Chern Number Computation

A discrete Chern algorithm sums gauge-invariant plaquette phases on a periodic momentum mesh. It can return an integer even on a coarse mesh, but integer output alone does not prove that the continuum invariant was resolved. Reliable use combines quantization, gap checks, overlap diagnostics, and refinement.

Helpful background. Berry-curvature numerics defines link phases and signs; Chern numbers in band theory explains the physical interpretation.

Periodic lattice formula

Use a right-handed $(k_x,k_y)$ mesh and $\mathcal A=i\langle u|\nabla u\rangle$. For forward normalized links $U_x,U_y$, the positively oriented plaquette product is

$$P_{ij}=U_x(i,j)U_y(i+1,j) U_x(i,j+1)^*U_y(i,j)^*.$$

With indices wrapped periodically,

$$C_{\rm mesh}=-\frac1{2\pi}\sum_{ij}\operatorname{Arg}P_{ij}.$$

The minus sign matches the connection convention. All links cancel in the product over plaquettes, forcing the sum of principal phases to be an integer multiple of $2\pi$ up to roundoff. This algebraic quantization is why a wrong integer can look exceptionally precise.

Do not duplicate both endpoints of a periodic grid as independent points. Include links across the seam using the correct reciprocal-lattice sewing transformation. In a periodic cell basis the seam may be identity; in another orbital gauge it need not be.

Occupied subspaces

For an isolated group of $r$ bands, store an orthonormal frame $V(\boldsymbol k)$ and use

$$U_\mu(\boldsymbol k)= \frac{\det[V(\boldsymbol k)^\dagger V(\boldsymbol k+\Delta_\mu)]} {|\det[V(\boldsymbol k)^\dagger V(\boldsymbol k+\Delta_\mu)]|}.$$

This computes the total Chern number of the subspace and is invariant under independent unitary frame rotations. Internal degeneracies are allowed; a gap separating the chosen group from excluded bands is required.

Use singular values or an SVD-based determinant phase to diagnose near-singular overlaps. A direct determinant can underflow for a large subspace even when its phase is formally defined.

In particular, if $M=V^\dagger V'=L\Sigma R^\dagger$ is nonsingular, its determinant phase is $\det(LR^\dagger)$ because $\det\Sigma>0$. The singular values remain essential diagnostics; replacing $M$ by its polar factor does not recover phase information from a zero singular value. Random independent $r\times r$ unitary frame rotations should leave the final integer invariant, even when all individual eigenvectors change.

Two-band lattice benchmark

Consider

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

The gap closes at $m=-2,0,2$. With the stated orientation and connection, the lower-band Chern number is

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

The signs can be checked by the four Dirac points. Their chiralities are $(+,-,-,+)$ and masses $(m+2,m,m,m-2)$; the lower-band degree is one half the chirality-weighted sum of mass signs.

At $m=-1$ and $m=1$, compare the mesh result with $+1$ and $-1$. Apply random eigenvector phases and verify invariance. At the critical masses, do not report a lower-band invariant: the isolated-band assumption fails.

A defensible convergence claim

Record the minimum direct gap on the mesh, and search between points where a gap minimum is suspected. Refine in both momentum directions. Track the largest absolute plaquette phase and smallest overlap singular value; a phase near the principal-branch cut requires further resolution.

Agreement of $C$ on two meshes is useful but not conclusive if both miss a narrow feature. Compare local curvature or adaptively refine near small gaps. Do not round a noninteger naive curvature integral without first controlling quadrature error.

The two-band model supplies a concrete failure case. At $m=-0.01$, sample $k_x,k_y=2\pi i/N$, $i=0,\ldots,N-1$, with the periodic cell basis and the formula above. Meshes $N=5,9,17$ each give $C_{\rm mesh}=-1$, while $N=31$ gives $+1$, the continuum value in $-2<m<0$. The true minimum gap is $0.02$ at $(\pi,0)$ and $(0,\pi)$, which these odd meshes do not sample exactly. Several agreeing integers therefore fail to certify the invariant. These values describe this particular mesh origin and model; a shifted mesh can fail differently.

A useful implementation reports the unrounded phase sum together with the largest plaquette phase, the least overlap singular value, and the minimum sampled direct gap. Repeat with a shifted mesh as well as a finer one. A rigorous global-gap certificate needs a separate bound or a validated minimization; a sampled minimum is only diagnostic.

Boundary conditions and band occupation are part of the definition. A metal without a globally isolated occupied subspace does not acquire an insulating Chern number by selecting a different number of occupied bands at each momentum.

Exercise

What should happen to the computed Chern number when $k_x$ and $k_y$ are exchanged?

Solution

The coordinate orientation reverses, so the Chern number changes sign. The spectrum remains unchanged under relabeling. This makes coordinate exchange a useful sign test that spectral benchmarks cannot provide.

References

• Fukui, T., Hatsugai, Y., and Suzuki, H., “Chern numbers in discretized Brillouin zone: efficient method of computing (spin) Hall conductances,” Journal of the Physical Society of Japan 74, 1674–1677 (2005), arXiv:cond-mat/0503172.
• Qi, X.-L., Wu, Y.-S., and Zhang, S.-C., “Topological quantization of the spin Hall effect in two-dimensional paramagnetic semiconductors,” Physical Review B 74, 085308 (2006), arXiv:cond-mat/0505308.