Chern Numbers in Band Theory

The Chern number of a two-dimensional band insulator is the quantized Berry-curvature flux of its occupied Bloch subspace over the Brillouin-zone torus. It is an integer that remains fixed under smooth deformations preserving the occupied–empty gap, and, with the response conventions stated below, it determines the zero-temperature filled-band Hall conductivity:

$$\sigma_{xy} = -\frac{e^2}{h}C_{\mathrm{occ}}.$$

The important object is $C_{\mathrm{occ}}$, not necessarily the Chern number of each energy-ordered band. Occupied bands may cross and mix among themselves while the total occupied subspace remains smooth and separated from empty states.

Required background. Bloch’s Theorem and Brillouin Zones supply the momentum-space bundle, while Berry Curvature and Chern Numbers supply the geometric flux and its quantization.

Helpful background. Topology in Quantum Matter explains why a gap-preserving deformation cannot change the phase.

Occupied-band curvature and the Chern convention

Consider a translation-invariant, noninteracting, two-dimensional crystal. Bloch’s theorem gives

$$\lvert\psi_{n\mathbf k}\rangle = e^{i\mathbf k\cdot\mathbf r} \lvert u_{n\mathbf k}\rangle,$$

with

$$H(\mathbf k) \lvert u_{n\mathbf k}\rangle = \varepsilon_n(\mathbf k) \lvert u_{n\mathbf k}\rangle.$$

The cell-periodic states are normalized in one primitive cell. Crystal momentum lies on the oriented Brillouin torus,

$$\mathrm{BZ}\simeq T^2, \qquad d^2k=dk_x\wedge dk_y.$$

For an isolated band, define

$$\mathcal A_{n,i}(\mathbf k) = i \langle u_{n\mathbf k} \vert \partial_{k_i}u_{n\mathbf k}\rangle$$

and

$$\Omega_n(\mathbf k) = \partial_{k_x}\mathcal A_{n,y} - \partial_{k_y}\mathcal A_{n,x}.$$

The band Chern number is

$$C_n = \frac{1}{2\pi} \int_{\mathrm{BZ}} \Omega_n(\mathbf k)\,d^2k.$$

Take the electron charge to be $-e$, with $e>0$, and use the conductivity tensor convention already fixed on Integer Quantum Hall Effect. With the Berry convention above,

$$\sigma_{xy} = -\frac{e^2}{h}C_{\mathrm{occ}}.$$

The minus sign is part of the complete convention block: it follows from the electron charge, the stated $+i$ Berry connection, the $(k_x,k_y)$ orientation, and $\sigma_{xy}=J_x/E_y$. Reversing the Brillouin-zone orientation reverses $C_{\mathrm{occ}}$. Changing the sign convention for the Berry connection also reverses it and the written Chern-to-Hall bridge. A consistent physical prediction is convention independent.

The Occupied Subspace

Suppose $N_{\mathrm{occ}}$ bands are filled. A direct gap separating occupied and empty states means

$$\Delta_{\mathrm{dir}} = \min_{\mathbf k} \left[ \varepsilon_{N_{\mathrm{occ}}+1}(\mathbf k) - \varepsilon_{N_{\mathrm{occ}}}(\mathbf k) \right] > 0.$$

The gauge-invariant occupied projector is

$$P(\mathbf k) = \sum_{n=1}^{N_{\mathrm{occ}}} \lvert u_{n\mathbf k}\rangle \langle u_{n\mathbf k}\rvert.$$

An arbitrary smooth unitary rotation among occupied states,

$$\lvert u_{n\mathbf k}\rangle \longrightarrow \sum_{m=1}^{N_{\mathrm{occ}}} \lvert u_{m\mathbf k}\rangle W_{mn}(\mathbf k),$$

leaves $P(\mathbf k)$ unchanged. The occupied Chern number can therefore be written without choosing separate band gauges:

$$C_{\mathrm{occ}} = \frac{i}{2\pi} \int_{\mathrm{BZ}} \operatorname{Tr} \left[ P \left[ \partial_{k_x}P, \partial_{k_y}P \right] \right] d^2k.$$

This form remains valid when occupied bands touch one another. Only the gap between the occupied and empty subspaces must remain open.

Non-Abelian connection

Collect the occupied eigenvectors into a matrix

$$U(\mathbf k) = \begin{pmatrix} \lvert u_{1\mathbf k}\rangle & \cdots & \lvert u_{N_{\mathrm{occ}}\mathbf k}\rangle \end{pmatrix}.$$

The occupied-space Berry connection and curvature are

$$\mathcal A_i = iU^\dagger\partial_{k_i}U,$$ $$\mathcal F_{xy} = \partial_{k_x}\mathcal A_y - \partial_{k_y}\mathcal A_x - i \left[ \mathcal A_x, \mathcal A_y \right].$$

Then

$$C_{\mathrm{occ}} = \frac{1}{2\pi} \int_{\mathrm{BZ}} \operatorname{Tr} \mathcal F_{xy}\,d^2k.$$

If every occupied band is isolated from every other band, this reduces to

$$C_{\mathrm{occ}} = \sum_{n\in\mathrm{occ}}C_n.$$

If two occupied bands cross, assigning separate $C_n$ values by instantaneous energy order can become meaningless even though their sum remains well defined.

Direct gap versus insulating filling

The direct gap above is enough to define a smooth occupied projector of fixed rank. A band insulator additionally needs an indirect gap,

$$\Delta_{\mathrm{ind}} = \min_{\mathbf k} \varepsilon_{N_{\mathrm{occ}}+1}(\mathbf k) - \max_{\mathbf k} \varepsilon_{N_{\mathrm{occ}}}(\mathbf k) > 0.$$

If

$$\Delta_{\mathrm{dir}}>0, \qquad \Delta_{\mathrm{ind}}<0,$$

the lower $N_{\mathrm{occ}}$ bands define a mathematical bundle, but a physical chemical potential at that filling intersects electron and hole pockets. The system is metallic, and the filled-band quantization formula does not describe its dc Hall response without further qualifications.

Gauge Patches on the Brillouin Torus

A Bloch eigenvector can be rephased locally:

$$\lvert u_{n\mathbf k}\rangle \longrightarrow e^{i\chi_n(\mathbf k)} \lvert u_{n\mathbf k}\rangle.$$

The connection changes,

$$\mathcal A_{n,i} \longrightarrow \mathcal A_{n,i} - \partial_{k_i}\chi_n,$$

while $\Omega_n$ is unchanged.

A nonzero Chern number obstructs one globally smooth, periodic eigenvector gauge over the Brillouin torus. One may use smooth gauges on overlapping patches $U_\alpha$ and $U_\beta$, related by

$$\lvert u_\beta\rangle = e^{i\chi_{\beta\alpha}} \lvert u_\alpha\rangle.$$

The Chern number is the winding of this transition phase around patch overlaps. The apparent singularity belongs to the gauge choice, not to the energy spectrum.

For an analytic isolated occupied projector in two dimensions, a nonzero $C_{\mathrm{occ}}$ obstructs an exponentially localized, translation-covariant orthonormal Wannier basis. Conversely, vanishing total Chern number removes this complex-bundle obstruction when no additional symmetry of the Wannier basis is required. Smoothness alone, without analytic regularity, does not guarantee exponential localization. Wannier Functions explains the composite-band Fourier construction and localization criterion. Wannierization Workflows explains numerical subspace construction; failure of one optimization does not by itself establish a topological obstruction.

This distinction matters numerically:

• a gauge discontinuity can occur in perfectly smooth, gapped physics;
• an energy degeneracy across the occupied–empty gap makes the occupied projector singular and can change $C_{\mathrm{occ}}$;
• a degeneracy within the occupied subspace invalidates separate-band gauges but not the total projector.

Opposite edges of a plotted Brillouin polygon are identified. A finite-difference calculation that treats those edges as unrelated open boundaries is not computing curvature on $T^2$.

A Lattice-Regulated Two-Band Example

Measure length in lattice spacings and energy in hopping units, so that $\mathbf k$ and $m$ below are dimensionless. Consider the square-lattice two-band Hamiltonian

$$H(\mathbf k;m)=\sin k_x\,\sigma_x+\sin k_y\,\sigma_y +(m+\cos k_x+\cos k_y)\sigma_z.$$

Its energies are

$$E_\pm(\mathbf k) = \pm \lvert\mathbf d(\mathbf k)\rvert,$$

where

$$\begin{aligned} d_x&=\sin k_x,\qquad d_y=\sin k_y,\\ d_z&=m+\cos k_x+\cos k_y. \end{aligned}$$

The gap closes only when all three components vanish. This occurs at four high-symmetry momenta:

Momentum	Dirac chirality	Mass
$\Gamma=(0,0)$	$+1$	$m+2$
$X=(\pi,0)$	$-1$	$m$
$Y=(0,\pi)$	$-1$	$m$
$M=(\pi,\pi)$	$+1$	$m-2$

For the occupied lower band and the orientation convention above,

$$C_- = \frac{1}{2} \sum_{a\in\{\Gamma,X,Y,M\}} \chi_a \operatorname{sgn}M_a,$$

away from the gap-closing points. Therefore

$$C_- = \begin{cases} 0, & m<-2,\\ +1, & -2<m<0,\\ -1, & 0<m<2,\\ 0, & m>2. \end{cases}$$

At $m=-2,0,2$, the lower-band Chern number is not defined because the gap closes. Topological Phase Transitions explains this invariant transfer through Berry monopoles in extended $(k_x,k_y,m)$ space and distinguishes the clean mechanism from mobility-gap and many-body transitions.

For a two-band model, the lower-band curvature can be written

$$\Omega_-(\mathbf k) = \frac{1}{2} \hat{\mathbf d} \boldsymbol{\cdot} \left( \partial_{k_x}\hat{\mathbf d} \mathbin{\times} \partial_{k_y}\hat{\mathbf d} \right),$$

so $C_-$ measures the degree of the map

$$\hat{\mathbf d}:T^2\longrightarrow S^2.$$

To see the origin of the half contributions, linearize near one closing: $\mathbf d=(v_xq_x,v_yq_y,M)$. The occupied curvature becomes

$$\Omega_-(\mathbf q) =\frac{M v_xv_y} {2(v_x^2q_x^2+v_y^2q_y^2+M^2)^{3/2}}.$$

Integrating over the plane gives $\tfrac12\operatorname{sgn}(v_xv_yM)$. This local continuum calculation fixes the change when $M$ reverses. Its half integer is not an isolated two-dimensional lattice band’s Chern number. The lattice supplies all four cones and the large-momentum completion needed for an integer (Kohmoto, 1985; Haldane, 1988).

The minimum band separation of this particular model is

$$\Delta_{\rm band}=2\min\{|m+2|,|m|,|m-2|\}.$$

It follows by writing $|\mathbf d|^2=m^2+2+2m(\cos k_x+\cos k_y) +2\cos k_x\cos k_y$: this bilinear function reaches a minimum at a corner of $[-1,1]^2$ in the variables $(\cos k_x,\cos k_y)$.

The upper panel locates the three gap closings and the four Chern sectors. The lower panel follows the reverse-overlap Wilson phase continuously around the $k_y$ cycle. Its winding is $-C$: the $m=-1$ band winds downward, the $m=+1$ band upward, and the $m=3$ band returns without winding. The curves come from the stated Hamiltonian, with 800 points on each $k_x$ loop and 241 values of $k_y$.

Hall Conductivity from Band Geometry

For the full cell-periodic Hamiltonian acting in a fixed Hilbert-space representation, the velocity operator in a Bloch fiber is

$$v_i(\mathbf k) = \frac{1}{\hbar} \partial_{k_i}H(\mathbf k).$$

A matrix in a $k$-dependent orbital basis needs the basis connection as well. For a complete orthonormal frame, or a model with consistently projected position, the corresponding expression is

$$\hbar v_i=\partial_{k_i}H-i[\mathcal A_i^{\rm orb},H].$$

For point orbitals in the cell-index Fourier convention, $\mathcal A_i^{\rm orb}=\operatorname{diag}(\tau_{\alpha i})$. Dropping this commutator can give incorrect interband optical matrix elements even when the energy bands are correct. Nonorthogonal or truncated bases require their own overlap and projection treatment; see Tight-Binding Models and Esteve-Paredes and Palacios (2023).

For a nondegenerate band, first-order eigenvector perturbation theory gives

$$\langle u_m \vert \partial_{k_i}u_n\rangle = \frac{ \langle u_m \vert \partial_{k_i}H \vert u_n\rangle }{ \varepsilon_n-\varepsilon_m }, \qquad m\ne n.$$

Write $v^i_{nm}=\langle u_n|v_i|u_m\rangle$. Substitution into the Berry curvature yields the interband formula

$$\Omega_n(\mathbf k) = -2\hbar^2 \operatorname{Im} \sum_{m\ne n} \frac{ v^x_{nm}v^y_{mn} }{ \left( \varepsilon_n-\varepsilon_m \right)^2 }.$$

For crossings within the occupied space, sum the occupied projector curvature or the occupied-to-empty interband expression; do not divide by vanishing occupied-to-occupied energy differences. The zero-temperature Kubo formula for a clean filled-band insulator then becomes

$$\sigma_{xy} = -\frac{e^2}{\hbar} \sum_{n\in\mathrm{occ}} \int_{\mathrm{BZ}} \frac{d^2k}{(2\pi)^2} \Omega_n(\mathbf k).$$

Using

$$\int_{\mathrm{BZ}} \Omega_n\,d^2k = 2\pi C_n$$

and $h=2\pi\hbar$ gives

$$\sigma_{xy} = -\frac{e^2}{h} \sum_{n\in\mathrm{occ}}C_n = -\frac{e^2}{h}C_{\mathrm{occ}}.$$

This is the Thouless–Kohmoto–Nightingale–den Nijs relation.

Metals and finite temperature

If the chemical potential intersects a band, let $f_n(\mathbf k)=f(\varepsilon_n(\mathbf k)-\mu)$ be its Fermi occupation. The intrinsic clean-limit contribution takes the weighted form

$$\sigma_{xy}^{\mathrm{int}} = -\frac{e^2}{\hbar} \sum_n \int_{\mathrm{BZ}} \frac{d^2k}{(2\pi)^2} f_n(\mathbf k) \Omega_n(\mathbf k).$$

The occupation factor no longer covers complete bands, so the result is generally not quantized. At nonzero temperature, even an insulator receives exponentially small carrier corrections when $k_{\mathrm B}T$ is much smaller than the distance from the chemical potential to either band edge; disorder, inelastic processes, and contacts add platform-specific corrections.

The Integer Quantum Hall Effect develops the mobility-gap and localization mechanism that turns the invariant into robust plateaus in a disordered device. Kubo Formula gives the general response derivation.

Symmetry Constraints

Time reversal

For a time-reversal-symmetric occupied subspace,

$$\Theta P(\mathbf k) \Theta^{-1} = P(-\mathbf k).$$

The trace Berry curvature is odd:

$$\operatorname{Tr} \mathcal F_{xy}(-\mathbf k) = - \operatorname{Tr} \mathcal F_{xy}(\mathbf k).$$

Because the Brillouin zone is inversion symmetric as an integration domain,

$$C_{\mathrm{occ}}=0.$$

This does not make every time-reversal-invariant insulator topologically trivial. Topological Insulators develops the $\mathbb Z_2$ invariant used by quantum spin Hall and three-dimensional strong and weak phases.

Spatial symmetries

A unitary orientation-reversing spatial symmetry of the two-dimensional Brillouin zone, such as an in-plane mirror, forces $C_{\mathrm{occ}}=0$. For an antiunitary symmetry there is an additional minus sign from complex conjugation. An antiunitary reflection can therefore allow a nonzero Chern number; treating every orientation-reversing action as a unitary mirror would give the wrong conclusion.

Inversion alone is orientation preserving in two dimensions:

$$(k_x,k_y) \longrightarrow (-k_x,-k_y),$$

whose Jacobian determinant is $+1$. It therefore does not forbid a Chern insulator. With both inversion and time reversal, the trace curvature of the full occupied subspace vanishes pointwise. For an isolated nondegenerate band, its own curvature vanishes. For a spinful Kramers-degenerate pair, separate band curvatures need not be well defined; the trace statement remains valid.

Broken time reversal is necessary for nonzero equilibrium charge Hall conductivity in the ordinary two-dimensional setting, but it is not sufficient. A magnetic band insulator can still have $C_{\mathrm{occ}}=0$.

Numerical Computation

Numerical work should compute the occupied subspace, not rely on visually smooth eigenvector phases.

Velocity-matrix integration

One route evaluates

$$\Omega_n(\mathbf k) = -2\hbar^2 \operatorname{Im} \sum_{m\ne n} \frac{ v^x_{nm}v^y_{mn} }{ \left( \varepsilon_n-\varepsilon_m \right)^2 }$$

and integrates it over a dense Brillouin-zone mesh. This can resolve curvature hot spots and is useful for physical interpretation. It becomes numerically delicate near small gaps and requires a consistent treatment of degenerate occupied multiplets.

Gauge-invariant plaquette method

Let $U(\mathbf k)$ contain an orthonormal basis for all occupied states at mesh point $\mathbf k$. For a step $\delta\mathbf k_\mu$, define the overlap matrix

$$M_\mu(\mathbf k) = U^\dagger( \mathbf k+\delta\mathbf k_\mu ) U(\mathbf k)$$

and its normalized determinant link

$$\mathcal U_\mu(\mathbf k) = \frac{ \det M_\mu(\mathbf k) }{ \left| \det M_\mu(\mathbf k) \right| }.$$

The orientation of this overlap is chosen to match

$$\mathcal A_i = iU^\dagger\partial_{k_i}U.$$

For the positively oriented plaquette based at $\mathbf k$, define

$$\Phi_\square(\mathbf k) = \operatorname{Arg} \Big[ \mathcal U_x(\mathbf k) \mathcal U_y( \mathbf k+\delta\mathbf k_x ) \times \mathcal U_x( \mathbf k+\delta\mathbf k_y )^{-1} \mathcal U_y(\mathbf k)^{-1} \Big],$$

where

$$\Phi_\square\in(-\pi,\pi].$$

Then

$$C_{\mathrm{mesh}} = \frac{1}{2\pi} \sum_{\square} \Phi_\square.$$

Gauge phases and arbitrary unitary rotations of the occupied basis cancel around every plaquette. At a reciprocal boundary, include the orbital sewing transformation. Plain array wrapping is sufficient only in a periodic basis with identity sewing.

The method is reliable when neighboring occupied subspaces have nonsingular overlap and no plaquette hides more curvature flux than the principal branch can resolve. A useful admissibility check is

$$\max_\square \left| \Phi_\square \right| \ll \pi.$$

Wilson-loop cross-check

At fixed $k_y$, replace each reverse-overlap matrix by its unitary polar factor $L_j$. Since it maps the frame at $j$ to the frame at $j+1$, the closed matrix product is $\mathcal W_x=L_{N-1}\cdots L_1L_0$, including sewing on the last link. The matrix order matters for more than one occupied band, although scalar determinants commute. With the link orientation above, the determinant phase

$$\vartheta(k_y) = \operatorname{Arg} \det\mathcal W_x(k_y)$$

tracks the total hybrid-Wannier center modulo $2\pi$. After phase unwrapping,

$$C_{\mathrm{occ}} = -\frac{ \vartheta(2\pi)-\vartheta(0) }{ 2\pi }.$$

The minus sign follows from the stated orientation: the reverse overlaps give $\vartheta=\oint\operatorname{Tr}\mathcal A_x\,dk_x$ modulo $2\pi$. Stokes’ theorem on the strip swept out as $k_y$ increases assigns its positive-$k_x$ top boundary the negative orientation. Thus this loop phase winds by $-2\pi C_{\mathrm{occ}}$. A Chern band exhibits spectral flow: the Wilson-loop phase cannot return without an integer winding.

A Wilson loop’s winding can be used without treating its intermediate coefficient phases as physical electric polarization. The latter also depends on orbital positions, as explained in Berry-Phase Polarization and Charge Pumping.

An exactly integer answer can still be wrong

For a periodic determinant-link mesh with all links defined, cancellation between neighboring plaquettes makes $C_{\rm mesh}$ an integer up to floating point error. Thus integrality is a property of the discretization; it does not establish agreement with the continuum band invariant (Fukui, Hatsugai and Suzuki, 2005).

For example, set $m=-0.03$ in the two-band model and use $k_j=2\pi(j+1/2)/N-\pi$ in each direction. The exact minimum gap is $0.06$ and the lower-band Chern number is $+1$. Direct diagonalization gives:

$N$	$C_{\rm mesh}$	Sampled minimum gap
5	$-1$	$1.21886$
9	$-1$	$0.68672$
17	$+1$	$0.36841$
65	$+1$	$0.11253$

The two coarse meshes agree on the wrong integer. They miss the small-gap regions near $X$ and $Y$. Even the finest sampled gap substantially exceeds the true minimum. A band-structure line plot or one mesh cannot certify a global gap. Locate possible closings, resolve concentrated curvature, monitor overlap singular values, and compare shifted or refined meshes and an independent curvature or Wilson calculation.

Random occupied-frame rotations test gauge invariance but cannot reveal a gap missed by every sample. Likewise, a small computed principal plaquette phase does not exclude unresolved additional flux inside that plaquette. Gap resolution and gauge covariance test different potential failures.

Exercises

1. Direct and indirect gaps

Suppose

$$\min_{\mathbf k} \left( \varepsilon_2-\varepsilon_1 \right) = 0.8\ \mathrm{eV},$$

but

$$\min_{\mathbf k}\varepsilon_2 - \max_{\mathbf k}\varepsilon_1 = -0.2\ \mathrm{eV}.$$

Can the lower-band Chern number be defined? Is the system a filled-band Chern insulator at one electron per cell?

Solution

The positive direct gap means band 1 is isolated from band 2 at every $\mathbf k$, so its line bundle and Chern number can be defined. The negative indirect gap means the top of band 1 lies above the bottom of band 2 at different momenta. At one electron per cell, electron and hole pockets generally occur, so there is no global chemical potential separating the two bands. The system is not a filled-band Chern insulator and its dc Hall conductivity is not forced to equal $-C_1e^2/h$ in this page’s convention.

2. Two-band phase diagram

For the regulated model in this page, evaluate

$$C_- = \frac12 \sum_a \chi_a\operatorname{sgn}M_a$$

at $m=-3,-1,1,3$.

Solution

Using

$$\begin{array}{c|cccc} a & \Gamma & X & Y & M\\ \hline \chi_a & +1 & -1 & -1 & +1\\ M_a & m+2 & m & m & m-2 \end{array}$$

gives:

$$\begin{array}{c|c} m & C_-\\ \hline -3 & 0\\ -1 & +1\\ +1 & -1\\ +3 & 0 \end{array}$$

The changes occur at $m=-2,0,2$, where one or two Dirac masses vanish and the bulk gap closes.

3. Sum over a complete finite-band basis

In a tight-binding model with a fixed finite set of localized orbitals per cell, explain why the sum of Chern numbers over all bands is zero.

Solution

The full orbital Hilbert space over each $\mathbf k$ has a global $\mathbf k$-independent basis supplied by the localized orbitals, so the complete vector bundle is trivial. Its projector is

$$P_{\mathrm{all}}(\mathbf k)=\mathbb I,$$

and therefore

$$\partial_{k_i}P_{\mathrm{all}}=0.$$

The projector Chern formula gives

$$C_{\mathrm{all}}=0.$$

When all individual bands are isolated, additivity implies

$$\sum_n C_n=0.$$

Nonzero Chern number is redistributed among bands rather than created for the complete basis.

4. An antiunitary reflection

For the two-band model, show that complex conjugation satisfies $H(k_x,k_y)^*=H(k_x,-k_y)$. Why does this symmetry permit the nonzero Chern numbers already calculated?

Solution

$\sigma_x$ and $\sigma_z$ are real, while $\sigma_y^*=-\sigma_y$. Only $\sin k_y\,\sigma_y$ changes sign, exactly as under $k_y\mapsto-k_y$. The momentum reflection reverses orientation, and complex conjugation reverses Berry curvature a second time. The signs cancel. In general a symmetry with two-dimensional momentum action $R$ implies

$$C=s\det(R)C,$$

where $s=+1$ for a unitary action and $s=-1$ for an antiunitary action. Here $s\det R=+1$, so there is no constraint forcing $C=0$. Ordinary time reversal has $R=-I$ and $\det R=+1$, giving $C=-C$ instead.

5. What a coarse mesh can establish

The $N=5$ and $N=9$ meshes in the table give the same integer and pass random gauge-rotation checks. Explain why these successes do not establish the physical invariant. What additional information in the analytic model immediately exposes the problem?

Solution

The link construction is integer valued and gauge invariant by construction when its links exist. Both coarse meshes miss the narrow curvature near the nearly massless $X$ and $Y$ cones. The exact gap formula gives $0.06$, whereas their sampled minima are more than ten times larger. Sampling around those known minima and comparing with the four Dirac masses resolves the discrepancy: the true invariant is $+1$. Integer stability across two inadequate meshes is weaker than convergence to the actual continuum projector.

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