---
title: Spin-Projected Berry Curvature
url: https://www.emergentmind.com/topics/spin-projected-berry-curvature
type: topic
---

# Spin-Projected Berry Curvature

Spin-projected Berry curvature is a spin-resolved generalization of Berry curvature in which the geometric response of Bloch states, quasiparticle states, or multicomponent wavefunctions is weighted by a chosen spin sector or spin operator. In the literature surveyed here, the construction appears in several closely related forms: as a Kubo kernel with one velocity vertex replaced by the spin-current operator \(J_i^s=\tfrac12\{\sigma_z,v_i\}\), as a band-resolved Berry curvature with spin-projection operators \(P_\uparrow=\tfrac12(1+\sigma_z)\) and \(P_\downarrow=\tfrac12(1-\sigma_z)\), and as a spin-channel decomposition of Berry curvature in block-diagonal Hamiltonians or even in real-space and nuclear-coordinate settings [1901.05651]. Across these formulations, the common purpose is to isolate how spin structure modifies geometric transport, symmetry, orbital moments, and topological indices [2201.12161].

## 1. Formal definitions and operator structures

In the intrinsic Kubo formalism for a Bloch band \(|n,\mathbf k\rangle\) with eigenvalue \(\epsilon_n(\mathbf k)\), the conventional Berry curvature is written as
\[
\Omega_{n,ij}(\mathbf k)
=
-2\hbar^2\sum_{m\neq n}
\frac{
\mathrm{Im}\bigl[
\langle n,\mathbf k|v_i|m,\mathbf k\rangle
\langle m,\mathbf k|v_j|n,\mathbf k\rangle
\bigr]
}{
\bigl(\epsilon_n(\mathbf k)-\epsilon_m(\mathbf k)\bigr)^2
}.
\]
Replacing the first velocity operator by the spin-current operator
\[
J_i^s\equiv v_i^3=\tfrac12\{\sigma_z,v_i\}
\]
gives the spin Berry curvature
\[
\Omega^s_{n,ij}(\mathbf k)
=
-2\hbar^2\sum_{m\neq n}
\frac{
\mathrm{Im}\bigl[
\langle n,\mathbf k|J_i^s|m,\mathbf k\rangle
\langle m,\mathbf k|v_j|n,\mathbf k\rangle
\bigr]
}{
\bigl(\epsilon_n(\mathbf k)-\epsilon_m(\mathbf k)\bigr)^2
},
\]
which is the formulation used for ferromagnetic \(L1_0\)-CoPt [1901.05651].

A second, explicitly spin-projected form inserts a projection operator \(P_s\) into the Kubo formula. In the band eigenbasis \(\{|\psi_n(\mathbf k)\rangle\}\),
\[
\Omega^s_n(\mathbf k)
=
-2\,\mathrm{Im}\sum_{m\neq n}
\frac{
\langle \psi_n(\mathbf k)|P_s\,v_x|\psi_m(\mathbf k)\rangle
\langle \psi_m(\mathbf k)|P_s\,v_y|\psi_n(\mathbf k)\rangle
}{
\bigl[E_n(\mathbf k)-E_m(\mathbf k)\bigr]^2
},
\]
with
\[
P_\uparrow=\tfrac12(1+\sigma_z),\qquad
P_\downarrow=\tfrac12(1-\sigma_z).
\]
The total spin-sourced Berry curvature is then
\[
\Omega^s(\mathbf k)=\sum_n f_n(\mathbf k)\,\Omega^s_n(\mathbf k),
\]
as used for the \((111)\) LaAlO\(_3\)/SrTiO\(_3\) interface and for bilayer kagome metals [2201.12161].

In block-diagonal spin systems, the same object can be written as the ordinary Berry curvature of a given spin channel. For a two-dimensional crystal with spin-\(z\) as a good quantum number,
\[
\Omega_{n\sigma}(\mathbf k)
=
i\,\epsilon_{ij}\,
\langle \partial_{k_i}u_{\mathbf k n\sigma}|\partial_{k_j}u_{\mathbf k n\sigma}\rangle,
\]
which directly enters spin-Chern constructions and spin-resolved ARPES protocols [1905.09404]. In interacting bosonic Mott insulators, analogous spin-resolved quantities are defined through
\[
A_{n,\sigma}(\mathbf k)=i\langle u_{n,\sigma}(\mathbf k)|\nabla_{\mathbf k}|u_{n,\sigma}(\mathbf k)\rangle,\qquad
\Omega_{n,\sigma}(\mathbf k)=\nabla_{\mathbf k}\times A_{n,\sigma}(\mathbf k),
\]
and these determine the many-body spin-Chern number through quasihole bands [1307.3594].

These definitions are not identical. The CoPt formulation emphasizes the spin-current operator in transport kernels, whereas the LaAlO\(_3\)/SrTiO\(_3\), kagome, ARPES, and bosonic formulations emphasize explicit spin-sector resolution. This suggests that “spin-projected Berry curvature” functions as a family of related constructions rather than a single universally normalized observable.

## 2. Relation to transport coefficients and topological invariants

In the CoPt Kubo formalism, the intrinsic anomalous Hall conductivity and spin Hall conductivity are Brillouin-zone integrals of the charge and spin Berry curvatures:
\[
\sigma_{ij}^{A}
=
-\frac{e^2}{\hbar}\int_{BZ}\frac{d^3k}{(2\pi)^3}\sum_n f_n(\mathbf k)\,\Omega_{n,ij}(\mathbf k),
\]
\[
\sigma_{ij}^{s}
=
-\frac{e^2}{\hbar}\int_{BZ}\frac{d^3k}{(2\pi)^3}\sum_n f_n(\mathbf k)\,\Omega^s_{n,ij}(\mathbf k).
\]
This construction makes the spin Berry curvature the intrinsic kernel for spin Hall response, while the ordinary Berry curvature controls the anomalous Hall response [1901.05651].

For the \((111)\) LaAlO\(_3\)/SrTiO\(_3\) two-dimensional electron system, the same logic appears in a two-dimensional form:
\[
\sigma_{xy}
=
\frac{e^2}{\hbar}
\sum_n
\int\frac{d^2k}{(2\pi)^2}\,
f_n(\mathbf k)\,
\Omega^s_n(\mathbf k).
\]
There, spin-projected Berry curvature is directly connected to the anomalous planar Hall effect, while a Berry-curvature dipole
\[
D_x
=
\int\frac{d^2k}{(2\pi)^2}\,
\partial_{k_x}\bigl[\Omega^s(\mathbf k)\bigr]
\]
controls the nonlinear Hall response [2201.12161].

In bosonic Mott insulators, the spin-resolved Berry curvatures of quasihole bands determine the many-body spin-Chern number:
\[
C_s=C_+-C_-,
\qquad
C_s=\frac{1}{2\pi}\int_{BZ}d^2k\,[\Omega_{h,\uparrow}(\mathbf k)-\Omega_{h,\downarrow}(\mathbf k)].
\]
The same work gives the hole-band curvature in terms of the on-shell spin-orbit field,
\[
\Omega_{h,s}(\mathbf k)=\frac{s}{2}\,\hat f\cdot
(\partial_{k_x}\hat f\times \partial_{k_y}\hat f),
\]
and adds an interaction-generated electric-field correction \(E^i_{h,s}(\mathbf k)\) contributing to the full Berry curvature via \(\Omega=B+v\times E\) [1307.3594].

In spin-resolved ARPES treatments of spin-Chern insulators, the partial Chern numbers
\[
C_\sigma=\frac{1}{2\pi}\int_{BZ}d^2k\,\Omega_{v\sigma}(\mathbf k)
\]
combine into
\[
C_s=\tfrac12(C_\uparrow-C_\downarrow),
\]
which provides a momentum-resolved topological interpretation for spin-projected curvature maps [1905.09404]. Across these settings, spin-projected Berry curvature is therefore both a transport kernel and a topological density.

## 3. Symmetry reduction and band-crossing physics in ferromagnetic CoPt

Ferromagnetic \(L1_0\)-CoPt provides a particularly sharp statement of the distinction between Berry curvature and spin Berry curvature. The crystal has a tetragonal \(C_{4v}\) point group about the \(c\)-axis. First-principles maps show that \(\Omega_{yx}^{00}(\mathbf k)\), the nonzero anomalous Hall component, is invariant under \(90^\circ\) rotation about \(z\), whereas \(\Omega_{yx}^{30}(\mathbf k)\), the spin Berry curvature with spin along \(z\), is invariant only under \(180^\circ\) rotations and mirror reflections. In the notation of the paper, the symmetry of the charge Berry curvature preserves \(C_{4v}\), while the symmetry of the spin Berry curvature reduces to \(C_{2v}\) [1901.05651].

The paper classifies band crossings at the Fermi surface into two types. In **Class I**, a pair of bands with the same spin character \((\uparrow\Leftrightarrow\uparrow\) or \(\downarrow\Leftrightarrow\downarrow)\) opens a spin-orbit gap, and the spin Berry curvature follows the same sign pattern as the charge Berry curvature under \(C_4\) rotation. In **Class II**, a pair of bands with opposite spin character \((\uparrow\Leftrightarrow\downarrow)\) opens a spin-orbit gap; the Bloch states are then strong \(\uparrow\)-\(\downarrow\) mixtures, off-diagonal matrix elements of \(v_i^\alpha\) appear, and peaks in \(\Omega^s_{yx}\) fail to map into one another under \(90^\circ\) rotation [1901.05651].

This mechanism is illustrated by the Rashba-plus-exchange model
\[
H(\mathbf k)=k^2+\alpha(\sigma_xk_y-\sigma_yk_x)+\beta\sigma_z,
\]
for which the lower-band curvatures are
\[
\Omega_{xy}^{00,-}(\mathbf k)=+\frac{\alpha^2\beta}{2\lambda^3(\mathbf k)},
\qquad
\Omega_{xy}^{30,-}(\mathbf k)=-\frac{\alpha^2k_x^2}{2\lambda^3(\mathbf k)},
\qquad
\Omega_{yx}^{30,-}(\mathbf k)=+\frac{\alpha^2k_y^2}{2\lambda^3(\mathbf k)},
\]
with \(\lambda(\mathbf k)=\sqrt{\alpha^2k^2+\beta^2}\). Under a \(90^\circ\) rotation \(\Lambda\),
\[
\Omega_{ij}^{00}(\Lambda\mathbf k)=\Lambda_{ir}\Lambda_{js}\Omega_{rs}^{00}(\mathbf k),
\]
whereas
\[
\Omega_{ij}^{30}(\Lambda\mathbf k)=\Lambda_{ir}\Lambda_{js}\Lambda_{3p}\Omega_{rs}^{p0}(\mathbf k),
\]
so full \(C_{4v}\) is retained for \(\Omega^{00}\) but only \(C_{2v}\) is obtained in general for \(\Omega^{30}\) [1901.05651].

The conductivity consequences are explicit. In the naive two-current model,
\[
\sigma^A=P\cdot \sigma^s,\qquad
P\equiv \frac{N_\uparrow-N_\downarrow}{N_\uparrow+N_\downarrow}\ \text{at}\ E_F.
\]
However, CoPt yields
\[
\sigma_{yx}^A\approx -3\,\mathrm{S/cm},\qquad
\sigma_{yx}^s\approx +787\,\mathrm{S/cm},
\]
implying a nearly zero ratio \(\sigma^A/\sigma^s\) despite strong spin polarization at \(E_F\). The stated interpretation is that the two-current picture fails when Class-II \(\uparrow\)-\(\downarrow\) gapped crossings dominate, because nonzero off-diagonal \(v_i^\alpha\) matrix elements mix spins in the intrinsic kernel [1901.05651].

## 4. Canonical model Hamiltonians and momentum-space textures

A widely used continuum model is the two-dimensional Rashba Hamiltonian with Zeeman splitting,
\[
H(\mathbf k)=\frac{k^2}{2m}\,1_{2\times2}
+\alpha(k_y\sigma_x-k_x\sigma_y)+\Delta\sigma_z,
\]
with helicity-band dispersions
\[
\epsilon_\pm(\mathbf k)=\frac{k^2}{2m}\pm\sqrt{\alpha^2k^2+\Delta^2}.
\]
Its Berry curvature is
\[
\Omega_\pm(\mathbf k)=
\pm\frac{\alpha^2\Delta}{2(\alpha^2k^2+\Delta^2)^{3/2}},
\]
which is axisymmetric, opposite for the two bands, finite at \(k\to0\), and decays as \(k^{-3}\) at large momentum. In the \(\Delta\to0\) limit the bands touch at \(k=0\) and the curvature formally approaches \(\pm \tfrac12\delta^2(\mathbf k)\) [1412.3638]. This model does not itself define a spin projection, but it provides the canonical Berry-curvature background against which spin-projected generalizations are often formulated.

At the \((111)\) LaAlO\(_3\)/SrTiO\(_3\) interface, the low-energy spin sector is modeled by
\[
H_{\rm spin}(\mathbf k)
=
\frac{k^2}{2m(k)}\,\mathbb1
-\alpha_R(\sigma_xk_y-\sigma_yk_x)
+\frac{\lambda}{2}(k_+^3+k_-^3)\sigma_z.
\]
Threefold-symmetry breaking is represented by
\[
\alpha_R(\sigma_xk_y-\sigma_yk_x)\longrightarrow
v_yk_y\sigma_x-v_xk_x\sigma_y,
\qquad v_x\neq v_y.
\]
The spin-projected Berry curvature then develops hot spots in the annular region between inner and outer Fermi lines. Without threefold-breaking perturbations, time reversal enforces \(\sum_n\Omega_n^s(\mathbf k)=0\) at each \(\mathbf k\); with \(v_x\neq v_y\), the hot spots no longer cancel perfectly, generating a finite dipole \(D_x\) directed along \([\bar110]\) [2201.12161].

In bilayer kagome metals, spin-projected Berry curvature is concentrated at spin-orbit-induced avoided crossings. One formulation is
\[
\Omega_n^s(\mathbf k)
=
-2\,\mathrm{Im}\sum_{m\neq n}
\frac{
\langle u_{n,\mathbf k}|P_s\,\partial_{k_x}H|u_{m,\mathbf k}\rangle
\langle u_{m,\mathbf k}|P_s\,\partial_{k_y}H|u_{n,\mathbf k}\rangle
}{
(E_{m,\mathbf k}-E_{n,\mathbf k})^2
},
\]
with an equivalent Wannier-interpolated expression involving \(\{\,\sigma_z,v_x\}/2\) and \(v_y\) [2305.15345]. In the absence of spin-orbit coupling, the kagome flat band touches a quadratic band at \(\Gamma\); spin-orbit coupling opens a direct gap
\[
\Delta_{\rm SOC}=E_{1,\Gamma}-E_{2,\Gamma}\approx 60\,\mathrm{meV},
\]
and the avoided crossing concentrates a peak of \(\Omega_n^s(\mathbf k)\) at \(\Gamma\) [2305.15345].

A closely related kagome example is TbV\(_6\)Sn\(_6\), where the V-\(d\) orbitals form gapped Dirac crossings at \(K\). There the spin-projected Berry curvature is defined through a spin operator \( \hat S_z \),
\[
\Omega_n^s(\mathbf k)
=
-2\,\mathrm{Im}\sum_{m\neq n}
\frac{
\langle u_n|\partial_{k_x}H|u_m\rangle
\langle u_m|\hat S_z\,\partial_{k_y}H|u_n\rangle
}{
[\varepsilon_m(\mathbf k)-\varepsilon_n(\mathbf k)]^2
},
\]
or equivalently through a spin-Berry connection
\[
A_{\alpha,n}^s(\mathbf k)=i\langle u_n(\mathbf k)|\hat S_z\,\partial_{k_\alpha}u_n(\mathbf k)\rangle.
\]
Near \(K\), a two-band massive-Dirac model yields
\[
\Omega_\pm^s(\mathbf q)\approx
\pm\frac{\Delta\,\hbar/2}{2(q^2v_D^2+\Delta^2)^{3/2}},
\]
with sharply concentrated curvature near the gapped Dirac point [2312.04445].

## 5. Experimental access: Hall probes, ARPES, STM, and nonlinear response

The \((111)\) LaAlO\(_3\)/SrTiO\(_3\) interface provides a direct transport probe of spin-projected Berry curvature through the anomalous planar Hall effect. In an in-plane magnetic field \(\mathbf B\) that breaks mirror symmetry \(\mathcal M_{[\bar110]}\), the Rashba bands anticross at a momentum \(\mathbf k^\star\). Once the anticrossing lies on the Fermi annulus, a net Berry-curvature flux is enclosed and a transverse Hall conductance appears even at zero Lorentz force. The measured transverse resistance \(R_{xy}(B)\equiv V_y/I_x\) is purely antisymmetric in \(B\) and switches on at a threshold field \(B^c\), which coincides with the field where the avoided crossing enters the annulus [2201.12161]. The same system exhibits a nonlinear Hall voltage \(V_{yxx}^{2\omega}\propto I_x^2\), from which the dipole is extracted through
\[
D_x=\frac{2\hbar^2}{e^3\tau}\,\chi_{yxx},
\qquad
\chi_{yxx}=\frac{V_{yxx}^{2\omega}\sigma_{xx}^3W}{|I_x^\omega|^2}.
\]
The extracted orbital-sourced dipole peaks at \(\approx70\) nm, while the spin-sourced dipole from the low-energy model is of order
\[
D_x^{\rm spin}\sim 10^{-2}\times k_F^{-1}\approx 0.01\text{--}0.05\,\mathrm{nm},
\]
two orders of magnitude smaller than the orbital-sourced dipole [2201.12161].

Circular-dichroic ARPES offers a momentum-resolved spectroscopic route. In a two-dimensional crystal with spin-\(z\) block diagonalization, the intrinsic orbital magnetic moment
\[
\ell_z(\mathbf k)
=
\frac{m}{\hbar}\,
\mathrm{Im}\,
\langle \partial_{k_x}u_{\mathbf k\alpha\sigma}|
(h_\sigma(\mathbf k)-\epsilon_{\mathbf k\alpha\sigma})
|\partial_{k_y}u_{\mathbf k\alpha\sigma}\rangle
\]
is related in the two-band limit to Berry curvature by
\[
\ell_z(\mathbf k)= -\frac{m}{\hbar}\,[\epsilon_{\mathbf k c}-\epsilon_{\mathbf k v}]\,\Omega_{v\sigma}(\mathbf k).
\]
Spin-resolved dichroism
\[
I_{\sigma,\mathrm{CD}}(\mathbf k)=I_\sigma^{(-)}(\mathbf k)-I_\sigma^{(+)}(\mathbf k)
\]
therefore yields \(\ell_{z,\sigma}(\mathbf k)\), hence \(\Omega_{v\sigma}(\mathbf k)\) up to the band-separation factor \([\epsilon_c-\epsilon_v]\). The operational extraction is
\[
\Omega_{v\sigma}(\mathbf k)\propto
-\frac{I_{\sigma,\mathrm{CD}}(\mathbf k)}
{I_{\sigma,\mathrm{tot}}(\mathbf k)\,[\epsilon_c(\mathbf k)-\epsilon_v(\mathbf k)]},
\]
with \(I_{\sigma,\mathrm{tot}}=I_\sigma^{(-)}+I_\sigma^{(+)}\) [1905.09404].

This spectroscopic logic has been applied directly to bilayer kagome metals. In the \(XV_6\)Sn\(_6\) family, the computed \(\Omega_z^{S_z}(\mathbf k)\) around the SOC gap shows a pronounced peak centered at \(\Gamma\), of order \(0.1\hbar^2/e^2\) in the paper’s units. Experimentally, circular-dichroic spin-ARPES yields \(|P_{S_z}^{\rm CD}(\Gamma)|\simeq90\%\) at \(-1\) eV and \(\sim60\%\) at \(-1.3\) eV, correlating with the computed spin-Berry-curvature maps [2305.15345].

In TbV\(_6\)Sn\(_6\), spectroscopic-imaging STM and quasiparticle-interference imaging resolve field-induced splitting of the gapped Dirac dispersion. For small fields,
\[
\Delta E(q,B)\approx g_*(q)\,\mu_B\,B,
\]
which defines an effective momentum-dependent \(g\)-factor. The paper states that \(g_*(q)=-(e/\hbar)\Omega_n^s(K+q)\), linking the extracted \(g\)-factor to the spin-Berry-curvature profile near \(K\) [2312.04445].

## 6. Extensions beyond Bloch electrons

Spin-projected Berry curvature is not restricted to weakly interacting electronic band theory. In bosonic Mott insulators with spin-orbit coupling, strong-coupling perturbation theory leads to an atomic propagator
\[
G^{-1}(\omega,\mathbf k)=g^{-1}(\omega)-h(\mathbf k),
\]
and a four-vector decomposition
\[
-G^{-1}(\omega,\mathbf k)=d_0(\omega,\mathbf k)\,1+\mathbf d(\omega,\mathbf k)\cdot\boldsymbol\sigma.
\]
The quasihole Berry curvature then follows from the on-shell spin texture \(\hat f(\mathbf k)\), and the many-body spin-Chern number is expressed as a Brillouin-zone integral over quasihole spin-resolved curvature. The same work describes experimental access via time-of-flight imaging, spin-resolved imaging, reconstruction of \(\hat f(\mathbf k)\), and wave-packet dynamics under weak external force [1307.3594].

In open molecular junctions, the analogous object appears in nuclear configuration space rather than momentum space. The antisymmetric part of the electronic friction tensor,
\[
\gamma_{\mu\nu}^{\rm A}=\tfrac12(\gamma-\gamma^T),
\]
is identified with a spin-dependent nuclear Berry curvature,
\[
\Omega_{\mu\nu}^{(s)}(\mathbf R)\equiv \gamma_{\mu\nu}^{\rm A,(s)}(\mathbf R).
\]
Because the two spin blocks \(\mathbf h^s(x,y)\) differ, \(\Omega^{(\uparrow)}\neq\Omega^{(\downarrow)}\), so nuclear wave packets experience distinct pseudo-magnetic forces and evolve into different steady-state distributions. The paper reports spin polarization
\[
P(V_{\rm sd})=
\frac{I^{(\downarrow)}-I^{(\uparrow)}}
{I^{(\downarrow)}+I^{(\uparrow)}},
\]
reaching \(\sim20\%\) at moderate biases and then decaying and oscillating in the large-voltage limit [2111.12815].

In photonic microcavity systems, one can also define a spin-projected Berry curvature by projecting onto circular polarization. For a two-band effective Hamiltonian
\[
H_{\rm eff}(\mathbf k)=d_0(\mathbf k)\sigma_0+\mathbf d(\mathbf k)\cdot\boldsymbol\sigma,
\]
with \(P_\pm=\tfrac12(\sigma_0\pm\sigma_z)\), the projected curvature is
\[
\Omega_s(\mathbf k)=\sum_{n=\pm}\langle u_n|P_s|u_n\rangle\,\Omega_n(\mathbf k),
\]
and in the model summarized in the data,
\[
\Omega_{s=+}(\mathbf k)=\frac{d_z}{|\mathbf d|}\,\Omega_+(\mathbf k).
\]
It is odd in \(k_y\), even in \(k_x\), and sharply peaked near the gapped former Dirac points [2009.07189].

A further generalization occurs in real-space two-component polariton fields. There one defines component-resolved Berry connections
\[
\mathbf A_\pm(\mathbf r)=i\,\psi_\pm^*(\mathbf r)\,\nabla\psi_\pm(\mathbf r),
\]
and spin-projected curvatures
\[
\boldsymbol\Omega_\pm(\mathbf r)=\nabla\times \mathbf A_\pm(\mathbf r).
\]
Using the pseudospin vector \(S(\mathbf r)\), the real-space Berry curvature becomes
\[
B_z(\mathbf r)
=
\frac12\,\sin\theta\,
(\partial_x\theta\,\partial_y\varphi-\partial_y\theta\,\partial_x\varphi)
=
\frac12\,S\cdot(\partial_xS\times\partial_yS),
\]
and the vortex-core velocity is
\[
v_{\rm core}(\mathbf r)=\frac{|\tilde\Omega_R|}{\sqrt{2\,B_z(\mathbf r)}},
\qquad
\tilde\Omega_R=\Omega_R+i\gamma_R.
\]
Here spin-projected curvature controls the spiral kinematics of coupled vortices in real space [2202.13210].

## 7. Conceptual scope, common themes, and recurring misconceptions

A recurring misconception is that spin-projected Berry curvature must inherit the full crystal symmetry of the band structure or of the ordinary Berry curvature. The CoPt analysis shows that this is not generally true: the form of the spin-current operator and velocity operator in the Kubo formula can reduce symmetry from \(C_{4v}\) to \(C_{2v}\) when opposite-spin band crossings are present [1901.05651].

A second misconception is that anomalous Hall and spin Hall conductivities should always be related by a single spin-polarization factor. The CoPt results explicitly contradict this expectation when class-II opposite-spin gapped crossings dominate, because spin mixing in the intrinsic kernel invalidates the naive two-current picture [1901.05651].

A third misconception is that spin-projected Berry curvature is a purely electronic momentum-space concept. The surveyed literature places analogous constructions in quasihole bands of interacting bosonic insulators, in nuclear-coordinate space through antisymmetric friction, in photonic pseudospin bands, and in real-space textures of two-component polaritons [1307.3594]. This suggests a broader organizing principle: whenever a multicomponent wavefunction admits a geometrical connection and a physically meaningful spin or pseudospin resolution, a spin-projected curvature can often be defined.

Another common theme is its concentration near avoided crossings, SOC gaps, and massive Dirac points. In LaAlO\(_3\)/SrTiO\(_3\), hot spots occur between inner and outer Fermi lines and acquire a finite dipole only when anisotropy prevents exact cancellation [2201.12161]. In bilayer kagome metals, the \(\Gamma\)-point SOC gap and deeper avoided crossings provide the dominant spectroscopic fingerprint [2305.15345]. In TbV\(_6\)Sn\(_6\), the gapped Dirac point at \(K\) hosts sharply peaked spin-Berry curvature and orbital moments as large as \(200\,\mu_B\) near the gap edge [2312.04445].

Finally, several works connect spin-projected curvature to orbital observables rather than to Hall response alone. Circular dichroism probes the orbital magnetic moment, which in two-band limits is strictly proportional to local Berry curvature [1905.09404]. In TbV\(_6\)Sn\(_6\), the useful spin-projected orbital moment
\[
m_n^{z,s}(\mathbf k)=-\frac{e}{\hbar}\,\Omega_n^s(\mathbf k)
\]
makes the connection explicit [2312.04445]. The resulting picture is that spin-projected Berry curvature is both a geometrical density and a practical diagnostic for spin-selective transport, dichroism, nonlinear response, orbital Zeeman physics, and topological characterization across electronic, bosonic, molecular, photonic, and real-space platforms.

Source: https://www.emergentmind.com/topics/spin-projected-berry-curvature