Orbital Edelstein Susceptibility

• Orbital Edelstein susceptibility is the linear-response coefficient that links applied electric fields or currents to induced orbital magnetization in noncentrosymmetric materials.
• It is derived using multiple frameworks, including Kubo linear response and semiclassical Boltzmann transport, and varies across platforms such as oxide interfaces, antiferromagnets, and superconductors.
• The effect is decoupled from the spin response, highlighting its potential for the design of gyrotropic and chiral magnetoelectric devices.

Searching arXiv for recent and foundational papers on orbital Edelstein susceptibility to ground the article in published work. Orbital Edelstein susceptibility is the linear-response coefficient that quantifies how an applied electric field, charge current, or superconducting phase gradient generates a nonequilibrium orbital magnetization or orbital angular momentum in a system with broken inversion symmetry. In its most common normal-state form, it is written as

$$M_i^{\mathrm{orb}}(\omega)=\sum_j \chi_{ij}^{\mathrm{orb}}(\omega) E_j(\omega),$$

while in superconductors it is often defined through the phase gradient,

$$\langle \hat{L}_\mu\rangle=-\frac{1}{2}\chi_{\mu\nu}q_\nu.$$

The literature shows that the quantity is not tied to a single microscopic mechanism or unit convention: it can be formulated for local atomic projections, unit-cell orbital moments, magnetization densities, or supercurrent-induced orbital polarization, and it has been evaluated with Kubo, semiclassical Boltzmann, modern orbital magnetization, and Bogoliubov–de Gennes response frameworks (Salemi et al., 2019, Chirolli et al., 2021, Göbel et al., 10 Apr 2025).

1. Definition and constitutive relations

The orbital Edelstein effect is the generation of a non-equilibrium orbital polarization by an applied electric field in noncentrosymmetric or gyrotropic materials. In the antiferromagnetic formulation of the generalized Rashba–Edelstein effect, the orbital susceptibility tensor is the coefficient in

$$M_i^{\mathrm{orb}}(\omega)=\sum_j \chi_{ij}^{\mathrm{orb}}(\omega)E_j(\omega),$$

with the full induced magnetic polarization given by $\delta M=\mu_B\delta(L+2S)$, so that orbital and spin susceptibilities can be discussed separately as $\chi^L_{ij}$ and $\chi^S_{ij}$ (Salemi et al., 2019).

The same object is written in several equivalent but context-dependent ways. Carbon nanotube work defines the observable as the non-equilibrium orbital magnetic moment per unit cell along the tube axis, $m_{L_z}^{uc}=\chi_z^{L_z}E_z$, and frequently reports the susceptibility per unit tube length, $\chi_z^{L_z}\cdot a/T$, in units of $\mu_B\cdot\mathrm{m}/\mathrm{V}$ (Göbel et al., 10 Apr 2025). Oxide-interface studies write the induced moment per 2D unit cell as $(A_0/A)m=[\chi^s+\chi^l]E\equiv \chi E$, with $$\langle \hat{L}_\mu\rangle=-\frac{1}{2}\chi_{\mu\nu}q_\nu.$$0 the orbital Edelstein efficiency tensor (Johansson et al., 2020). In bilayer Rashba systems, the same distinction appears between a “per-field” susceptibility, $$\langle \hat{L}_\mu\rangle=-\frac{1}{2}\chi_{\mu\nu}q_\nu.$$1, and a “per-current” susceptibility, $$\langle \hat{L}_\mu\rangle=-\frac{1}{2}\chi_{\mu\nu}q_\nu.$$2, related by $$\langle \hat{L}_\mu\rangle=-\frac{1}{2}\chi_{\mu\nu}q_\nu.$$3 (M. et al., 2023).

A central point in the literature is that orbital Edelstein susceptibility is not reducible to spin Edelstein susceptibility with a different prefactor. In noncentrosymmetric antiferromagnets, the dominant staggered orbital response remains finite and essentially unchanged when spin–orbit coupling is turned off, while the spin susceptibility vanishes with spin–orbit coupling (Salemi et al., 2019). In Cu$$\langle \hat{L}_\mu\rangle=-\frac{1}{2}\chi_{\mu\nu}q_\nu.$$4WSe$$\langle \hat{L}_\mu\rangle=-\frac{1}{2}\chi_{\mu\nu}q_\nu.$$5, the orbital Edelstein effect is described as current-induced orbital magnetization in a noncentrosymmetric conductor, with constitutive relation $$\langle \hat{L}_\mu\rangle=-\frac{1}{2}\chi_{\mu\nu}q_\nu.$$6 and units of $$\langle \hat{L}_\mu\rangle=-\frac{1}{2}\chi_{\mu\nu}q_\nu.$$7 for $$\langle \hat{L}_\mu\rangle=-\frac{1}{2}\chi_{\mu\nu}q_\nu.$$8 (Nakazawa et al., 18 Dec 2025). This diversity of definitions reflects different observables, not a disagreement about the underlying response.

2. Microscopic formulations

Two broad microscopic routes dominate the subject. The first is Kubo-type linear response, where the orbital operator and the current or velocity operator appear explicitly in interband and intraband terms. For noncentrosymmetric antiferromagnets, the susceptibility is evaluated from a Kubo-like expression with

$$\langle \hat{L}_\mu\rangle=-\frac{1}{2}\chi_{\mu\nu}q_\nu.$$9

where $$M_i^{\mathrm{orb}}(\omega)=\sum_j \chi_{ij}^{\mathrm{orb}}(\omega)E_j(\omega),$$0 or $$M_i^{\mathrm{orb}}(\omega)=\sum_j \chi_{ij}^{\mathrm{orb}}(\omega)E_j(\omega),$$1, $$M_i^{\mathrm{orb}}(\omega)=\sum_j \chi_{ij}^{\mathrm{orb}}(\omega)E_j(\omega),$$2, and a single phenomenological broadening $$M_i^{\mathrm{orb}}(\omega)=\sum_j \chi_{ij}^{\mathrm{orb}}(\omega)E_j(\omega),$$3 is used in both channels (Salemi et al., 2019). In the diffusive parabolic-band problem with an asymmetric scalar potential, the orbital Edelstein effect is also derived from a Kubo bubble between the current vertex and the orbital magnetic moment operator, yielding a result that scales as a cube of the momentum relaxation time and does not rely on spin–orbit coupling (Ado et al., 2024).

The second route is semiclassical Boltzmann transport in the relaxation-time approximation. In the oxide two-dimensional electron gas at AlO$$M_i^{\mathrm{orb}}(\omega)=\sum_j \chi_{ij}^{\mathrm{orb}}(\omega)E_j(\omega),$$4/SrTiO$$M_i^{\mathrm{orb}}(\omega)=\sum_j \chi_{ij}^{\mathrm{orb}}(\omega)E_j(\omega),$$5, the orbital susceptibility is computed as

$$M_i^{\mathrm{orb}}(\omega)=\sum_j \chi_{ij}^{\mathrm{orb}}(\omega)E_j(\omega),$$6

with $$M_i^{\mathrm{orb}}(\omega)=\sum_j \chi_{ij}^{\mathrm{orb}}(\omega)E_j(\omega),$$7 and the orbital moment built from atomic-like orbital angular momentum expectation values in a realistic eight-band $$M_i^{\mathrm{orb}}(\omega)=\sum_j \chi_{ij}^{\mathrm{orb}}(\omega)E_j(\omega),$$8 model (Johansson et al., 2020). The (111) LaAlO$$M_i^{\mathrm{orb}}(\omega)=\sum_j \chi_{ij}^{\mathrm{orb}}(\omega)E_j(\omega),$$9/SrTiO$\delta M=\mu_B\delta(L+2S)$0 interface uses an analogous Boltzmann formula based on $\delta M=\mu_B\delta(L+2S)$1 and the shifted distribution function, evaluated at $\delta M=\mu_B\delta(L+2S)$2 and $\delta M=\mu_B\delta(L+2S)$3 (Trama et al., 2022). Chiral CNTs, Cu$\delta M=\mu_B\delta(L+2S)$4WSe$\delta M=\mu_B\delta(L+2S)$5, analytically solvable helical models for chirality-induced spin selectivity, and gated monolayer TMDs all employ related Boltzmann expressions, although the orbital moment entering the transport kernel may come from modern orbital magnetization rather than from an atomic operator (Göbel et al., 10 Apr 2025, Nakazawa et al., 18 Dec 2025, Göbel et al., 7 Feb 2025, Gautam et al., 30 Sep 2025).

A further distinction concerns how the orbital moment itself is represented. Some works use the expectation value of an orbital angular momentum operator in a restricted orbital basis, as in oxide $\delta M=\mu_B\delta(L+2S)$6 models (Johansson et al., 2020, Trama et al., 2022). Others use the gauge-invariant Bloch-band orbital magnetic moment,

$\delta M=\mu_B\delta(L+2S)$7

or its modern-theory counterpart in terms of interband velocity matrix elements (Nakazawa et al., 18 Dec 2025, Göbel et al., 10 Apr 2025). Edge-state studies emphasize that the relevant orbital degree of freedom can be entirely inter-atomic, with the orbital angular momentum per Bloch eigenstate computed from velocity matrix elements and no intra-atomic contribution in the tight-binding basis (Lee et al., 2024).

In superconductors the susceptibility is formulated against a supercurrent source rather than an electric field. For multiorbital, inversion-breaking superconductors with orbital Rashba coupling, the static susceptibility is extracted from Bogoliubov–de Gennes theory and has the band-resolved form

$\delta M=\mu_B\delta(L+2S)$8

which makes explicit the role of interband matrix elements and superconducting coherence factors (Ando et al., 2024). Closely related work on non-centrosymmetric superconductors writes the response as $\delta M=\mu_B\delta(L+2S)$9 and finds a supercurrent-induced orbital magnetization more than one order of magnitude greater than the spin contribution (Chirolli et al., 2021).

3. Symmetry, gyrotropy, and what is actually required

Broken inversion symmetry is the recurring symmetry condition, but the literature repeatedly distinguishes inversion breaking from chirality and from spin–orbit coupling. In Cu$\chi^L_{ij}$0WSe$\chi^L_{ij}$1, which has space group $\chi^L_{ij}$2 and point group $\chi^L_{ij}$3, the orbital Edelstein tensor has the symmetry-allowed form

$\chi^L_{ij}$4

so that $\chi^L_{ij}$5, $\chi^L_{ij}$6, and all off-diagonal components vanish. The same work stresses that chirality is not required for $\chi^L_{ij}$7; gyrotropy is the essential requirement (Nakazawa et al., 18 Dec 2025).

Other settings enforce different tensor structures. In the (111) LaAlO$\chi^L_{ij}$8/SrTiO$\chi^L_{ij}$9 interface with point group $\chi^S_{ij}$0, the in-plane susceptibility is antisymmetric, $\chi^S_{ij}$1, so only $\chi^S_{ij}$2 needs to be evaluated (Trama et al., 2022). Gated monolayer TMDs with $\chi^S_{ij}$3 symmetry likewise allow only antisymmetric in-plane off-diagonal components,

$\chi^S_{ij}$4

so that $\chi^S_{ij}$5 and $\chi^S_{ij}$6 (Gautam et al., 30 Sep 2025). In the analytically solvable chiral-helix model and in chiral CNTs, the response is effectively one-dimensional, and only the longitudinal component along the chiral axis is nonzero (Göbel et al., 7 Feb 2025, Göbel et al., 10 Apr 2025).

In antiferromagnets, symmetry must also be resolved at the sublattice level. For CuMnAs and Mn$\chi^S_{ij}$7Au, the orbital susceptibility decomposes naturally into staggered and uniform parts,

$\chi^S_{ij}$8

with the staggered in-plane components controlling sublattice torques and the uniform components tilting moments (Salemi et al., 2019). By contrast, edge-state work shows that even a fixed bulk Hamiltonian does not determine whether an orbital Edelstein response exists at all: zigzag-type edges exhibit an $\chi^S_{ij}$9 texture and hence an edge susceptibility, whereas straight edges can give $m_{L_z}^{uc}=\chi_z^{L_z}E_z$0. The paper explicitly characterizes this as an absence of bulk–boundary correspondence in the accumulation process (Lee et al., 2024).

Several common misconceptions are therefore incorrect. Spin–orbit coupling is not universally required; chirality is not universally required; and bulk orbital Berry curvature is not, by itself, a predictor of edge Edelstein accumulation. These conclusions are explicit in the antiferromagnetic, scalar-potential, Cu$m_{L_z}^{uc}=\chi_z^{L_z}E_z$1WSe$m_{L_z}^{uc}=\chi_z^{L_z}E_z$2, and edge-state formulations (Salemi et al., 2019, Ado et al., 2024, Nakazawa et al., 18 Dec 2025, Lee et al., 2024).

4. Representative material platforms and magnitudes

The reported susceptibility depends strongly on dimensionality, symmetry class, and choice of observable. Representative results span antiferromagnets, oxide interfaces, superconductors, chiral nanotubes, chiral helices, semiconductors, and topological materials.

Platform	Representative response statement	Paper
CuMnAs, Mn$m_{L_z}^{uc}=\chi_z^{L_z}E_z$3Au	Orbital response dominates spin; in CuMnAs with moments along $m_{L_z}^{uc}=\chi_z^{L_z}E_z$4, $m_{L_z}^{uc}=\chi_z^{L_z}E_z$5	(Salemi et al., 2019)
AlO$m_{L_z}^{uc}=\chi_z^{L_z}E_z$6/SrTiO$m_{L_z}^{uc}=\chi_z^{L_z}E_z$7 2DEG	Orbital Edelstein effect exceeds spin Edelstein effect by more than one order of magnitude	(Johansson et al., 2020)
(111) LaAlO$m_{L_z}^{uc}=\chi_z^{L_z}E_z$8/SrTiO$m_{L_z}^{uc}=\chi_z^{L_z}E_z$9	Orbital Edelstein effect an order of magnitude larger then the spin one	(Trama et al., 2022)
Non-centrosymmetric superconductors	Supercurrent-induced orbital magnetization more than one order of magnitude greater; up to approximately 25 times and approximately 60 times larger in compared cases	(Chirolli et al., 2021)
Metallic chiral CNTs	$\chi_z^{L_z}\cdot a/T$0 reaches up to about $\chi_z^{L_z}\cdot a/T$1 in the low-energy range and $\chi_z^{L_z}\cdot a/T$2 at higher energies	(Göbel et al., 10 Apr 2025)
Chiral-helix model for Te	$\chi_z^{L_z}\cdot a/T$3 spinless; $\chi_z^{L_z}\cdot a/T$4 including spin degeneracy	(Göbel et al., 7 Feb 2025)

Several platform-specific features recur. In noncentrosymmetric antiferromagnets, the dominant components are staggered off-diagonal terms, and the induced local magnetization can contain both staggered in-plane components and non-staggered out-of-plane components (Salemi et al., 2019). In oxide interfaces, the enhancement is traced to multi-orbital hybridization and unequal orbital moment magnitudes within Rashba-split band pairs, rather than simply to large spin–orbit coupling (Johansson et al., 2020, Trama et al., 2022). In bilayer Rashba models, the sign of the orbital response can be enhanced, suppressed, or reversed depending on the relation of the effective Rashba parameters of each layer, and a sign change is related to an interchange of the corresponding layer localization of the states (M. et al., 2023).

Chiral systems furnish especially transparent scaling laws. In metallic chiral CNTs near the Fermi level,

$\chi_z^{L_z}\cdot a/T$5

the susceptibility is an odd function of chirality angle, and it vanishes for zigzag and armchair tubes (Göbel et al., 10 Apr 2025). In the analytically solvable chiral-helix model,

$\chi_z^{L_z}\cdot a/T$6

so the response is linear in the chirality parameter $\chi_z^{L_z}\cdot a/T$7, linear in $\chi_z^{L_z}\cdot a/T$8, and maximal near the band center (Göbel et al., 7 Feb 2025).

Recent first-principles work extends these trends to gated monolayer TMDs, where the gate-induced broken mirror symmetry produces a Rashba-type chiral spin/orbital angular momentum texture and the predicted OEE is described as an order of magnitude larger than values reported in previously studied systems (Gautam et al., 30 Sep 2025). In optically driven monolayer jacutingaite, the orbital Edelstein susceptibility instead becomes a probe of a Floquet-engineered topological transition: it vanishes at the onset of the semimetallic regime because the orbital magnetic moment is proportional to the light-controlled mass term (Bau et al., 9 May 2025).

5. Nonlinear, reciprocal, and edge-resolved extensions

The susceptibility concept has expanded beyond linear dc bulk response. In few-layer WTe$\chi_z^{L_z}\cdot a/T$9, the reported effect is explicitly nonlinear:

$\mu_B\cdot\mathrm{m}/\mathrm{V}$0

with the second-harmonic component under an ac drive given by

$\mu_B\cdot\mathrm{m}/\mathrm{V}$1

The out-of-plane magnetization is explained by current-induced orbital magnetization via the Berry connection polarizability tensors in WTe$\mu_B\cdot\mathrm{m}/\mathrm{V}$2, and the orbital degree of freedom plays the primary role in the observed nonlinear Edelstein effect (Ye et al., 2024).

Reciprocity has also moved to the forefront. Nonlocal measurements on an orbital Edelstein system based on oxidized Cu demonstrate that the direct and inverse orbital-charge conversion processes produce identical electric voltages, confirming Onsager reciprocity. At $\mu_B\cdot\mathrm{m}/\mathrm{V}$3 with FM = Co$\mu_B\cdot\mathrm{m}/\mathrm{V}$4Fe$\mu_B\cdot\mathrm{m}/\mathrm{V}$5 and $\mu_B\cdot\mathrm{m}/\mathrm{V}$6, $\mu_B\cdot\mathrm{m}/\mathrm{V}$7 and $\mu_B\cdot\mathrm{m}/\mathrm{V}$8, while the lateral orbital decay length is $\mu_B\cdot\mathrm{m}/\mathrm{V}$9 at room temperature and is independent of Cu thickness (Gao et al., 16 Feb 2025). A plausible implication is that orbital Edelstein susceptibility is not only a local constitutive parameter but also part of a nonlocal transport reciprocity structure once orbital accumulation and its conjugate chemical potential are resolved experimentally.

Edge-local susceptibility provides another extension. For finite slabs, the nonequilibrium OAM profile is written as

$(A_0/A)m=[\chi^s+\chi^l]E\equiv \chi E$0

so that the local orbital Edelstein susceptibility can be identified as $(A_0/A)m=[\chi^s+\chi^l]E\equiv \chi E$1. The effect is controlled by the existence of an $(A_0/A)m=[\chi^s+\chi^l]E\equiv \chi E$2 texture in edge-localized states and can therefore be present for zigzag or irregular terminations and absent for straight edges of the same bulk model (Lee et al., 2024).

Superconducting and driven systems show that the “source” field need not be a dc electric field. In superconductors it is the phase gradient $(A_0/A)m=[\chi^s+\chi^l]E\equiv \chi E$3 or the supercurrent; in Floquet-engineered jacutingaite it is a dc probe response carried by a light-modified band structure; in WTe$(A_0/A)m=[\chi^s+\chi^l]E\equiv \chi E$4 it is a quadratic response to an ac current. Orbital Edelstein susceptibility is therefore best understood as a family of symmetry-allowed magnetoelectric coefficients rather than a single fixed transport number (Chirolli et al., 2021, Bau et al., 9 May 2025, Ye et al., 2024).

6. Experimental probes, open problems, and current research directions

The experimental literature and the theory papers converge on a set of favored probes. Magnetometry under applied electric field or current, Kerr or Faraday rotation, XMCD, torque magnetometry, NV-center magnetometry, lateral SQUID magnetoscopy, and nonlocal electrical detection all appear as viable approaches depending on whether the target is bulk magnetization, site-resolved orbital moments, or reciprocal charge–orbital conversion (Nakazawa et al., 18 Dec 2025, Chirolli et al., 2021, Johansson et al., 2020, Trama et al., 2022, Gao et al., 16 Feb 2025). In few-layer WTe$(A_0/A)m=[\chi^s+\chi^l]E\equiv \chi E$5, the nonlinear orbital Edelstein effect is electrically probed using an Fe$(A_0/A)m=[\chi^s+\chi^l]E\equiv \chi E$6GeTe$(A_0/A)m=[\chi^s+\chi^l]E\equiv \chi E$7 electrode with perpendicular magnetic anisotropy, and the second-harmonic response scales quadratically with the applied current (Ye et al., 2024).

Several unresolved issues are explicit in the literature. For antiferromagnets, open questions include how efficiently orbital polarization transfers torque to spins, how to quantify and experimentally detect OREE versus SREE contributions at optical and THz frequencies, and how uniform components affect switching pathways (Salemi et al., 2019). In reciprocal nonlocal transport, the experiment constrains the product of interfacial charge-to-orbital conversion and orbital-to-charge detection efficiencies, but does not isolate an absolute $(A_0/A)m=[\chi^s+\chi^l]E\equiv \chi E$8 because the detector calibration is not independently known (Gao et al., 16 Feb 2025). In WTe$(A_0/A)m=[\chi^s+\chi^l]E\equiv \chi E$9, the experiment establishes quadratic scaling, symmetry, and temperature dependence, but does not convert the second-harmonic voltage into an absolute material $$\langle \hat{L}_\mu\rangle=-\frac{1}{2}\chi_{\mu\nu}q_\nu.$$00 because interfacial orbital-to-spin and spin-to-voltage conversion factors are not known (Ye et al., 2024).

A further open problem concerns the appropriate microscopic orbital variable. Some calculations use atomic-like $$\langle \hat{L}_\mu\rangle=-\frac{1}{2}\chi_{\mu\nu}q_\nu.$$01 operators; others use gauge-invariant Bloch orbital moments; still others emphasize purely inter-site self-rotation or effective orbital moments generated by Berry connection polarizability. This suggests that “orbital Edelstein susceptibility” is not a single microscopic observable but a response class whose concrete realization depends on the effective low-energy description. A plausible implication is that cross-comparison between materials requires care not only with units but also with which orbital quantity is being transported or accumulated (Johansson et al., 2020, Nakazawa et al., 18 Dec 2025, Lee et al., 2024, Ye et al., 2024).

Taken together, the published results establish a consistent core picture. Orbital Edelstein susceptibility is the response coefficient linking inversion-breaking electronic structure to current- or field-induced orbital polarization. Its dominant tensor components are fixed by crystal or boundary symmetry; its microscopic value depends on orbital texture, interband hybridization, and scattering; it can persist without spin–orbit coupling; and in multiple concrete systems it is reported as larger than the corresponding spin Edelstein response (Salemi et al., 2019, Chirolli et al., 2021, Johansson et al., 2020, Göbel et al., 7 Feb 2025).