Orbital Edelstein Effect (OEE)

• Orbital Edelstein effect is the dc electric field-induced generation of orbital angular momentum in gyrotropic conductors, independent of spin-orbit coupling.
• It employs current-induced redistribution of Bloch electrons and orbital texture, yielding significant orbital magnetization in materials like Cu₂WSe₄ and gated TMDs.
• Studies focus on symmetry, microscopic mechanisms, and charge-orbital reciprocity to advance orbitronic responses and device applications.

The orbital Edelstein effect (OEE) is the nonequilibrium generation of orbital angular momentum density or orbital magnetization by a dc electric field in a conductor with broken inversion symmetry. In the narrow usage adopted in much of the recent literature, it is the dissipative, Fermi-surface, relaxation-time-dependent orbital analog of the spin Edelstein effect; in broader reviews it is often grouped with orbital magneto-electric effects more generally (Atencia et al., 2024). Across current theory and experiment, OEE is formulated either as a current-induced orbital magnetization, $M_i^{\rm O}=\alpha^{\rm O}_{ij}E_j$, or as an induced orbital angular momentum density, $L_i=\chi_{ij}E_j$, depending on whether one emphasizes magnetic moment or angular momentum. Its central ingredients are orbital texture in Bloch states, gyrotropic symmetry, and a nonequilibrium shift of the electronic distribution; unlike the spin Edelstein effect, it need not rely on spin-orbit coupling (SOC), and in several multi-orbital or chiral systems it exceeds the spin contribution by more than one order of magnitude (Nakazawa et al., 18 Dec 2025, Johansson et al., 2020).

1. Linear-response formulation

A standard formulation writes the Edelstein response as

$M_i=\alpha_{ij}E_j\equiv M_i^{\rm S}+M_i^{\rm O},$

where $M_i^{\rm S}$ and $M_i^{\rm O}$ are the spin and orbital contributions. In the relaxation-time approximation for a uniform dc field,

$$\alpha^{\rm O}_{ij}=\frac{e\tau}{V}\sum_{n,\mathbf{k}}\left(-\frac{\partial f}{\partial \varepsilon}\right) m^{\rm O}_{n\mathbf{k},i}\,v_{n\mathbf{k},j},$$

with $$v_{n\mathbf{k},j}=\hbar^{-1}\partial_{k_j}\varepsilon_{n\mathbf{k}}$$ and $m^{\rm O}_{n\mathbf{k}}$ the orbital magnetic moment of the Bloch state. In the same framework, the induced nonequilibrium distribution is

$$\delta f_{n\mathbf{k}}\sim e\tau\,(\mathbf{E}\cdot\mathbf{v}_{n\mathbf{k}})\left(-\frac{\partial f}{\partial \varepsilon}\right),$$

so the OEE is a Fermi-surface response governed by the correlation between orbital moment and velocity (Nakazawa et al., 18 Dec 2025).

The orbital magnetic moment entering this expression is the modern-theory quantity

$$\mathbf{m}^{\rm O}_{n\mathbf{k}}=-\frac{e}{2\hbar}\,\mathrm{Im}\Big[\langle \boldsymbol{\nabla}_{\mathbf{k}}u_{n\mathbf{k}}|\times\big(\hat H(\mathbf{k})-\varepsilon_{n\mathbf{k}}\big)|\boldsymbol{\nabla}_{\mathbf{k}}u_{n\mathbf{k}}\rangle\Big].$$

Equivalent formulations use orbital angular momentum rather than orbital magnetization. In one-dimensional chiral models and carbon nanotubes, for example, one writes $L_i=\chi_{ij}E_j$0 or $L_i=\chi_{ij}E_j$1, with $L_i=\chi_{ij}E_j$2 obtained from Boltzmann transport and band-resolved orbital angular momentum expectation values (Göbel et al., 10 Apr 2025, Göbel et al., 7 Feb 2025).

A useful distinction in the review literature is between the dissipative $L_i=\chi_{ij}E_j$3 term and an intrinsic $L_i=\chi_{ij}E_j$4 term that is independent of disorder strength. The former is what is usually meant by the Edelstein effect proper; the latter is often discussed under the broader label orbital magneto-electric effect, particularly in $L_i=\chi_{ij}E_j$5-symmetric antiferromagnets (Atencia et al., 2024).

2. Symmetry: inversion breaking, gyrotropy, and chirality

Broken inversion symmetry is essential. In inversion-symmetric crystals, contributions from $L_i=\chi_{ij}E_j$6 and $L_i=\chi_{ij}E_j$7 cancel, and the orbital Edelstein tensor vanishes. More specifically, the response tensor connecting a polar vector to an axial vector is allowed only in gyrotropic point groups. In the chiral-crystal formulation of Yoda, Yokoyama, and Murakami, nonzero orbital Edelstein tensors are allowed in 18 of the 21 noncentrosymmetric point groups (Yoda et al., 2017).

A central clarification of the recent literature is that chirality is not required for a finite OEE. Cu$L_i=\chi_{ij}E_j$8WSe$L_i=\chi_{ij}E_j$9 has point group $M_i=\alpha_{ij}E_j\equiv M_i^{\rm S}+M_i^{\rm O},$0, which is gyrotropic but achiral because it contains an $M_i=\alpha_{ij}E_j\equiv M_i^{\rm S}+M_i^{\rm O},$1 rotoreflection axis. Its Edelstein tensor has the symmetry

$M_i=\alpha_{ij}E_j\equiv M_i^{\rm S}+M_i^{\rm O},$2

so a strong OEE is symmetry-allowed even though the crystal is achiral (Nakazawa et al., 18 Dec 2025). This directly refutes the common identification of OEE with chirality alone.

What chirality changes is the behavior of the trace of a gyrotropic tensor. In Cu$M_i=\alpha_{ij}E_j\equiv M_i^{\rm S}+M_i^{\rm O},$3WSe$M_i=\alpha_{ij}E_j\equiv M_i^{\rm S}+M_i^{\rm O},$4, $M_i=\alpha_{ij}E_j\equiv M_i^{\rm S}+M_i^{\rm O},$5, while the authors propose $M_i=\alpha_{ij}E_j\equiv M_i^{\rm S}+M_i^{\rm O},$6 for a generic gyrotropic tensor as a “chirality indicator”: $M_i=\alpha_{ij}E_j\equiv M_i^{\rm S}+M_i^{\rm O},$7 implies chirality, and its sign flips between left- and right-handed enantiomers. In their formulation, the traces of the Edelstein and nonlinear chiral thermoelectric tensors reduce to Brillouin-zone integrals of $M_i=\alpha_{ij}E_j\equiv M_i^{\rm S}+M_i^{\rm O},$8 or $M_i=\alpha_{ij}E_j\equiv M_i^{\rm S}+M_i^{\rm O},$9, linking chirality to monopole-like singularities of magnetic moments in $M_i^{\rm S}$0-space (Nakazawa et al., 18 Dec 2025).

The same symmetry logic appears in two-dimensional systems. Pristine monolayer 2H-TMDs have point group $M_i^{\rm S}$1, for which in-plane Edelstein response is forbidden; a perpendicular gate field lowers the symmetry to $M_i^{\rm S}$2, allowing

$M_i^{\rm S}$3

so an in-plane electric field produces an in-plane magnetization perpendicular to the field (Yao et al., 30 Sep 2025). In strained TMDCs, symmetry lowering to $M_i^{\rm S}$4 likewise activates orbital magneto-electric response (Atencia et al., 2024).

3. Microscopic mechanisms

One broad mechanism is orbital texture of Bloch electrons in noncentrosymmetric bands. In Cu$M_i^{\rm S}$5WSe$M_i^{\rm S}$6, the OEE is controlled directly by $M_i^{\rm S}$7; Berry curvature is computed but does not enter the linear Edelstein coefficient $M_i^{\rm S}$8 directly. This sharply contrasts with the nonlinear Hall and nonlinear chiral thermoelectric responses in the same material, which are dominated by the Berry-curvature dipole (Nakazawa et al., 18 Dec 2025).

A second mechanism is inter-site or itinerant orbital circulation imposed by crystal geometry. In the analytically solvable chiral helix model developed for tellurium-like systems, the orbital Edelstein susceptibility is

$M_i^{\rm S}$9

so the sign is set by chirality through $M_i^{\rm O}$0. The same work shows that the orbital contribution can surpass the spin contribution by orders of magnitude, while SOC is needed only to convert orbital polarization into spin polarization (Göbel et al., 7 Feb 2025). In carbon nanotubes, the OEE becomes chirality-induced orbital selectivity: the susceptibility is odd in the chirality angle, vanishes for zigzag and armchair tubes, scales linearly with circumference in the low-energy Dirac regime, and increases quadratically with energy near the Fermi level for metallic tubes (Göbel et al., 10 Apr 2025).

A third mechanism is interaction-driven symmetry reduction. In the density-wave model of Massarelli, Wu, and Paramekanti, a spinless line-node semimetal has no OEE in its centrosymmetric normal phase, but a sublattice-staggered charge-density-wave order breaks inversion, reduces the point group to $M_i^{\rm O}$1, gaps the nodal line, and activates an orbital Edelstein tensor with $M_i^{\rm O}$2. The resulting OEE is non-monotonic in temperature: zero above the transition, maximal in the partially gapped regime, and vanishing again at low temperature when the system becomes fully insulating (Massarelli et al., 2019).

A fourth mechanism is inversion breaking by a scalar-potential gradient. In a single parabolic band with no SOC, an asymmetric scalar potential $M_i^{\rm O}$3 generates a density gradient, and a dc field along $M_i^{\rm O}$4 produces an orbital magnetization along $M_i^{\rm O}$5. The resulting OEE scales as $M_i^{\rm O}$6,

$M_i^{\rm O}$7

which the authors argue can exceed the spin Edelstein effect by orders of magnitude in sufficiently clean systems with weak SOC (Ado et al., 2024).

A fifth mechanism is edge-localized itinerant OAM. In $M_i^{\rm O}$8-orbital lattice models with zigzag or irregular edges, edge states acquire inter-atomic orbital angular momentum $M_i^{\rm O}$9 and yield an edge OEE under an in-plane field, even though straight edges can give zero edge OAM. This work shows that edge OAM accumulation does not obey a simple bulk-boundary correspondence with the orbital Hall effect; geometry and “wiggling” real-space trajectories at the boundary are decisive (Lee et al., 2024).

4. Representative material platforms

Comparative studies show that the hierarchy between orbital and spin Edelstein responses is strongly material dependent, but the orbital channel is often dominant. The following platforms illustrate the present landscape.

Platform	Salient OEE result	Reference
Cu$$\alpha^{\rm O}_{ij}=\frac{e\tau}{V}\sum_{n,\mathbf{k}}\left(-\frac{\partial f}{\partial \varepsilon}\right) m^{\rm O}_{n\mathbf{k},i}\,v_{n\mathbf{k},j},$$0WSe$$\alpha^{\rm O}_{ij}=\frac{e\tau}{V}\sum_{n,\mathbf{k}}\left(-\frac{\partial f}{\partial \varepsilon}\right) m^{\rm O}_{n\mathbf{k},i}\,v_{n\mathbf{k},j},$$1	Valence regime $$\alpha^{\rm O}_{ij}=\frac{e\tau}{V}\sum_{n,\mathbf{k}}\left(-\frac{\partial f}{\partial \varepsilon}\right) m^{\rm O}_{n\mathbf{k},i}\,v_{n\mathbf{k},j},$$2; conduction regime $$\alpha^{\rm O}_{ij}=\frac{e\tau}{V}\sum_{n,\mathbf{k}}\left(-\frac{\partial f}{\partial \varepsilon}\right) m^{\rm O}_{n\mathbf{k},i}\,v_{n\mathbf{k},j},$$3; SOC minor in valence	(Nakazawa et al., 18 Dec 2025)
AlO$$\alpha^{\rm O}_{ij}=\frac{e\tau}{V}\sum_{n,\mathbf{k}}\left(-\frac{\partial f}{\partial \varepsilon}\right) m^{\rm O}_{n\mathbf{k},i}\,v_{n\mathbf{k},j},$$4/SrTiO$$\alpha^{\rm O}_{ij}=\frac{e\tau}{V}\sum_{n,\mathbf{k}}\left(-\frac{\partial f}{\partial \varepsilon}\right) m^{\rm O}_{n\mathbf{k},i}\,v_{n\mathbf{k},j},$$5 2DEG	Orbital Edelstein effect exceeds spin by more than one order of magnitude	(Johansson et al., 2020)
Gated monolayer TMDs	$$\alpha^{\rm O}_{ij}=\frac{e\tau}{V}\sum_{n,\mathbf{k}}\left(-\frac{\partial f}{\partial \varepsilon}\right) m^{\rm O}_{n\mathbf{k},i}\,v_{n\mathbf{k},j},$$6; one–two orders above previously studied systems	(Yao et al., 30 Sep 2025)
Janus monolayer TMDs	OEE about one order larger than SEE; strongest for large internal field $$\alpha^{\rm O}_{ij}=\frac{e\tau}{V}\sum_{n,\mathbf{k}}\left(-\frac{\partial f}{\partial \varepsilon}\right) m^{\rm O}_{n\mathbf{k},i}\,v_{n\mathbf{k},j},$$7	(Sahu et al., 13 Nov 2025)
Chiral carbon nanotubes	$$\alpha^{\rm O}_{ij}=\frac{e\tau}{V}\sum_{n,\mathbf{k}}\left(-\frac{\partial f}{\partial \varepsilon}\right) m^{\rm O}_{n\mathbf{k},i}\,v_{n\mathbf{k},j},$$8; metallic tubes show $$\alpha^{\rm O}_{ij}=\frac{e\tau}{V}\sum_{n,\mathbf{k}}\left(-\frac{\partial f}{\partial \varepsilon}\right) m^{\rm O}_{n\mathbf{k},i}\,v_{n\mathbf{k},j},$$9 near $$v_{n\mathbf{k},j}=\hbar^{-1}\partial_{k_j}\varepsilon_{n\mathbf{k}}$$0	(Göbel et al., 10 Apr 2025)
Chiral helix / Te model	$$v_{n\mathbf{k},j}=\hbar^{-1}\partial_{k_j}\varepsilon_{n\mathbf{k}}$$1; orbital exceeds spin by orders of magnitude	(Göbel et al., 7 Feb 2025)

In Cu$$v_{n\mathbf{k},j}=\hbar^{-1}\partial_{k_j}\varepsilon_{n\mathbf{k}}$$2WSe$$v_{n\mathbf{k},j}=\hbar^{-1}\partial_{k_j}\varepsilon_{n\mathbf{k}}$$3, the orbital channel dominates in the hole-doped valence region because the relevant Cu-$$v_{n\mathbf{k},j}=\hbar^{-1}\partial_{k_j}\varepsilon_{n\mathbf{k}}$$4/Se-$$v_{n\mathbf{k},j}=\hbar^{-1}\partial_{k_j}\varepsilon_{n\mathbf{k}}$$5 states have relatively weak SOC but strong orbital character, while conduction bands involve W-$$v_{n\mathbf{k},j}=\hbar^{-1}\partial_{k_j}\varepsilon_{n\mathbf{k}}$$6 states and bring the spin contribution closer to the orbital one. Calculations with and without SOC show that the valence-band OEE is almost unchanged, establishing that the effect is primarily orbital and structural rather than SOC-driven (Nakazawa et al., 18 Dec 2025).

At oxide interfaces, the multi-orbital $$v_{n\mathbf{k},j}=\hbar^{-1}\partial_{k_j}\varepsilon_{n\mathbf{k}}$$7 structure of AlO$$v_{n\mathbf{k},j}=\hbar^{-1}\partial_{k_j}\varepsilon_{n\mathbf{k}}$$8/SrTiO$$v_{n\mathbf{k},j}=\hbar^{-1}\partial_{k_j}\varepsilon_{n\mathbf{k}}$$9 produces unequal orbital moments in Rashba-split band pairs, reducing cancellation between branches and yielding an orbital Edelstein efficiency that exceeds the spin one by more than an order of magnitude (Johansson et al., 2020).

In gated monolayer TMDs, gate-induced mirror-symmetry breaking generates Rashba-type chiral orbital and spin textures. The calculated orbital Edelstein susceptibility reaches $m^{\rm O}_{n\mathbf{k}}$0, with electron doping dominated by the orbital channel and hole doping showing comparable orbital and spin contributions that can be strongly enhanced by strain through $m^{\rm O}_{n\mathbf{k}}$1- versus $m^{\rm O}_{n\mathbf{k}}$2-valley shifts (Yao et al., 30 Sep 2025). Janus TMDs realize an intrinsic version of the same idea: the built-in out-of-plane electric field $m^{\rm O}_{n\mathbf{k}}$3 mixes $m^{\rm O}_{n\mathbf{k}}$4 with $m^{\rm O}_{n\mathbf{k}}$5, produces robust orbital textures around $m^{\rm O}_{n\mathbf{k}}$6, $m^{\rm O}_{n\mathbf{k}}$7, and $m^{\rm O}_{n\mathbf{k}}$8, and yields OEE about one order larger than SEE (Sahu et al., 13 Nov 2025).

5. Interfaces, nonlocal transport, optics, and superconductivity

At interfaces, the relevant language is often orbital Rashba Edelstein effect (OREE): charge current produces interfacial orbital accumulation, or its inverse converts orbital accumulation into charge current. In Pt/CuO$m^{\rm O}_{n\mathbf{k}}$9-based nonlocal magnon devices, adding the Pt/CuO$$\delta f_{n\mathbf{k}}\sim e\tau\,(\mathbf{E}\cdot\mathbf{v}_{n\mathbf{k}})\left(-\frac{\partial f}{\partial \varepsilon}\right),$$0 interface enhances first-harmonic electrical magnon signals from about $$\delta f_{n\mathbf{k}}\sim e\tau\,(\mathbf{E}\cdot\mathbf{v}_{n\mathbf{k}})\left(-\frac{\partial f}{\partial \varepsilon}\right),$$1 to about $$\delta f_{n\mathbf{k}}\sim e\tau\,(\mathbf{E}\cdot\mathbf{v}_{n\mathbf{k}})\left(-\frac{\partial f}{\partial \varepsilon}\right),$$2 and second-harmonic thermal magnon signals from about $$\delta f_{n\mathbf{k}}\sim e\tau\,(\mathbf{E}\cdot\mathbf{v}_{n\mathbf{k}})\left(-\frac{\partial f}{\partial \varepsilon}\right),$$3 to about $$\delta f_{n\mathbf{k}}\sim e\tau\,(\mathbf{E}\cdot\mathbf{v}_{n\mathbf{k}})\left(-\frac{\partial f}{\partial \varepsilon}\right),$$4 for $$\delta f_{n\mathbf{k}}\sim e\tau\,(\mathbf{E}\cdot\mathbf{v}_{n\mathbf{k}})\left(-\frac{\partial f}{\partial \varepsilon}\right),$$5 nm. After normalization, the enhancement factors are $$\delta f_{n\mathbf{k}}\sim e\tau\,(\mathbf{E}\cdot\mathbf{v}_{n\mathbf{k}})\left(-\frac{\partial f}{\partial \varepsilon}\right),$$6 and $$\delta f_{n\mathbf{k}}\sim e\tau\,(\mathbf{E}\cdot\mathbf{v}_{n\mathbf{k}})\left(-\frac{\partial f}{\partial \varepsilon}\right),$$7, and the extracted asymmetry

$$\delta f_{n\mathbf{k}}\sim e\tau\,(\mathbf{E}\cdot\mathbf{v}_{n\mathbf{k}})\left(-\frac{\partial f}{\partial \varepsilon}\right),$$8

implies that inverse OREE is about $$\delta f_{n\mathbf{k}}\sim e\tau\,(\mathbf{E}\cdot\mathbf{v}_{n\mathbf{k}})\left(-\frac{\partial f}{\partial \varepsilon}\right),$$9 stronger than direct OREE (Mendoza-Rodarte et al., 2024).

A more explicit reciprocity test was achieved in Al$$\mathbf{m}^{\rm O}_{n\mathbf{k}}=-\frac{e}{2\hbar}\,\mathrm{Im}\Big[\langle \boldsymbol{\nabla}_{\mathbf{k}}u_{n\mathbf{k}}|\times\big(\hat H(\mathbf{k})-\varepsilon_{n\mathbf{k}}\big)|\boldsymbol{\nabla}_{\mathbf{k}}u_{n\mathbf{k}}\rangle\Big].$$0O$$\mathbf{m}^{\rm O}_{n\mathbf{k}}=-\frac{e}{2\hbar}\,\mathrm{Im}\Big[\langle \boldsymbol{\nabla}_{\mathbf{k}}u_{n\mathbf{k}}|\times\big(\hat H(\mathbf{k})-\varepsilon_{n\mathbf{k}}\big)|\boldsymbol{\nabla}_{\mathbf{k}}u_{n\mathbf{k}}\rangle\Big].$$1/CuO$$\mathbf{m}^{\rm O}_{n\mathbf{k}}=-\frac{e}{2\hbar}\,\mathrm{Im}\Big[\langle \boldsymbol{\nabla}_{\mathbf{k}}u_{n\mathbf{k}}|\times\big(\hat H(\mathbf{k})-\varepsilon_{n\mathbf{k}}\big)|\boldsymbol{\nabla}_{\mathbf{k}}u_{n\mathbf{k}}\rangle\Big].$$2/Cu structures using nonlocal measurements of direct and inverse orbital Edelstein effect. The measured magnetization-odd resistances satisfy

$$\mathbf{m}^{\rm O}_{n\mathbf{k}}=-\frac{e}{2\hbar}\,\mathrm{Im}\Big[\langle \boldsymbol{\nabla}_{\mathbf{k}}u_{n\mathbf{k}}|\times\big(\hat H(\mathbf{k})-\varepsilon_{n\mathbf{k}}\big)|\boldsymbol{\nabla}_{\mathbf{k}}u_{n\mathbf{k}}\rangle\Big].$$3

with a representative value $$\mathbf{m}^{\rm O}_{n\mathbf{k}}=-\frac{e}{2\hbar}\,\mathrm{Im}\Big[\langle \boldsymbol{\nabla}_{\mathbf{k}}u_{n\mathbf{k}}|\times\big(\hat H(\mathbf{k})-\varepsilon_{n\mathbf{k}}\big)|\boldsymbol{\nabla}_{\mathbf{k}}u_{n\mathbf{k}}\rangle\Big].$$4 and $$\mathbf{m}^{\rm O}_{n\mathbf{k}}=-\frac{e}{2\hbar}\,\mathrm{Im}\Big[\langle \boldsymbol{\nabla}_{\mathbf{k}}u_{n\mathbf{k}}|\times\big(\hat H(\mathbf{k})-\varepsilon_{n\mathbf{k}}\big)|\boldsymbol{\nabla}_{\mathbf{k}}u_{n\mathbf{k}}\rangle\Big].$$5 at $$\mathbf{m}^{\rm O}_{n\mathbf{k}}=-\frac{e}{2\hbar}\,\mathrm{Im}\Big[\langle \boldsymbol{\nabla}_{\mathbf{k}}u_{n\mathbf{k}}|\times\big(\hat H(\mathbf{k})-\varepsilon_{n\mathbf{k}}\big)|\boldsymbol{\nabla}_{\mathbf{k}}u_{n\mathbf{k}}\rangle\Big].$$6 nm, confirming Onsager reciprocity for charge-orbital conversion. The lateral orbital decay length is approximately $$\mathbf{m}^{\rm O}_{n\mathbf{k}}=-\frac{e}{2\hbar}\,\mathrm{Im}\Big[\langle \boldsymbol{\nabla}_{\mathbf{k}}u_{n\mathbf{k}}|\times\big(\hat H(\mathbf{k})-\varepsilon_{n\mathbf{k}}\big)|\boldsymbol{\nabla}_{\mathbf{k}}u_{n\mathbf{k}}\rangle\Big].$$7 nm at room temperature, is independent of Cu thickness, and decreases upon cooling, in clear contrast to spin diffusion in conventional nonlocal spin valves (Gao et al., 16 Feb 2025).

Optically, OEE appears as a current-induced, time-reversal-odd change in the ac conductivity near metallic surfaces. In Pt thin films, first-principles electro-optic calculations show that electric-field-induced Kerr rotation contains comparable contributions from OEE and a surface Pockels effect. The OEE contribution arises from the dc-field-induced change of the electron distribution function and is antisymmetric in the relevant conductivity indices; it gives similar Kerr rotations for $$\mathbf{m}^{\rm O}_{n\mathbf{k}}=-\frac{e}{2\hbar}\,\mathrm{Im}\Big[\langle \boldsymbol{\nabla}_{\mathbf{k}}u_{n\mathbf{k}}|\times\big(\hat H(\mathbf{k})-\varepsilon_{n\mathbf{k}}\big)|\boldsymbol{\nabla}_{\mathbf{k}}u_{n\mathbf{k}}\rangle\Big].$$8- and $$\mathbf{m}^{\rm O}_{n\mathbf{k}}=-\frac{e}{2\hbar}\,\mathrm{Im}\Big[\langle \boldsymbol{\nabla}_{\mathbf{k}}u_{n\mathbf{k}}|\times\big(\hat H(\mathbf{k})-\varepsilon_{n\mathbf{k}}\big)|\boldsymbol{\nabla}_{\mathbf{k}}u_{n\mathbf{k}}\rangle\Big].$$9-polarized light, $L_i=\chi_{ij}E_j$00 and $L_i=\chi_{ij}E_j$01. The surface Pockels contribution instead arises mainly from dc-field-induced changes of the wave functions and yields opposing $L_i=\chi_{ij}E_j$02 and $L_i=\chi_{ij}E_j$03. The active region lies within a couple of nanometers of the surface, with a fitted characteristic length below $L_i=\chi_{ij}E_j$04 nm in the truncation analysis (Mahfouzi et al., 26 Oct 2025).

In superconductors, the OEE becomes a non-dissipative magneto-electric effect driven by supercurrent rather than normal current. In a non-centrosymmetric multi-orbital superconductor with orbital Rashba coupling, the induced magnetization obeys $L_i=\chi_{ij}E_j$05, and the orbital moment can exceed the spin moment by a factor of about $L_i=\chi_{ij}E_j$06 for $L_i=\chi_{ij}E_j$07 and about $L_i=\chi_{ij}E_j$08 for $L_i=\chi_{ij}E_j$09, for equal coupling constants. The sign is tunable by moving the Fermi level near avoided crossings, and phase inhomogeneities produce coherence-length-scale domains with opposite orbital moment orientation; these features survive self-consistent real-space treatment of the superconducting order parameter (Chirolli et al., 2021).

6. Conceptual boundaries, misconceptions, and open problems

Several conceptual points now appear settled. First, OEE is not equivalent to chirality. Achiral gyrotropic crystals such as Cu$L_i=\chi_{ij}E_j$10WSe$L_i=\chi_{ij}E_j$11 can show a strong OEE, whereas chirality enters more sharply through traced gyrotropic tensors and enantiomer-sensitive signs (Nakazawa et al., 18 Dec 2025). Second, OEE does not require SOC. This is established independently in gyrotropic achiral Cu$L_i=\chi_{ij}E_j$12WSe$L_i=\chi_{ij}E_j$13, in the density-wave line-node model, in the chiral helix model, in Janus and gated TMD analyses where orbital texture exists prior to spin conversion, and in the scalar-potential-gradient theory where spin remains entirely passive (Nakazawa et al., 18 Dec 2025, Massarelli et al., 2019, Göbel et al., 7 Feb 2025, Ado et al., 2024). Third, edge OAM accumulation is not automatically the boundary image of a bulk orbital Hall current; edge geometry can generate an independent edge OEE without a simple bulk-boundary correspondence (Lee et al., 2024).

At the same time, several issues remain active. The review literature highlights the need for a nonequilibrium extension of the modern theory of orbital magnetization that systematically incorporates disorder, boundaries, and the coexistence of Fermi-surface and Fermi-sea contributions. Interface charge-orbital-spin conversion is still not microscopically settled, even though it is central to orbital torques, inverse orbital Edelstein conversion, and orbital pumping. Experimentally, separating spin and orbital channels remains nontrivial; proposed routes include XMCD at oxide interfaces, polarization-resolved Kerr measurements that exploit the distinct $L_i=\chi_{ij}E_j$14 signatures of OEE and surface Pockels effects, and light-element platforms where spin Hall backgrounds are weak (Atencia et al., 2024, Johansson et al., 2020, Mahfouzi et al., 26 Oct 2025).

Taken together, the present literature defines OEE as a unifying orbitronic response: a dc field or supercurrent redistributes Bloch-electron orbital angular momentum in a gyrotropic environment, generating bulk or interfacial orbital magnetization that can dominate over the spin Edelstein effect, survive without SOC, exhibit chirality-dependent sign structure, and couple naturally to nonlinear Hall phenomena, magneto-optics, nonlocal transport, and superconducting phase textures (Atencia et al., 2024).