Colinear Edelstein Effect (CEE)

• Colinear Edelstein effect (CEE) is a spin–orbit-induced phenomenon where an electric field or current generates a spin polarization aligned parallel to the driving direction.
• CEE arises when symmetry permits diagonal magnetoelectric tensor components, as seen in systems like twisted graphene/TMD heterostructures, chiral metals, and oxide interfaces.
• Microscopic models using Kubo and Boltzmann approaches, along with superconducting and multi-orbital enhancements, highlight practical routes to measurable longitudinal magnetoelectric responses.

The colinear Edelstein effect (CEE) is the longitudinal variant of the Edelstein, or inverse spin galvanic, effect: a non-equilibrium spin polarization or magnetization is generated by an applied electric field or current such that the induced spin is parallel or antiparallel to the driving direction, i.e. $\mathbf{S}\parallel \mathbf{E}$ or $\mathbf{S}\parallel \mathbf{J}$. In linear response one writes $S_i=\sum_j \chi_{ij}E_j$; CEE corresponds to dominant diagonal components such as $\chi_{xx}$, in contrast to the conventional Rashba geometry where off-diagonal components dominate and the response is transverse (Veneri et al., 2022). Across normal metals, superconductors, chiral conductors, correlated $f$-electron systems, oxide interfaces, and non-relativistic magnets, the central issues are the same: which tensor components are symmetry-allowed, what microscopic spin or orbital texture prevents cancellation, and how transport, superconductivity, or dissipation converts those textures into a measurable longitudinal magnetoelectric response (Yuan, 2023).

1. Definition and tensor structure

The Edelstein effect is the linear-response generation of a non-equilibrium spin polarization by an applied electric field in a system with spin–orbit coupling and broken inversion symmetry. In tensor form,

$S_i \equiv M_i = \sum_j \chi_{ij}E_j,$

or, in one notation used for correlated $f$-electron systems,

$M_y = \Upsilon_{yx} E_x.$

The distinction between transverse and colinear response is entirely tensorial: a conventional Rashba system is characterized by off-diagonal components such as $\chi_{yx}$, whereas CEE requires diagonal components such as $\chi_{xx}$ (Peters et al., 2018).

A concise superconducting formulation makes the same point in current language: $\mathbf{S}\parallel \mathbf{J}$0 and the colinear case is explicitly the diagonal one, e.g. $\mathbf{S}\parallel \mathbf{J}$1 (Yuan, 2023). In a transformed longitudinal frame, this can also be written as

$\mathbf{S}\parallel \mathbf{J}$2

with the longitudinal axis set by the spin–orbit tensor and field geometry (Yuan, 2023).

The microscopic response is commonly written through Kubo or Boltzmann expressions. A general Kubo form for the Edelstein susceptibility is

$\mathbf{S}\parallel \mathbf{J}$3

so the passage from transverse to colinear response is formally just the replacement of $\mathbf{S}\parallel \mathbf{J}$4 by $\mathbf{S}\parallel \mathbf{J}$5 when evaluating $\mathbf{S}\parallel \mathbf{J}$6 instead of $\mathbf{S}\parallel \mathbf{J}$7 (Peters et al., 2018). In semiclassical transport, the same physics appears as a Fermi-surface integral of spin or orbital expectation values weighted by the drifted distribution function (Trama et al., 2022).

A persistent misconception is that “Edelstein effect” is synonymous with the Rashba transverse response. Several of the systems discussed below show that the direction of the induced polarization is not fixed by the concept of the Edelstein effect itself, but by the symmetry of the response tensor and by the momentum-space spin or orbital texture (Veneri et al., 2022).

2. Symmetry conditions for a colinear response

In high-symmetry Rashba systems, mirror symmetry forbids diagonal in-plane components. For a mirror plane $\mathbf{S}\parallel \mathbf{J}$8, one has $\mathbf{S}\parallel \mathbf{J}$9 while $S_i=\sum_j \chi_{ij}E_j$0, so a term $S_i=\sum_j \chi_{ij}E_j$1 must vanish. This is why an aligned graphene/TMD bilayer with $S_i=\sum_j \chi_{ij}E_j$2 symmetry has $S_i=\sum_j \chi_{ij}E_j$3 and only the perpendicular response survives (Veneri et al., 2022). Likewise, the (111) LaAlO$S_i=\sum_j \chi_{ij}E_j$4/SrTiO$S_i=\sum_j \chi_{ij}E_j$5 interface has an antisymmetric in-plane tensor,

$S_i=\sum_j \chi_{ij}E_j$6

so the induced in-plane magnetization is necessarily perpendicular to the electric field (Trama et al., 2022).

The same restriction appears in superconducting Rashba models. In a 2D Rashba superconductor with a single polar axis $S_i=\sum_j \chi_{ij}E_j$7, the magnetoelectric tensor takes the antisymmetric form

$S_i=\sum_j \chi_{ij}E_j$8

equivalent to

$S_i=\sum_j \chi_{ij}E_j$9

so the induced magnetization is transverse rather than colinear (Chirolli et al., 2021). For current along $\chi_{xx}$0, the induced magnetization is along $\chi_{xx}$1.

CEE becomes symmetry-allowed when these constraints are relaxed. Several routes recur across the literature:

Symmetry setting	Allowed geometry	Representative consequence
High-symmetry Rashba / antisymmetric tensor	Transverse only	$\chi_{xx}$2, $\chi_{xx}$3 (Veneri et al., 2022)
Lower symmetry or mirror breaking	Diagonal terms allowed	finite $\chi_{xx}$4 becomes possible (Peters et al., 2018)
Chiral cubic symmetry in CoSi	Isotropic diagonal response	$\chi_{xx}$5 (Xu et al., 17 Jan 2025)
Twisted vdW heterostructure	Generic in-plane orientation	mirror breaking allows $\chi_{xx}$6 (Veneri et al., 2022)

Lower point-group symmetry, more general antisymmetric spin–orbit coupling, or combined Rashba–Dresselhaus-like structures can all lift the cancellations that force a purely transverse response. The correlated $\chi_{xx}$7-electron analysis makes this explicit: with pure Rashba $\chi_{xx}$8, $\chi_{xx}$9 vanishes by symmetry, but if $f$0 contains components such as $f$1 or the crystal symmetry is lowered, diagonal entries like $f$2 can be finite (Peters et al., 2018). This suggests that CEE is best regarded as a symmetry-allowed tensor component rather than a separate microscopic mechanism.

3. Microscopic mechanisms in normal-state systems

A direct route to CEE is to engineer the momentum-space spin texture so that the electrically shifted Fermi surface produces a net spin parallel to the drift direction. Twisted graphene/TMD heterostructures provide the clearest explicit realization. In that system the proximity-induced spin–orbit coupling contains a twist-dependent Rashba phase $f$3, and the in-plane spin response obeys

$f$4

At the critical twist angle defined by $f$5, the perpendicular component vanishes and the response is purely collinear, $f$6 (Veneri et al., 2022). For graphene/WSe$f$7, the predicted critical angle is $f$8, and the paper states that the effect is robust against twist-angle disorder and remains substantial up to room temperature (Veneri et al., 2022).

The microscopic picture is a twist-controlled evolution of the spin texture from conventional helical Rashba locking, $f$9, to a “hedgehog” or Weyl-type texture in which the spins are approximately radial. Under a current along $S_i \equiv M_i = \sum_j \chi_{ij}E_j,$0, such a radial texture produces a net $S_i \equiv M_i = \sum_j \chi_{ij}E_j,$1, i.e. a genuine CEE (Veneri et al., 2022).

Chiral metal surfaces provide another pathway. For a two-dimensional chiral surface with anisotropic spin–orbit coupling

$S_i \equiv M_i = \sum_j \chi_{ij}E_j,$2

the component $S_i \equiv M_i = \sum_j \chi_{ij}E_j,$3 allows a spin response parallel to the current direction, unlike pure Rashba locking. The induced longitudinal spin density takes the form

$S_i \equiv M_i = \sum_j \chi_{ij}E_j,$4

for electric field and current along $S_i \equiv M_i = \sum_j \chi_{ij}E_j,$5, explicitly realizing a spin polarization colinear with the current when $S_i \equiv M_i = \sum_j \chi_{ij}E_j,$6 is appreciable (Suzuki et al., 2022). In the strongly anisotropic limit, one spin component becomes nearly conserved, and the same eigenmode analysis that defines $S_i \equiv M_i = \sum_j \chi_{ij}E_j,$7 and $S_i \equiv M_i = \sum_j \chi_{ij}E_j,$8 shows a long-lived longitudinal spin mode (Suzuki et al., 2022).

A third, symmetry-distinct route does not require spin–orbit coupling at all. In coplanar $S_i \equiv M_i = \sum_j \chi_{ij}E_j,$9-wave magnets, the non-relativistic Edelstein effect is generated by exchange-driven spin splitting in a non-collinear magnetic texture. The response tensor in the minimal model has only

$f$0

with all other components vanishing by symmetry, so an electric field $f$1 induces a spin density strictly along $f$2 (Chakraborty et al., 2024). In CeNiAsO, the dominant component is likewise $f$3, and the first-principles analysis reports a response “25 times larger” than the maximally achieved relativistic EE quoted for comparison (Chakraborty et al., 2024). Strictly speaking, this is colinear with a fixed crystal axis rather than with the current itself, but it shows that CEE-like directional purity can arise from spin-space symmetry alone.

4. Superconducting CEE and the supercurrent diode effect

In superconductors, the Edelstein effect is the magnetoelectric coupling between supercurrent and spin polarization. The fundamental variable is the Cooper-pair momentum $f$4, which enters the free energy together with the Zeeman field $f$5. In a quasi-1D model with

$f$6

the depairing energy contains the term

$f$7

and $f$8 is identified as the microscopic Edelstein contribution (Yuan, 2023). The corresponding Ginzburg–Landau kernel is

$f$9

with

$M_y = \Upsilon_{yx} E_x.$0

as the Edelstein parameter. In this geometry the current direction, Zeeman spin axis, and induced spin polarization are effectively colinear (Yuan, 2023).

The central consequence is the supercurrent diode effect. In 1D the diode coefficient is

$M_y = \Upsilon_{yx} E_x.$1

and in 2D with linear in-plane SOC

$M_y = \Upsilon_{yx} E_x.$2

The paper states that $M_y = \Upsilon_{yx} E_x.$3 is strictly proportional to $M_y = \Upsilon_{yx} E_x.$4, so the diode effect vanishes if the Edelstein effect is turned off (Yuan, 2023). This makes the supercurrent diode effect a direct probe of the superconducting CEE.

The 2D formulation clarifies the geometry further. For

$M_y = \Upsilon_{yx} E_x.$5

the free-energy kernel contains

$M_y = \Upsilon_{yx} E_x.$6

and minimizing $M_y = \Upsilon_{yx} E_x.$7 gives

$M_y = \Upsilon_{yx} E_x.$8

The equilibrium superconducting state therefore chooses $M_y = \Upsilon_{yx} E_x.$9, i.e. the colinear configuration is selected automatically by the magnetoelectric coupling (Yuan, 2023). In this framework, CEE is not an accidental alignment but the longitudinal component of the superconducting Edelstein tensor in the transformed SOC frame.

A complementary superconducting literature emphasizes that not every superconducting Edelstein effect is colinear. In non-centrosymmetric orbital-Rashba superconductors,

$\chi_{yx}$0

and the response is strictly transverse by $\chi_{yx}$1 symmetry (Chirolli et al., 2021). What distinguishes the colinear case is therefore again the tensor structure allowed by symmetry, not the presence or absence of superconductivity.

5. Correlations, orbital physics, and enhancement mechanisms

The correlated $\chi_{yx}$2-electron study shows that strong correlations can greatly amplify any symmetry-allowed Edelstein component. In a periodic Anderson lattice with intra-orbital and inter-orbital antisymmetric spin–orbit coupling, the computed magnetoelectric ratio $\chi_{yx}$3 has a sharp maximum near the coherence temperature, where $\chi_{yx}$4-electrons cross over from localized to itinerant (Peters et al., 2018). The enhancement originates from two linked effects: inter-orbital antisymmetric spin–orbit coupling generates an effective spin texture in the conduction band via virtual $\chi_{yx}$5 processes, and incoherent $\chi_{yx}$6-states suppress the cancellation between opposite-helicity Fermi sheets that normally keeps the Edelstein effect small (Peters et al., 2018).

The paper gives explicit enhancement scales. For realistic parameters it finds a maximum $\chi_{yx}$7 more than $\chi_{yx}$8 larger than the non-interacting value near the coherence temperature, and at high temperatures the enhancement can be on the order of $\chi_{yx}$9 compared to a non-interacting Rashba-like system (Peters et al., 2018). The analysis is performed for the transverse component $\chi_{xx}$0 from $\chi_{xx}$1, but the Kubo formalism is general. This suggests that a symmetry-allowed colinear component $\chi_{xx}$2 would be amplified by the same localized–itinerant crossover and by the same removal of helicity cancellation.

Orbital Edelstein physics supplies another enhancement channel. In orbital-Rashba superconductors, the induced orbital magnetization can exceed the spin Edelstein response by more than an order of magnitude. The paper reports that for $\chi_{xx}$3 the maximum orbital magnetization is about $\chi_{xx}$4 times larger than the spin magnetization at equal $\chi_{xx}$5, and for $\chi_{xx}$6 it can be up to about $\chi_{xx}$7 times larger (Chirolli et al., 2021). The microscopic origin is multi-orbital avoided crossings, where the interband matrix element

$\chi_{xx}$8

is enhanced and changes sign as the orbital character switches across the crossing (Chirolli et al., 2021). Although the specific model remains transverse by symmetry, the paper explicitly notes that the same multi-orbital mechanism would carry over to a colinear response if $\chi_{xx}$9 were symmetry-allowed (Chirolli et al., 2021).

At oxide interfaces, the same orbital logic appears in transport rather than superconductivity. At the (111) LaAlO$\mathbf{S}\parallel \mathbf{J}$00/SrTiO$\mathbf{S}\parallel \mathbf{J}$01 interface, the orbital Edelstein susceptibility is typically about one order of magnitude larger than the spin susceptibility, and the spin contribution changes sign with chemical potential while the orbital part remains large (Trama et al., 2022). In the ideal trigonal geometry the in-plane tensor is antisymmetric, so no in-plane CEE occurs, but the work identifies the crucial ingredients for engineering one: generalized Rashba textures, strong orbital pseudospin, and multiband hybridization (Trama et al., 2022).

6. Chiral, nonlinear, and optical manifestations

In chiral crystals the linear Edelstein tensor can itself be purely diagonal. CoSi, with cubic space group $\mathbf{S}\parallel \mathbf{J}$02, is the cleanest example. The linear response is

$\mathbf{S}\parallel \mathbf{J}$03

and symmetry reduces the tensor to a single isotropic diagonal component,

$\mathbf{S}\parallel \mathbf{J}$04

with all off-diagonal components vanishing (Xu et al., 17 Jan 2025). In this case the induced magnetization is strictly parallel to the electric field: $\mathbf{S}\parallel \mathbf{J}$05 This is a genuine bulk colinear Edelstein effect enforced by cubic chirality rather than by low symmetry (Xu et al., 17 Jan 2025).

The same work emphasizes the symmetry distinction between linear and nonlinear Edelstein effects. The linear response is inversion-odd, so the linear coefficients of left- and right-handed CoSi have opposite signs, whereas the second-order nonlinear coefficients are inversion-even and are identical in the two enantiomers (Xu et al., 17 Jan 2025). The nonlinear response is not strictly colinear because its tensor structure involves $\mathbf{S}\parallel \mathbf{J}$06, but the linear chiral case shows that CEE can arise even in a highly symmetric crystal if the symmetry is chiral rather than mirror-protected.

Recent nonlinear theory extends the Edelstein family further by introducing the nonlinear magnetoelectric Edelstein effect,

$\mathbf{S}\parallel \mathbf{J}$07

Its intrinsic part is $\mathbf{S}\parallel \mathbf{J}$08-even but $\mathbf{S}\parallel \mathbf{J}$09-odd, so it can exist in noncentrosymmetric $\mathbf{S}\parallel \mathbf{J}$10-invariant materials, including insulators, where the usual intrinsic Edelstein effect is forbidden (Jia et al., 31 Jul 2025). The explicit model calculations mostly generate spins perpendicular to the plane for in-plane $\mathbf{S}\parallel \mathbf{J}$11 and $\mathbf{S}\parallel \mathbf{J}$12, rather than $\mathbf{S}\parallel \mathbf{J}$13, but the tensor analysis identifies the symmetry conditions under which more CEE-like field alignments could occur (Jia et al., 31 Jul 2025).

Optical detection has likewise become part of the CEE landscape. A recent first-principles analysis of electric-field-induced Kerr rotation on metallic Pt surfaces separates two contributions linear in the dc field: a time-reversal-odd orbital Edelstein contribution arising from the nonequilibrium occupation, and a time-reversal-even surface Pockels contribution arising from wave-function modification (Mahfouzi et al., 26 Oct 2025). The paper shows that the orbital Edelstein effect yields similar $\mathbf{S}\parallel \mathbf{J}$14 and $\mathbf{S}\parallel \mathbf{J}$15, while the surface Pockels effect leads to opposing values of $\mathbf{S}\parallel \mathbf{J}$16 and $\mathbf{S}\parallel \mathbf{J}$17 (Mahfouzi et al., 26 Oct 2025). This provides a practical way to distinguish magnetization-like Edelstein signals from purely electro-optic backgrounds in optical measurements, and it plausibly extends to future CEE-specific Kerr geometries.

7. Experimental probes, materials, and conceptual boundaries

Several experimental strategies recur across the literature. In superconductors, the most direct probe is the supercurrent diode effect, because its amplitude is proportional to the Edelstein parameter $\mathbf{S}\parallel \mathbf{J}$18 and its angular dependence tracks the longitudinal magnetoelectric coupling (Yuan, 2023). In twisted graphene/TMD bilayers, the proposed all-electrical “X-protocol” isolates the reciprocal collinear spin–galvanic effect in a lateral spin-valve geometry, thereby detecting the CEE without direct spin imaging (Veneri et al., 2022). In chiral surfaces and interfaces, transfer-matrix and Onsager formulations relate local spin accumulation, spin current, and charge current across the interface, making charge–spin conversion efficiencies experimentally accessible (Suzuki et al., 2022).

Candidate materials span distinct mechanisms. Graphene/WSe$\mathbf{S}\parallel \mathbf{J}$19 provides a twist-tunable room-temperature CEE at a critical angle (Veneri et al., 2022). CoSi provides a bulk chiral diagonal response $\mathbf{S}\parallel \mathbf{J}$20 (Xu et al., 17 Jan 2025). CeNiAsO exemplifies a non-relativistic, nearly axis-colinear current-induced spin polarization without spin–orbit coupling (Chakraborty et al., 2024). Correlated heavy-fermion compounds such as CeRhSi$\mathbf{S}\parallel \mathbf{J}$21, CeIrSi$\mathbf{S}\parallel \mathbf{J}$22, and CePt$\mathbf{S}\parallel \mathbf{J}$23Si are identified as promising correlated platforms in which any symmetry-allowed Edelstein component can be strongly enhanced near the coherence temperature (Peters et al., 2018).

Two conceptual boundaries are especially important. First, CEE is not synonymous with “large Edelstein effect”: large orbital or correlated enhancements can remain strictly transverse if the point group enforces $\mathbf{S}\parallel \mathbf{J}$24 (Chirolli et al., 2021). Second, CEE is not synonymous with “spin–orbit-driven”: non-relativistic $\mathbf{S}\parallel \mathbf{J}$25-wave magnets realize a highly directional Edelstein effect without SOC, with the response set by spin-space symmetry and exchange splitting (Chakraborty et al., 2024).

Taken together, the current literature establishes CEE as a symmetry-selected longitudinal member of the broader Edelstein family. Its realizations range from twist-engineered radial spin textures and chiral diagonal tensors to superconducting free-energy couplings, correlation-enhanced heavy-fermion responses, orbital magnetoelectric effects, and non-relativistic magnetic textures. The unifying principle is simple but restrictive: once the crystal, magnetic, or superconducting symmetry permits a diagonal magnetoelectric tensor element and the electronic structure avoids cancellation between opposite textures, a colinear current-induced polarization can emerge and, in several known platforms, become unusually large (Veneri et al., 2022).