Spin-Projected Berry Curvature

• Spin-projected Berry curvature is a spin-resolved generalization of Berry curvature that weights the geometric response of Bloch states by a selected spin sector, offering insights into spin transport and symmetry.
• It is formulated using Kubo kernels with spin-current or spin-projection operators, revealing reduced symmetry and spin mixing at band crossings in materials like ferromagnetic CoPt.
• This concept underpins experimental techniques such as spin Hall measurements, ARPES, and STM, and is crucial for understanding topological invariants and orbital observables in diverse systems.

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 (Qu et al., 2019). Across these formulations, the common purpose is to isolate how spin structure modifies geometric transport, symmetry, orbital moments, and topological indices (Lesne et al., 2022).

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 (Qu et al., 2019).

A second, explicitly spin-projected form inserts a projection operator $P_s$ into the Kubo formula. In the band eigenbasis $P_\uparrow=\tfrac12(1+\sigma_z)$0,

$P_\uparrow=\tfrac12(1+\sigma_z)$1

with

$P_\uparrow=\tfrac12(1+\sigma_z)$2

The total spin-sourced Berry curvature is then

$P_\uparrow=\tfrac12(1+\sigma_z)$3

as used for the $P_\uparrow=\tfrac12(1+\sigma_z)$4 LaAlO$P_\uparrow=\tfrac12(1+\sigma_z)$5/SrTiO$P_\uparrow=\tfrac12(1+\sigma_z)$6 interface and for bilayer kagome metals (Lesne et al., 2022).

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-$P_\uparrow=\tfrac12(1+\sigma_z)$7 as a good quantum number,

$P_\uparrow=\tfrac12(1+\sigma_z)$8

which directly enters spin-Chern constructions and spin-resolved ARPES protocols (Schüler et al., 2019). In interacting bosonic Mott insulators, analogous spin-resolved quantities are defined through

$P_\uparrow=\tfrac12(1+\sigma_z)$9

and these determine the many-body spin-Chern number through quasihole bands (Wong et al., 2013).

These definitions are not identical. The CoPt formulation emphasizes the spin-current operator in transport kernels, whereas the LaAlO$P_\downarrow=\tfrac12(1-\sigma_z)$0/SrTiO$P_\downarrow=\tfrac12(1-\sigma_z)$1, 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: $P_\downarrow=\tfrac12(1-\sigma_z)$2

$P_\downarrow=\tfrac12(1-\sigma_z)$3

This construction makes the spin Berry curvature the intrinsic kernel for spin Hall response, while the ordinary Berry curvature controls the anomalous Hall response (Qu et al., 2019).

For the $P_\downarrow=\tfrac12(1-\sigma_z)$4 LaAlO$P_\downarrow=\tfrac12(1-\sigma_z)$5/SrTiO$P_\downarrow=\tfrac12(1-\sigma_z)$6 two-dimensional electron system, the same logic appears in a two-dimensional form: $P_\downarrow=\tfrac12(1-\sigma_z)$7 There, spin-projected Berry curvature is directly connected to the anomalous planar Hall effect, while a Berry-curvature dipole

$P_\downarrow=\tfrac12(1-\sigma_z)$8

controls the nonlinear Hall response (Lesne et al., 2022).

In bosonic Mott insulators, the spin-resolved Berry curvatures of quasihole bands determine the many-body spin-Chern number: $P_\downarrow=\tfrac12(1-\sigma_z)$9 The same work gives the hole-band curvature in terms of the on-shell spin-orbit field,

$|n,\mathbf k\rangle$0

and adds an interaction-generated electric-field correction $|n,\mathbf k\rangle$1 contributing to the full Berry curvature via $|n,\mathbf k\rangle$2 (Wong et al., 2013).

In spin-resolved ARPES treatments of spin-Chern insulators, the partial Chern numbers

$|n,\mathbf k\rangle$3

combine into

$|n,\mathbf k\rangle$4

which provides a momentum-resolved topological interpretation for spin-projected curvature maps (Schüler et al., 2019). 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 $|n,\mathbf k\rangle$5-CoPt provides a particularly sharp statement of the distinction between Berry curvature and spin Berry curvature. The crystal has a tetragonal $|n,\mathbf k\rangle$6 point group about the $|n,\mathbf k\rangle$7-axis. First-principles maps show that $|n,\mathbf k\rangle$8, the nonzero anomalous Hall component, is invariant under $|n,\mathbf k\rangle$9 rotation about $\epsilon_n(\mathbf k)$0, whereas $\epsilon_n(\mathbf k)$1, the spin Berry curvature with spin along $\epsilon_n(\mathbf k)$2, is invariant only under $\epsilon_n(\mathbf k)$3 rotations and mirror reflections. In the notation of the paper, the symmetry of the charge Berry curvature preserves $\epsilon_n(\mathbf k)$4, while the symmetry of the spin Berry curvature reduces to $\epsilon_n(\mathbf k)$5 (Qu et al., 2019).

The paper classifies band crossings at the Fermi surface into two types. In Class I, a pair of bands with the same spin character $\epsilon_n(\mathbf k)$6 or $\epsilon_n(\mathbf k)$7 opens a spin-orbit gap, and the spin Berry curvature follows the same sign pattern as the charge Berry curvature under $\epsilon_n(\mathbf k)$8 rotation. In Class II, a pair of bands with opposite spin character $\epsilon_n(\mathbf k)$9 opens a spin-orbit gap; the Bloch states are then strong $$\Omega_{n,ij}(\mathbf k) = -2\hbar^2\sum_{m\neq n} \frac{ \mathrm{Im}\bigl[ \langle n,\mathbf k|v_i|m,\mathbf k\rangle \langle m,\mathbf k|v_j|n,\mathbf k\rangle \bigr] }{ \bigl(\epsilon_n(\mathbf k)-\epsilon_m(\mathbf k)\bigr)^2 }.$$0-$$\Omega_{n,ij}(\mathbf k) = -2\hbar^2\sum_{m\neq n} \frac{ \mathrm{Im}\bigl[ \langle n,\mathbf k|v_i|m,\mathbf k\rangle \langle m,\mathbf k|v_j|n,\mathbf k\rangle \bigr] }{ \bigl(\epsilon_n(\mathbf k)-\epsilon_m(\mathbf k)\bigr)^2 }.$$1 mixtures, off-diagonal matrix elements of $$\Omega_{n,ij}(\mathbf k) = -2\hbar^2\sum_{m\neq n} \frac{ \mathrm{Im}\bigl[ \langle n,\mathbf k|v_i|m,\mathbf k\rangle \langle m,\mathbf k|v_j|n,\mathbf k\rangle \bigr] }{ \bigl(\epsilon_n(\mathbf k)-\epsilon_m(\mathbf k)\bigr)^2 }.$$2 appear, and peaks in $$\Omega_{n,ij}(\mathbf k) = -2\hbar^2\sum_{m\neq n} \frac{ \mathrm{Im}\bigl[ \langle n,\mathbf k|v_i|m,\mathbf k\rangle \langle m,\mathbf k|v_j|n,\mathbf k\rangle \bigr] }{ \bigl(\epsilon_n(\mathbf k)-\epsilon_m(\mathbf k)\bigr)^2 }.$$3 fail to map into one another under $$\Omega_{n,ij}(\mathbf k) = -2\hbar^2\sum_{m\neq n} \frac{ \mathrm{Im}\bigl[ \langle n,\mathbf k|v_i|m,\mathbf k\rangle \langle m,\mathbf k|v_j|n,\mathbf k\rangle \bigr] }{ \bigl(\epsilon_n(\mathbf k)-\epsilon_m(\mathbf k)\bigr)^2 }.$$4 rotation (Qu et al., 2019).

This mechanism is illustrated by the Rashba-plus-exchange model

$$\Omega_{n,ij}(\mathbf k) = -2\hbar^2\sum_{m\neq n} \frac{ \mathrm{Im}\bigl[ \langle n,\mathbf k|v_i|m,\mathbf k\rangle \langle m,\mathbf k|v_j|n,\mathbf k\rangle \bigr] }{ \bigl(\epsilon_n(\mathbf k)-\epsilon_m(\mathbf k)\bigr)^2 }.$$5

for which the lower-band curvatures are

$$\Omega_{n,ij}(\mathbf k) = -2\hbar^2\sum_{m\neq n} \frac{ \mathrm{Im}\bigl[ \langle n,\mathbf k|v_i|m,\mathbf k\rangle \langle m,\mathbf k|v_j|n,\mathbf k\rangle \bigr] }{ \bigl(\epsilon_n(\mathbf k)-\epsilon_m(\mathbf k)\bigr)^2 }.$$6

with $$\Omega_{n,ij}(\mathbf k) = -2\hbar^2\sum_{m\neq n} \frac{ \mathrm{Im}\bigl[ \langle n,\mathbf k|v_i|m,\mathbf k\rangle \langle m,\mathbf k|v_j|n,\mathbf k\rangle \bigr] }{ \bigl(\epsilon_n(\mathbf k)-\epsilon_m(\mathbf k)\bigr)^2 }.$$7. Under a $$\Omega_{n,ij}(\mathbf k) = -2\hbar^2\sum_{m\neq n} \frac{ \mathrm{Im}\bigl[ \langle n,\mathbf k|v_i|m,\mathbf k\rangle \langle m,\mathbf k|v_j|n,\mathbf k\rangle \bigr] }{ \bigl(\epsilon_n(\mathbf k)-\epsilon_m(\mathbf k)\bigr)^2 }.$$8 rotation $$\Omega_{n,ij}(\mathbf k) = -2\hbar^2\sum_{m\neq n} \frac{ \mathrm{Im}\bigl[ \langle n,\mathbf k|v_i|m,\mathbf k\rangle \langle m,\mathbf k|v_j|n,\mathbf k\rangle \bigr] }{ \bigl(\epsilon_n(\mathbf k)-\epsilon_m(\mathbf k)\bigr)^2 }.$$9,

$J_i^s\equiv v_i^3=\tfrac12\{\sigma_z,v_i\}$0

whereas

$J_i^s\equiv v_i^3=\tfrac12\{\sigma_z,v_i\}$1

so full $J_i^s\equiv v_i^3=\tfrac12\{\sigma_z,v_i\}$2 is retained for $J_i^s\equiv v_i^3=\tfrac12\{\sigma_z,v_i\}$3 but only $J_i^s\equiv v_i^3=\tfrac12\{\sigma_z,v_i\}$4 is obtained in general for $J_i^s\equiv v_i^3=\tfrac12\{\sigma_z,v_i\}$5 (Qu et al., 2019).

The conductivity consequences are explicit. In the naive two-current model,

$J_i^s\equiv v_i^3=\tfrac12\{\sigma_z,v_i\}$6

However, CoPt yields

$J_i^s\equiv v_i^3=\tfrac12\{\sigma_z,v_i\}$7

implying a nearly zero ratio $J_i^s\equiv v_i^3=\tfrac12\{\sigma_z,v_i\}$8 despite strong spin polarization at $J_i^s\equiv v_i^3=\tfrac12\{\sigma_z,v_i\}$9. The stated interpretation is that the two-current picture fails when Class-II $$\Omega^s_{n,ij}(\mathbf k) = -2\hbar^2\sum_{m\neq n} \frac{ \mathrm{Im}\bigl[ \langle n,\mathbf k|J_i^s|m,\mathbf k\rangle \langle m,\mathbf k|v_j|n,\mathbf k\rangle \bigr] }{ \bigl(\epsilon_n(\mathbf k)-\epsilon_m(\mathbf k)\bigr)^2 },$$0-$$\Omega^s_{n,ij}(\mathbf k) = -2\hbar^2\sum_{m\neq n} \frac{ \mathrm{Im}\bigl[ \langle n,\mathbf k|J_i^s|m,\mathbf k\rangle \langle m,\mathbf k|v_j|n,\mathbf k\rangle \bigr] }{ \bigl(\epsilon_n(\mathbf k)-\epsilon_m(\mathbf k)\bigr)^2 },$$1 gapped crossings dominate, because nonzero off-diagonal $$\Omega^s_{n,ij}(\mathbf k) = -2\hbar^2\sum_{m\neq n} \frac{ \mathrm{Im}\bigl[ \langle n,\mathbf k|J_i^s|m,\mathbf k\rangle \langle m,\mathbf k|v_j|n,\mathbf k\rangle \bigr] }{ \bigl(\epsilon_n(\mathbf k)-\epsilon_m(\mathbf k)\bigr)^2 },$$2 matrix elements mix spins in the intrinsic kernel (Qu et al., 2019).

4. Canonical model Hamiltonians and momentum-space textures

A widely used continuum model is the two-dimensional Rashba Hamiltonian with Zeeman splitting,

$$\Omega^s_{n,ij}(\mathbf k) = -2\hbar^2\sum_{m\neq n} \frac{ \mathrm{Im}\bigl[ \langle n,\mathbf k|J_i^s|m,\mathbf k\rangle \langle m,\mathbf k|v_j|n,\mathbf k\rangle \bigr] }{ \bigl(\epsilon_n(\mathbf k)-\epsilon_m(\mathbf k)\bigr)^2 },$$3

with helicity-band dispersions

$$\Omega^s_{n,ij}(\mathbf k) = -2\hbar^2\sum_{m\neq n} \frac{ \mathrm{Im}\bigl[ \langle n,\mathbf k|J_i^s|m,\mathbf k\rangle \langle m,\mathbf k|v_j|n,\mathbf k\rangle \bigr] }{ \bigl(\epsilon_n(\mathbf k)-\epsilon_m(\mathbf k)\bigr)^2 },$$4

Its Berry curvature is

$$\Omega^s_{n,ij}(\mathbf k) = -2\hbar^2\sum_{m\neq n} \frac{ \mathrm{Im}\bigl[ \langle n,\mathbf k|J_i^s|m,\mathbf k\rangle \langle m,\mathbf k|v_j|n,\mathbf k\rangle \bigr] }{ \bigl(\epsilon_n(\mathbf k)-\epsilon_m(\mathbf k)\bigr)^2 },$$5

which is axisymmetric, opposite for the two bands, finite at $$\Omega^s_{n,ij}(\mathbf k) = -2\hbar^2\sum_{m\neq n} \frac{ \mathrm{Im}\bigl[ \langle n,\mathbf k|J_i^s|m,\mathbf k\rangle \langle m,\mathbf k|v_j|n,\mathbf k\rangle \bigr] }{ \bigl(\epsilon_n(\mathbf k)-\epsilon_m(\mathbf k)\bigr)^2 },$$6, and decays as $$\Omega^s_{n,ij}(\mathbf k) = -2\hbar^2\sum_{m\neq n} \frac{ \mathrm{Im}\bigl[ \langle n,\mathbf k|J_i^s|m,\mathbf k\rangle \langle m,\mathbf k|v_j|n,\mathbf k\rangle \bigr] }{ \bigl(\epsilon_n(\mathbf k)-\epsilon_m(\mathbf k)\bigr)^2 },$$7 at large momentum. In the $$\Omega^s_{n,ij}(\mathbf k) = -2\hbar^2\sum_{m\neq n} \frac{ \mathrm{Im}\bigl[ \langle n,\mathbf k|J_i^s|m,\mathbf k\rangle \langle m,\mathbf k|v_j|n,\mathbf k\rangle \bigr] }{ \bigl(\epsilon_n(\mathbf k)-\epsilon_m(\mathbf k)\bigr)^2 },$$8 limit the bands touch at $$\Omega^s_{n,ij}(\mathbf k) = -2\hbar^2\sum_{m\neq n} \frac{ \mathrm{Im}\bigl[ \langle n,\mathbf k|J_i^s|m,\mathbf k\rangle \langle m,\mathbf k|v_j|n,\mathbf k\rangle \bigr] }{ \bigl(\epsilon_n(\mathbf k)-\epsilon_m(\mathbf k)\bigr)^2 },$$9 and the curvature formally approaches $L1_0$0 (Price et al., 2014). 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 $L1_0$1 LaAlO$L1_0$2/SrTiO$L1_0$3 interface, the low-energy spin sector is modeled by

$L1_0$4

Threefold-symmetry breaking is represented by

$L1_0$5

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 $L1_0$6 at each $L1_0$7; with $L1_0$8, the hot spots no longer cancel perfectly, generating a finite dipole $L1_0$9 directed along $P_s$0 (Lesne et al., 2022).

In bilayer kagome metals, spin-projected Berry curvature is concentrated at spin-orbit-induced avoided crossings. One formulation is

$P_s$1

with an equivalent Wannier-interpolated expression involving $P_s$2 and $P_s$3 (Sante et al., 2023). In the absence of spin-orbit coupling, the kagome flat band touches a quadratic band at $P_s$4; spin-orbit coupling opens a direct gap

$P_s$5

and the avoided crossing concentrates a peak of $P_s$6 at $P_s$7 (Sante et al., 2023).

A closely related kagome example is TbV$P_s$8Sn$P_s$9, where the V-$P_\uparrow=\tfrac12(1+\sigma_z)$00 orbitals form gapped Dirac crossings at $P_\uparrow=\tfrac12(1+\sigma_z)$01. There the spin-projected Berry curvature is defined through a spin operator $P_\uparrow=\tfrac12(1+\sigma_z)$02,

$P_\uparrow=\tfrac12(1+\sigma_z)$03

or equivalently through a spin-Berry connection

$P_\uparrow=\tfrac12(1+\sigma_z)$04

Near $P_\uparrow=\tfrac12(1+\sigma_z)$05, a two-band massive-Dirac model yields

$P_\uparrow=\tfrac12(1+\sigma_z)$06

with sharply concentrated curvature near the gapped Dirac point (Li et al., 2023).

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

The $P_\uparrow=\tfrac12(1+\sigma_z)$07 LaAlO$P_\uparrow=\tfrac12(1+\sigma_z)$08/SrTiO$P_\uparrow=\tfrac12(1+\sigma_z)$09 interface provides a direct transport probe of spin-projected Berry curvature through the anomalous planar Hall effect. In an in-plane magnetic field $P_\uparrow=\tfrac12(1+\sigma_z)$10 that breaks mirror symmetry $P_\uparrow=\tfrac12(1+\sigma_z)$11, the Rashba bands anticross at a momentum $P_\uparrow=\tfrac12(1+\sigma_z)$12. 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 $P_\uparrow=\tfrac12(1+\sigma_z)$13 is purely antisymmetric in $P_\uparrow=\tfrac12(1+\sigma_z)$14 and switches on at a threshold field $P_\uparrow=\tfrac12(1+\sigma_z)$15, which coincides with the field where the avoided crossing enters the annulus (Lesne et al., 2022). The same system exhibits a nonlinear Hall voltage $P_\uparrow=\tfrac12(1+\sigma_z)$16, from which the dipole is extracted through

$P_\uparrow=\tfrac12(1+\sigma_z)$17

The extracted orbital-sourced dipole peaks at $P_\uparrow=\tfrac12(1+\sigma_z)$18 nm, while the spin-sourced dipole from the low-energy model is of order

$P_\uparrow=\tfrac12(1+\sigma_z)$19

two orders of magnitude smaller than the orbital-sourced dipole (Lesne et al., 2022).

Circular-dichroic ARPES offers a momentum-resolved spectroscopic route. In a two-dimensional crystal with spin-$P_\uparrow=\tfrac12(1+\sigma_z)$20 block diagonalization, the intrinsic orbital magnetic moment

$P_\uparrow=\tfrac12(1+\sigma_z)$21

is related in the two-band limit to Berry curvature by

$P_\uparrow=\tfrac12(1+\sigma_z)$22

Spin-resolved dichroism

$P_\uparrow=\tfrac12(1+\sigma_z)$23

therefore yields $P_\uparrow=\tfrac12(1+\sigma_z)$24, hence $P_\uparrow=\tfrac12(1+\sigma_z)$25 up to the band-separation factor $P_\uparrow=\tfrac12(1+\sigma_z)$26. The operational extraction is

$P_\uparrow=\tfrac12(1+\sigma_z)$27

with $P_\uparrow=\tfrac12(1+\sigma_z)$28 (Schüler et al., 2019).

This spectroscopic logic has been applied directly to bilayer kagome metals. In the $P_\uparrow=\tfrac12(1+\sigma_z)$29Sn$P_\uparrow=\tfrac12(1+\sigma_z)$30 family, the computed $P_\uparrow=\tfrac12(1+\sigma_z)$31 around the SOC gap shows a pronounced peak centered at $P_\uparrow=\tfrac12(1+\sigma_z)$32, of order $P_\uparrow=\tfrac12(1+\sigma_z)$33 in the paper’s units. Experimentally, circular-dichroic spin-ARPES yields $P_\uparrow=\tfrac12(1+\sigma_z)$34 at $P_\uparrow=\tfrac12(1+\sigma_z)$35 eV and $P_\uparrow=\tfrac12(1+\sigma_z)$36 at $P_\uparrow=\tfrac12(1+\sigma_z)$37 eV, correlating with the computed spin-Berry-curvature maps (Sante et al., 2023).

In TbV$P_\uparrow=\tfrac12(1+\sigma_z)$38Sn$P_\uparrow=\tfrac12(1+\sigma_z)$39, spectroscopic-imaging STM and quasiparticle-interference imaging resolve field-induced splitting of the gapped Dirac dispersion. For small fields,

$P_\uparrow=\tfrac12(1+\sigma_z)$40

which defines an effective momentum-dependent $P_\uparrow=\tfrac12(1+\sigma_z)$41-factor. The paper states that $P_\uparrow=\tfrac12(1+\sigma_z)$42, linking the extracted $P_\uparrow=\tfrac12(1+\sigma_z)$43-factor to the spin-Berry-curvature profile near $P_\uparrow=\tfrac12(1+\sigma_z)$44 (Li et al., 2023).

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

$P_\uparrow=\tfrac12(1+\sigma_z)$45

and a four-vector decomposition

$P_\uparrow=\tfrac12(1+\sigma_z)$46

The quasihole Berry curvature then follows from the on-shell spin texture $P_\uparrow=\tfrac12(1+\sigma_z)$47, 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 $P_\uparrow=\tfrac12(1+\sigma_z)$48, and wave-packet dynamics under weak external force (Wong et al., 2013).

In open molecular junctions, the analogous object appears in nuclear configuration space rather than momentum space. The antisymmetric part of the electronic friction tensor,

$P_\uparrow=\tfrac12(1+\sigma_z)$49

is identified with a spin-dependent nuclear Berry curvature,

$P_\uparrow=\tfrac12(1+\sigma_z)$50

Because the two spin blocks $P_\uparrow=\tfrac12(1+\sigma_z)$51 differ, $P_\uparrow=\tfrac12(1+\sigma_z)$52, so nuclear wave packets experience distinct pseudo-magnetic forces and evolve into different steady-state distributions. The paper reports spin polarization

$P_\uparrow=\tfrac12(1+\sigma_z)$53

reaching $P_\uparrow=\tfrac12(1+\sigma_z)$54 at moderate biases and then decaying and oscillating in the large-voltage limit (Teh et al., 2021).

In photonic microcavity systems, one can also define a spin-projected Berry curvature by projecting onto circular polarization. For a two-band effective Hamiltonian

$P_\uparrow=\tfrac12(1+\sigma_z)$55

with $P_\uparrow=\tfrac12(1+\sigma_z)$56, the projected curvature is

$P_\uparrow=\tfrac12(1+\sigma_z)$57

and in the model summarized in the data,

$P_\uparrow=\tfrac12(1+\sigma_z)$58

It is odd in $P_\uparrow=\tfrac12(1+\sigma_z)$59, even in $P_\uparrow=\tfrac12(1+\sigma_z)$60, and sharply peaked near the gapped former Dirac points (Kokhanchik et al., 2020).

A further generalization occurs in real-space two-component polariton fields. There one defines component-resolved Berry connections

$P_\uparrow=\tfrac12(1+\sigma_z)$61

and spin-projected curvatures

$P_\uparrow=\tfrac12(1+\sigma_z)$62

Using the pseudospin vector $P_\uparrow=\tfrac12(1+\sigma_z)$63, the real-space Berry curvature becomes

$P_\uparrow=\tfrac12(1+\sigma_z)$64

and the vortex-core velocity is

$P_\uparrow=\tfrac12(1+\sigma_z)$65

Here spin-projected curvature controls the spiral kinematics of coupled vortices in real space (Dominici et al., 2022).

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 $P_\uparrow=\tfrac12(1+\sigma_z)$66 to $P_\uparrow=\tfrac12(1+\sigma_z)$67 when opposite-spin band crossings are present (Qu et al., 2019).

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 (Qu et al., 2019).

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 (Wong et al., 2013). 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$P_\uparrow=\tfrac12(1+\sigma_z)$68/SrTiO$P_\uparrow=\tfrac12(1+\sigma_z)$69, hot spots occur between inner and outer Fermi lines and acquire a finite dipole only when anisotropy prevents exact cancellation (Lesne et al., 2022). In bilayer kagome metals, the $P_\uparrow=\tfrac12(1+\sigma_z)$70-point SOC gap and deeper avoided crossings provide the dominant spectroscopic fingerprint (Sante et al., 2023). In TbV$P_\uparrow=\tfrac12(1+\sigma_z)$71Sn$P_\uparrow=\tfrac12(1+\sigma_z)$72, the gapped Dirac point at $P_\uparrow=\tfrac12(1+\sigma_z)$73 hosts sharply peaked spin-Berry curvature and orbital moments as large as $P_\uparrow=\tfrac12(1+\sigma_z)$74 near the gap edge (Li et al., 2023).

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 (Schüler et al., 2019). In TbV$P_\uparrow=\tfrac12(1+\sigma_z)$75Sn$P_\uparrow=\tfrac12(1+\sigma_z)$76, the useful spin-projected orbital moment

$P_\uparrow=\tfrac12(1+\sigma_z)$77

makes the connection explicit (Li et al., 2023). 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.