spin–momentum locking rooted in helical geometry and multichannel orbital structure, rather than in magnetism or dephasing (Utsumi et al., 2020).

A third symmetry-centered formulation introduces pseudo-angular momentum (PAM) generated by nonsymmorphic screw symmetry. Along screw-invariant lines, Bloch states satisfy

$PT$1

with $PT$2 the PAM quantum number modulo $PT$3. Counterpropagating states carry opposite PAM, and at chiral–achiral interfaces this PAM polarization is converted into spin polarization through wavefunction matching. In that account, the most accurate characterization is symmetry-protected CISS with quantized PAM rather than a bulk topological phase established by an explicit invariant (Wang et al., 2023).

3. Non-Hermitian exchange and the $PT$4-symmetric TCISS paradigm

The most explicit use of TCISS as a topological non-Hermitian theory appears in the proposal that structurally chiral many-electron systems enforce a chirality-preserving twin-pair exchange. When all mirror symmetries are broken, a single pair exchange flips the enantiomer, whereas twin-pair exchanges preserve global chirality. Summing over the relevant even-permutation subgroup yields a chirality-dependent imaginary term that maps to a $PT$5 contribution in the Dirac description and, in the nonrelativistic limit, to the effective Hamiltonian

$PT$6

Here $PT$7 is an exchange energy scale estimated as $PT$8–$PT$9 eV in organic systems. The term $$P_z(E)=\frac{T_{\uparrow\uparrow}(E)+T_{\uparrow\downarrow}(E)-T_{\downarrow\uparrow}(E)-T_{\downarrow\downarrow}(E)}{T_{\uparrow\uparrow}(E)+T_{\uparrow\downarrow}(E)+T_{\downarrow\uparrow}(E)+T_{\downarrow\downarrow}(E)}.$$0 breaks parity $$P_z(E)=\frac{T_{\uparrow\uparrow}(E)+T_{\uparrow\downarrow}(E)-T_{\downarrow\uparrow}(E)-T_{\downarrow\downarrow}(E)}{T_{\uparrow\uparrow}(E)+T_{\uparrow\downarrow}(E)+T_{\downarrow\uparrow}(E)+T_{\downarrow\downarrow}(E)}.$$1 and time reversal $$P_z(E)=\frac{T_{\uparrow\uparrow}(E)+T_{\uparrow\downarrow}(E)-T_{\downarrow\uparrow}(E)-T_{\downarrow\downarrow}(E)}{T_{\uparrow\uparrow}(E)+T_{\uparrow\downarrow}(E)+T_{\downarrow\uparrow}(E)+T_{\downarrow\downarrow}(E)}.$$2 individually, while preserving combined $$P_z(E)=\frac{T_{\uparrow\uparrow}(E)+T_{\uparrow\downarrow}(E)-T_{\downarrow\uparrow}(E)-T_{\downarrow\downarrow}(E)}{T_{\uparrow\uparrow}(E)+T_{\uparrow\downarrow}(E)+T_{\downarrow\uparrow}(E)+T_{\downarrow\downarrow}(E)}.$$3 symmetry, since $$P_z(E)=\frac{T_{\uparrow\uparrow}(E)+T_{\uparrow\downarrow}(E)-T_{\downarrow\uparrow}(E)-T_{\downarrow\downarrow}(E)}{T_{\uparrow\uparrow}(E)+T_{\uparrow\downarrow}(E)+T_{\downarrow\uparrow}(E)+T_{\downarrow\downarrow}(E)}.$$4, $$P_z(E)=\frac{T_{\uparrow\uparrow}(E)+T_{\uparrow\downarrow}(E)-T_{\downarrow\uparrow}(E)-T_{\downarrow\downarrow}(E)}{T_{\uparrow\uparrow}(E)+T_{\uparrow\downarrow}(E)+T_{\downarrow\uparrow}(E)+T_{\downarrow\downarrow}(E)}.$$5, but $$P_z(E)=\frac{T_{\uparrow\uparrow}(E)+T_{\uparrow\downarrow}(E)-T_{\downarrow\uparrow}(E)-T_{\downarrow\downarrow}(E)}{T_{\uparrow\uparrow}(E)+T_{\uparrow\downarrow}(E)+T_{\downarrow\uparrow}(E)+T_{\downarrow\downarrow}(E)}.$$6 (Theiler et al., 9 May 2025).

In the one-dimensional analytic model, the Hamiltonian reduces in each spin sector to a Hatano–Nelson-type tridiagonal Toeplitz matrix with asymmetric hoppings. The open-boundary spectrum is real in the $$P_z(E)=\frac{T_{\uparrow\uparrow}(E)+T_{\uparrow\downarrow}(E)-T_{\downarrow\uparrow}(E)-T_{\downarrow\downarrow}(E)}{T_{\uparrow\uparrow}(E)+T_{\uparrow\downarrow}(E)+T_{\downarrow\uparrow}(E)+T_{\downarrow\downarrow}(E)}.$$7-unbroken regime, with threshold condition $$P_z(E)=\frac{T_{\uparrow\uparrow}(E)+T_{\uparrow\downarrow}(E)-T_{\downarrow\uparrow}(E)-T_{\downarrow\downarrow}(E)}{T_{\uparrow\uparrow}(E)+T_{\uparrow\downarrow}(E)+T_{\downarrow\uparrow}(E)+T_{\downarrow\downarrow}(E)}.$$8. Right eigenvectors exhibit a non-Hermitian skin effect: opposite spins localize exponentially at opposite boundaries even though the energies remain spin-degenerate. In this picture, spin selectivity arises not from an energy splitting but from spin-dependent spatial support, interfacial accumulation, and nonreciprocal drift. The transport polarization

$$P_z(E)=\frac{T_{\uparrow\uparrow}(E)+T_{\uparrow\downarrow}(E)-T_{\downarrow\uparrow}(E)-T_{\downarrow\downarrow}(E)}{T_{\uparrow\uparrow}(E)+T_{\uparrow\downarrow}(E)+T_{\downarrow\uparrow}(E)+T_{\downarrow\downarrow}(E)}.$$9

therefore increases with the non-Hermitian exchange strength $\sigma_l$0, the length of the chiral region, and the extent to which open boundaries are imposed by interfaces.

This framework also links TCISS to exceptional points and generalized bulk–boundary behavior. Open boundaries collapse periodic-boundary complex-energy loops into boundary-localized modes, while the exceptional-point condition marks the transition out of the real-spectrum phase. The paper identifies the skin effect and $\sigma_l$1 phase structure as the topological content, but does not compute explicit non-Hermitian invariants for realistic materials. It therefore presents TCISS as a non-Hermitian topological phenomenon whose formal core is symmetry-enforced many-body exchange, while leaving realistic transport theory beyond Hermitian Landauer–Büttiker or standard NEGF as an open problem (Theiler et al., 9 May 2025).

4. Chiral crystals, moiré materials, and orbital-to-spin conversion

In conductive chiral crystals, TCISS is formulated in terms of intrinsic SOC, homochiral bulk structure, and often topological band features such as multifold fermion crossings, large Chern numbers, and Fermi arcs. In two-terminal Landauer–Büttiker transport, electrons transmitted through such crystals acquire both spin and orbital polarization along the transport direction; both polarizations reverse sign between enantiomers, and both transmitted and reflected electrons show the same type of polarization. For TaSi$\sigma_l$2, the spin polarization $\sigma_l$3 increases with thickness and saturates by $\sigma_l$4 unit cells; at charge neutrality, $\sigma_l$5 and $\sigma_l$6, while the maximum $\sigma_l$7 reaches $\sigma_l$8 at $\sigma_l$9 eV. Across TMSi$P_z=0$0 with TM = V, Nb, Ta, the spin polarization at $P_z=0$1 eV rises from $P_z=0$2 to $P_z=0$3 to $P_z=0$4, whereas orbital polarization is largely insensitive to SOC. In Te, a direct gap of $P_z=0$5 eV and valence-band spin splitting of $P_z=0$6 eV at H yield $P_z=0$7 near the valence-band edge. In RhSi, the magneto-current ratio reaches $P_z=0$8 around $P_z=0$9 meV for $C_{nv}$0, while $C_{nv}$1 at the Fermi energy, showing that magneto-transport and spin polarization are related but distinct observables (Yang et al., 2023).

Twisted homobilayer transition-metal dichalcogenides furnish a different condensed-matter platform. There, the reported giant spin polarization arises from structural chirality plus spin-flipping SOC, not from demonstrated topological protection. Reversing the twist angle changes the sign of the polarization, and the untwisted case $C_{nv}$2 gives $C_{nv}$3 even when inversion symmetry is broken by finite $C_{nv}$4, so chirality is essential while inversion breaking alone is insufficient. The effect is tunable with twist angle and SOC strength: at $C_{nv}$5, the maximum $C_{nv}$6 reaches approximately $C_{nv}$7 in MoS$C_{nv}$8, $C_{nv}$9 in MoSe$D_{nh}$0, $D_{nh}$1 in MoTe$D_{nh}$2, $D_{nh}$3 in WS$D_{nh}$4, $D_{nh}$5 in WSe$D_{nh}$6, and $D_{nh}$7 in WTe$D_{nh}$8. The same work emphasizes that standard linear-response two-terminal magnetoresistance cannot directly reveal this spin-polarized current because of reciprocity constraints (Menichetti et al., 2023).

A further usage of TCISS centers on “topological orbital” texture rather than on a topological band invariant. In chiral molecular spin valves, experiments comparing Au and Al electrodes show that a heavy-metal electrode supplies the SOC required to convert chirality-induced orbital polarization into spin polarization through $D_{nh}$9. In (Ga,Mn)As/AHPA-L/Au devices, the extracted $D_{nd}$0 spans roughly $D_{nd}$1–$D_{nd}$2S as current increases from $D_{nd}$3 to $D_{nd}$4A, whereas (Ga,Mn)As/AHPA-L/Al yields only about $D_{nd}$5–$D_{nd}$6S over comparable currents. In the Simmons-type tunneling model with magnetochiral barrier modulation, the fitted modulation amplitude $D_{nd}$7 is larger for Au than for Al by a factor of about $D_{nd}$8–$D_{nd}$9, supporting the view that the electrode, not the light-atom molecule, supplies the decisive SOC. In this vocabulary, TCISS means chirality-generated orbital texture converted into spin selectivity at the contact (Adhikari et al., 2022).

5. Engineered topological devices and edge-state realizations

A controlled device-level realization of TCISS has been proposed in an InAs/GaSb quantum spin Hall quantum well. The topological regime provides helical edge states, while geometric chirality is engineered by boundary asymmetry and layer stacking, and dephasing is introduced through Büttiker voltage probes attached only to one boundary. In the bulk gap, transport is edge dominated: dephasing selectively attenuates one helical branch while the opposite branch remains nearly quantized. For a lower-edge chain of $P_z=0$0 voltage probes, the analytic result is

$P_z=0$1

so the predicted plateaus are $P_z=0$2 for one probe, $P_z=0$3 for two probes, and $P_z=0$4 for three probes. The sign reverses when chirality is flipped, symmetric edge dephasing gives $P_z=0$5, and distributed bulk dephasing also gives approximately zero spin selectivity. The polarization remains large and nearly unchanged up to Anderson disorder strength $P_z=0$6, with $P_z=0$7 meV for the chosen lattice mapping (Liu et al., 23 Mar 2026).

This platform is significant because it separates three ingredients that are often entangled in molecular CISS: SOC, chirality, and dephasing. Here SOC and band inversion create the quantum spin Hall phase, chirality is encoded by device geometry rather than molecular stereochemistry, and dephasing is a tunable control rather than an uncontrolled environment. The work explicitly treats helical-edge protection as the source of disorder robustness, and it presents achiral controls that eliminate spin selectivity, thereby ruling out trivial filtering within the model (Liu et al., 23 Mar 2026).

6. Geometric-current and knot-topology mechanisms

A different route to TCISS begins from relativistic four-current dynamics. In that formulation, the conserved four-current is $P_z=0$8, and the Gordon decomposition produces a spin-dependent paramagnetic current term

$P_z=0$9

For helicenes, curvature induces helical current patterns and a handedness-dependent axial magnetization. The estimated internal magnetic field is of order $\Delta k=\pm G$0 T per single helicene strand, with sign reversal between enantiomers. This leads to the suggestion that CISS can originate from intrinsic relativistic curvature-induced currents and their self-generated magnetic fields, without relying on interfacial effects or strong Rashba-like SOC (Zheng et al., 3 Apr 2025).

The most literal topological use of TCISS concerns molecular trefoil knots. There the chirality is a global property of a nontrivial closed loop rather than a local helical arrangement. The trefoil carries $\Delta k=\pm G$1 rotational symmetry and a curvature $\Delta k=\pm G$2 with $\Delta k=\pm G$3 periodicity, and the geometric SOC is written as

$\Delta k=\pm G$4

In the discretized model, the default choice is $\Delta k=\pm G$5, which for $\Delta k=\pm G$6 eV gives $\Delta k=\pm G$7 eV. Spin filtering requires asymmetric multichannel transport: a symmetric two-terminal trefoil or an opened single-channel wire gives negligible polarization, whereas asymmetric three-terminal and four-terminal geometries do not. For a three-terminal trefoil with $\Delta k=\pm G$8, contact pairs $\Delta k=\pm G$9, $PT$00, and $PT$01 yield about $PT$02, a larger value for the more asymmetric middle case, and approximately $PT$03, respectively. A four-terminal trefoil with $PT$04 gives about $PT$05, and for fixed molecular length the polarization stays near $PT$06 as $PT$07 is increased from $PT$08 to $PT$09. Lateral and vertical strain do not significantly reduce this value. When a trefoil is tuned through a topology-to-structure transition into a trivial loop, the polarization drops from about $PT$10 to about $PT$11, and further planarization drives $PT$12. The same work cites experiments on trefoil-knot monolayers reporting nearly $PT$13 spin polarization, conductivity increased by two orders of magnitude, and stability up to $PT$14C (Sun et al., 27 Jul 2025).

These geometric-current and knot-based proposals share a common claim: chirality can act directly on the electron current distribution and not merely on a static scalar potential. They differ, however, in what they take to be the robust object. In the four-current theory it is the helical current and axial magnetization; in knot-driven TCISS it is the nontrivial knot topology together with curvature-modulated multichannel interference.

7. Conceptual tensions, misconceptions, and open problems

The first recurring misconception is that TCISS names a single settled mechanism. The literature instead contains several non-equivalent programs. In one, TCISS is global point-group chirality of the entire junction; in another, it is a non-Hermitian $PT$15-symmetric boundary phenomenon; in another, it is the use of topological solids as chiral spin polarizers; and in yet another, it is literal knot topology. The term therefore has descriptive value, but not yet a universally fixed microscopic meaning (Dednam et al., 2022, Theiler et al., 9 May 2025, Yang et al., 2023, Sun et al., 27 Jul 2025).

A second misconception is that inversion breaking alone explains the effect. The record is more restrictive and more subtle. The group-theoretic transport theory shows that inversion symmetry alone does not force $PT$16, so breaking inversion is not necessary for spin-polarized transport. Conversely, in twisted TMDs, finite inversion breaking at $PT$17 still gives $PT$18, so inversion breaking by itself is not sufficient there (Dednam et al., 2022, Menichetti et al., 2023).

A third misconception is that strong SOC or dephasing is universally required. Several proposals make SOC central: intrinsic SOC in chiral crystals, spin-flipping SOC in twisted TMDs, heavy-metal SOC in molecular spin valves, and SOC-assisted helical edge transport in topological quantum wells. Other proposals instead claim SOC-free or effectively SOC-independent routes based on non-Hermitian exchange, PAM selection at interfaces, or relativistic current self-organization. Dephasing is likewise framework dependent: it is essential in the InAs/GaSb quantum-well realization and for magnetoresistance in the EMCA setting, but not in the time-reversal-symmetric two-orbital helix or in the non-Hermitian exchange theory (Liu et al., 23 Mar 2026, Adhikari et al., 2022, Zheng et al., 3 Apr 2025, Utsumi et al., 2020).

The principal open problems remain methodological and classificatory. Explicit non-Hermitian topological invariants are not computed in the $PT$19-symmetric exchange framework for realistic chiral crystals. The chiral-crystal literature emphasizes topological band structures, Berry-curvature-related responses, and orbital magnetization, but does not derive a unique microscopic formula that directly links those quantities to CISS observables. PAM-based interface theories establish symmetry protection but not a bulk invariant with a bulk–boundary correspondence. Knot-based TCISS achieves unusually large and robust polarization in model calculations, but requires more microscopic parameter extraction and systematic extension beyond the trefoil. Across all formulations, separating bulk chirality, interface conversion, dephasing, and electrode effects remains the central challenge for turning TCISS from a family resemblance into a unified theory (Theiler et al., 9 May 2025, Yang et al., 2023, Wang et al., 2023, Sun et al., 27 Jul 2025).