Patch Euler Number in Kagome Systems

• Patch Euler number is a local topological invariant defined on a Brillouin zone subregion where the two principal bands are isolated from the third.
• It is computed by integrating the SO(2) curvature over the patch, with gauge corrections ensuring a robust measure of net vortex charge.
• The invariant connects to quaternionic charges and the second Stiefel–Whitney class, providing insights into non-abelian braiding and photonic realizations.

The patch Euler number is a local topological invariant defined for a pair of bands in a three-band, real-symmetric Hamiltonian with time-reversal symmetry $T^2=+1$ when the corresponding global Euler number is ill-defined on the full Brillouin zone. In the kagome-lattice setting, it is attached to a subregion $R\subset BZ$ on which the principal bands $2$–$3$ are isolated from band $1$, and it measures the net $SO(2)$ vortex charge of a real two-band frame over that patch. Its central use is diagnostic: if the patch Euler number is non-zero, the enclosed principal Dirac points cannot annihilate under continuous deformations that keep $R$ free of additional degeneracies with the third band (Finck et al., 25 Jul 2025).

1. Topological setting in the three-band kagome model

In the “base II” representation, the kagome tight-binding Bloch Hamiltonian is a $3\times 3$ real-symmetric matrix,

$$H(k)=- \begin{bmatrix} E_A & t_{AB}\cos(k\cdot\delta_1) & t_{AC}\cos(k\cdot\delta_2)\ t_{AB}\cos(k\cdot\delta_1) & E_B & t_{BC}\cos(k\cdot\delta_3)\ t_{AC}\cos(k\cdot\delta_2) & t_{BC}\cos(k\cdot\delta_3) & E_C \end{bmatrix},$$

with $k=(k_x,k_y)$, nearest-neighbor vectors $R\subset BZ$0, and tunable parameters $R\subset BZ$1 and $R\subset BZ$2. Because $R\subset BZ$3 is real, time-reversal symmetry takes the form $R\subset BZ$4, and the eigenvectors can be chosen real and orthonormal at each $R\subset BZ$5 (Finck et al., 25 Jul 2025).

The bands are labeled by increasing energy,

$R\subset BZ$6

The model generically features Dirac points between bands $R\subset BZ$7–$R\subset BZ$8, termed principal Dirac points, and also between bands $R\subset BZ$9–$2$0, termed adjacent Dirac points. The motion of the adjacent Dirac points is not ancillary: it determines whether the direct sum of bands $2$1–$2$2 defines a globally meaningful rank-2 real bundle and, consequently, whether a pair of principal Dirac points can be annihilated.

The ordinary Euler number of two bands is globally defined only when those bands remain isolated from the third band over the entire Brillouin zone. In the kagome model, parameter regimes exist where bands $2$3–$2$4 touch band $2$5 at adjacent Dirac points somewhere in $2$6. In that situation the global two-band bundle over the full $2$7-torus does not exist, so the global Euler number is ill-defined. The patch Euler number is the local replacement for that missing invariant.

2. Real two-band geometry on a subregion

To define the invariant, one fixes a subregion $2$8 such that the principal bands $2$9–$3$0 are isolated from band $3$1 for all $3$2. Equivalently, there is no degeneracy between $3$3 and band $3$4 anywhere in $3$5. The direct sum of the eigenspaces of bands $3$6 and $3$7 then defines a rank-2 real subbundle $3$8 (Finck et al., 25 Jul 2025).

On this patch one chooses a smooth, real, orthonormal frame $3$9 spanning $1$0. In that frame, the real Berry connection is

$1$1

and the $1$2 Berry, or Euler, connection is its off-diagonal component,

$1$3

In differential-form notation, $1$4 is the only independent component of the skew-symmetric connection matrix.

The associated $1$5 curvature is

$1$6

or, equivalently, $1$7. Under a local $1$8 frame rotation $1$9 with $SO(2)$0 a local rotation by $SO(2)$1, the connection transforms as $SO(2)$2, while the curvature remains gauge invariant, $SO(2)$3. This local $SO(2)$4 geometry is the geometric input from which the patch Euler number is extracted.

The restriction to a patch is not merely technical. On $SO(2)$5, one can choose a smooth, real, orthonormal frame except possibly along gauge branch cuts, or Dirac strings, that connect Dirac points. The curvature remains well defined and gauge invariant on $SO(2)$6, and if $SO(2)$7 is smooth on $SO(2)$8, Stokes’s theorem converts the area integral into a boundary integral.

3. Definition and computation

For a smooth gauge on $SO(2)$9 and its boundary, the patch Euler number is

$R$0

When the gauge is smooth on $R$1, Stokes’s theorem gives the equivalent boundary formula

$R$2

If unavoidable gauge discontinuities occur along $R$3, for example because a Dirac string crosses the boundary, the gauge-covariant expression is

$R$4

In the formulation used in the paper, this subtraction produces a gauge-invariant integer or half-integer result. When a globally smooth $R$5 exists on $R$6, the boundary term vanishes and the simple area integral is sufficient (Finck et al., 25 Jul 2025).

A practical numerical workflow is explicitly given:

1. Diagonalize $R$7 on a mesh in $R$8 to obtain real, normalized states for the principal bands.
2. Enforce a continuous gauge on $R$9 as much as possible and record Dirac strings if they are unavoidable.
3. Compute $3\times 3$0 numerically, for example by finite differences.
4. Form $3\times 3$1 and $3\times 3$2.
5. Integrate $3\times 3$3 over $3\times 3$4, and if the gauge is not smooth along $3\times 3$5, subtract the boundary contribution.

The sign of $3\times 3$6 is gauge dependent, but its vanishing or non-vanishing in a fixed setup is the physically relevant datum for annihilation. The formalism also admits half-integer values in certain conventions when adjacent Dirac strings pierce the patch and boundary corrections are essential.

4. Dirac points, vortex charges, and annihilation obstruction

Locally, the $3\times 3$7 frame can be parameterized by an angle $3\times 3$8, so that $3\times 3$9 and $$H(k)=- \begin{bmatrix} E_A & t_{AB}\cos(k\cdot\delta_1) & t_{AC}\cos(k\cdot\delta_2)\ t_{AB}\cos(k\cdot\delta_1) & E_B & t_{BC}\cos(k\cdot\delta_3)\ t_{AC}\cos(k\cdot\delta_2) & t_{BC}\cos(k\cdot\delta_3) & E_C \end{bmatrix},$$0. Around an isolated principal Dirac point, $$H(k)=- \begin{bmatrix} E_A & t_{AB}\cos(k\cdot\delta_1) & t_{AC}\cos(k\cdot\delta_2)\ t_{AB}\cos(k\cdot\delta_1) & E_B & t_{BC}\cos(k\cdot\delta_3)\ t_{AC}\cos(k\cdot\delta_2) & t_{BC}\cos(k\cdot\delta_3) & E_C \end{bmatrix},$$1 winds by $$H(k)=- \begin{bmatrix} E_A & t_{AB}\cos(k\cdot\delta_1) & t_{AC}\cos(k\cdot\delta_2)\ t_{AB}\cos(k\cdot\delta_1) & E_B & t_{BC}\cos(k\cdot\delta_3)\ t_{AC}\cos(k\cdot\delta_2) & t_{BC}\cos(k\cdot\delta_3) & E_C \end{bmatrix},$$2 along a small loop $$H(k)=- \begin{bmatrix} E_A & t_{AB}\cos(k\cdot\delta_1) & t_{AC}\cos(k\cdot\delta_2)\ t_{AB}\cos(k\cdot\delta_1) & E_B & t_{BC}\cos(k\cdot\delta_3)\ t_{AC}\cos(k\cdot\delta_2) & t_{BC}\cos(k\cdot\delta_3) & E_C \end{bmatrix},$$3, yielding

$$H(k)=- \begin{bmatrix} E_A & t_{AB}\cos(k\cdot\delta_1) & t_{AC}\cos(k\cdot\delta_2)\ t_{AB}\cos(k\cdot\delta_1) & E_B & t_{BC}\cos(k\cdot\delta_3)\ t_{AC}\cos(k\cdot\delta_2) & t_{BC}\cos(k\cdot\delta_3) & E_C \end{bmatrix},$$4

Each principal Dirac point therefore contributes an integer vortex charge $$H(k)=- \begin{bmatrix} E_A & t_{AB}\cos(k\cdot\delta_1) & t_{AC}\cos(k\cdot\delta_2)\ t_{AB}\cos(k\cdot\delta_1) & E_B & t_{BC}\cos(k\cdot\delta_3)\ t_{AC}\cos(k\cdot\delta_2) & t_{BC}\cos(k\cdot\delta_3) & E_C \end{bmatrix},$$5. The sign convention is explicit: counterclockwise increase of $$H(k)=- \begin{bmatrix} E_A & t_{AB}\cos(k\cdot\delta_1) & t_{AC}\cos(k\cdot\delta_2)\ t_{AB}\cos(k\cdot\delta_1) & E_B & t_{BC}\cos(k\cdot\delta_3)\ t_{AC}\cos(k\cdot\delta_2) & t_{BC}\cos(k\cdot\delta_3) & E_C \end{bmatrix},$$6 gives $$H(k)=- \begin{bmatrix} E_A & t_{AB}\cos(k\cdot\delta_1) & t_{AC}\cos(k\cdot\delta_2)\ t_{AB}\cos(k\cdot\delta_1) & E_B & t_{BC}\cos(k\cdot\delta_3)\ t_{AC}\cos(k\cdot\delta_2) & t_{BC}\cos(k\cdot\delta_3) & E_C \end{bmatrix},$$7, while clockwise gives $$H(k)=- \begin{bmatrix} E_A & t_{AB}\cos(k\cdot\delta_1) & t_{AC}\cos(k\cdot\delta_2)\ t_{AB}\cos(k\cdot\delta_1) & E_B & t_{BC}\cos(k\cdot\delta_3)\ t_{AC}\cos(k\cdot\delta_2) & t_{BC}\cos(k\cdot\delta_3) & E_C \end{bmatrix},$$8 (Finck et al., 25 Jul 2025).

When $$H(k)=- \begin{bmatrix} E_A & t_{AB}\cos(k\cdot\delta_1) & t_{AC}\cos(k\cdot\delta_2)\ t_{AB}\cos(k\cdot\delta_1) & E_B & t_{BC}\cos(k\cdot\delta_3)\ t_{AC}\cos(k\cdot\delta_2) & t_{BC}\cos(k\cdot\delta_3) & E_C \end{bmatrix},$$9 encloses only principal Dirac points and remains free of third-band degeneracies, the patch Euler number reduces to the charge sum

$k=(k_x,k_y)$0

Adjacent Dirac strings that cross $k=(k_x,k_y)$1 or $k=(k_x,k_y)$2 can reverse one local contribution by flipping the gauge; the boundary correction in the definition of $k=(k_x,k_y)$3 precisely accounts for this. In that sense the invariant measures the net $k=(k_x,k_y)$4 vortex charge of the two-band frame on the chosen patch.

The obstruction criterion is direct. If

$k=(k_x,k_y)$5

then the enclosed principal Dirac points cannot be pairwise annihilated by any continuous parameter deformation that keeps $k=(k_x,k_y)$6 free of additional degeneracies with the third band. If two principal Dirac points were to annihilate and open a gap between bands $k=(k_x,k_y)$7 and $k=(k_x,k_y)$8 over $k=(k_x,k_y)$9, the rank-2 bundle over $R\subset BZ$00 would have to deform into a trivial direct sum of two rank-1 trivial bundles, a triviality guaranteed by $R\subset BZ$01. A non-zero patch Euler number signals a twisting that cannot be unwound without crossing a degeneracy with band $R\subset BZ$02.

5. Quaternionic charges, non-abelian braiding, and parity

The paper gives a second description in homotopy-theoretic terms. The space of real orthonormal frames of the three-band system, modulo sign flips of individual eigenvectors, is homotopy equivalent to $R\subset BZ$03, where

$R\subset BZ$04

Loops in $R\subset BZ$05-space that encircle Dirac points are therefore assigned quaternionic charges. The identifications used are (Finck et al., 25 Jul 2025):

• Principal Dirac point: $R\subset BZ$06
• Adjacent Dirac point: $R\subset BZ$07
• One principal and one adjacent in the loop: $R\subset BZ$08
• Two same-type non-annihilable points in the loop: $R\subset BZ$09
• Trivial loop: $R\subset BZ$10

In this language, non-abelian braiding in $R\subset BZ$11-space acts by conjugation,

$R\subset BZ$12

where $R\subset BZ$13 is the charge of the braid path around the other Dirac point. Braiding a principal Dirac point around an adjacent one can therefore change the sign or type of the principal charge. This explains why a configuration that was previously non-annihilable may become annihilable after adjacent strings cross the patch.

The annihilation criterion becomes an ordered-product condition: for a collection of Dirac points inside a loop $R\subset BZ$14, the ordered product of their charges along $R\subset BZ$15 must equal $R\subset BZ$16. If the product is $R\subset BZ$17, corresponding to two same-type, same-orientation points, annihilation is obstructed. The gauge-sensitive flips of $R\subset BZ$18 induced by adjacent Dirac strings are thus the local, $R\subset BZ$19-frame manifestation of a non-abelian quaternionic algebra.

The same section relates the Euler class to the second Stiefel–Whitney class. For an oriented rank-2 real bundle,

$R\subset BZ$20

Over a patch, odd $R\subset BZ$21 gives a non-zero $R\subset BZ$22, signaling a parity obstruction consistent with the $R\subset BZ$23 product in the quaternionic picture. Even $R\subset BZ$24, by contrast, has $R\subset BZ$25, which allows trivialization and hence annihilation if no other obstruction is present.

6. Kagome tuning protocols and photonic realization

The kagome tight-binding model admits concrete parameter protocols in which the patch Euler number changes as adjacent Dirac points are created and their strings reconfigure. In the simplest nearest-neighbor limit, equal hoppings $R\subset BZ$26 and zero on-site energies give a real-symmetric, $R\subset BZ$27-invariant Bloch Hamiltonian $R\subset BZ$28. Deforming $R\subset BZ$29 and $R\subset BZ$30 then moves both principal and adjacent Dirac points (Finck et al., 25 Jul 2025).

Two protocols are described explicitly:

Protocol	Tuning	Outcome
Six-panel sequence (Fig. 6)	Start at $R\subset BZ$31, $R\subset BZ$32, $R\subset BZ$33, $R\subset BZ$34, $R\subset BZ$35, $R\subset BZ$36; vary $R\subset BZ$37, then $R\subset BZ$38	Principal Dirac points first merge and bounce with $R\subset BZ$39; after adjacent Dirac points appear and an adjacent Dirac string crosses $R\subset BZ$40, $R\subset BZ$41 and annihilation becomes allowed
Alternative photonic-friendly protocol	Tune only $R\subset BZ$42 from $R\subset BZ$43 to $R\subset BZ$44	Principal Dirac points merge and initially bounce, then annihilate at the $R\subset BZ$45 edge thanks to periodicity; $R\subset BZ$46 changes as adjacent strings reconfigure

The figure sequence gives the clearest explicit demonstration. In Fig. 6(a$R\subset BZ$47c), principal Dirac points merge but bounce and no gap opens; the chosen rectangular patch $R\subset BZ$48 has $R\subset BZ$49. In Fig. 6(d$R\subset BZ$50f), after tuning $R\subset BZ$51, adjacent Dirac points appear; an adjacent Dirac string crosses $R\subset BZ$52, one principal charge is reversed, $R\subset BZ$53, and the principal Dirac points annihilate.

The paper also argues that the same deformation can be implemented in realistic photonic systems. In photonic kagome lattices, specifically microcavity pillar arrays or EIT-induced lattices, $R\subset BZ$54 can be tuned in situ by optical pumping and effective hoppings can be controlled through geometry. Beyond tight-binding numerics, the authors solve a $R\subset BZ$55D stationary Schrödinger equation with a structured potential $R\subset BZ$56. The resulting dispersions show a double Dirac point at $R\subset BZ$57 in the unperturbed case and separated Dirac points when $R\subset BZ$58, reproducing the annihilation sequence. In that setting, selecting a momentum-space patch $R\subset BZ$59 and computing $R\subset BZ$60 from the photonic bands provides a direct topological readout of whether annihilation is obstructed or allowed.

7. Terminological ambiguity and non-equivalent usages

The expression “patch Euler number” is specific in the kagome-band-topology setting, but the phrase is not standard in the vortex-patch literature on the $R\subset BZ$61D incompressible Euler equations. One source states this explicitly and notes that the relevant object there is the Euler equation for vortex patches rather than the topological Euler characteristic or Euler number (Li, 2017).

This ambiguity matters because several fluid-dynamical papers involve both “patch” and “Euler” while referring to entirely different quantities. In one interpretive usage, the phrase is attached to the patch problem for $R\subset BZ$62D Euler as a quantitative measure of time analyticity, namely the radius $R\subset BZ$63 in bounds of the form $R\subset BZ$64 for derivatives of the Lagrangian flow (Burgués et al., 2019). Other works on Euler patches study winding numbers for particle trajectories in disk-like vortex patches (Choi et al., 2020), or global regularity of patch solutions for Loglog–Euler type active scalar equations (Tan et al., 21 Oct 2025). These are mathematically separate from the momentum-space invariant $R\subset BZ$65.

In the topological usage treated here, the patch Euler number is therefore not a fluid-dynamical observable attached to a moving characteristic-function domain. It is a local Euler invariant of a real rank-2 band bundle on a chosen region of the Brillouin zone, designed precisely for situations in which the full-zone Euler number cannot be defined because the principal two-band sector is not globally isolated from the third band.