Chern insulator phase diagram

Physical phenomenon

Why is a topological integer stable until two energy bands touch?

An isolated two-dimensional Bloch band can carry a nonzero Chern number: the integral of Berry curvature over the entire Brillouin-zone torus. Local curvature need not be quantized and can change strongly as the Hamiltonian parameters vary. Yet the integrated integer remains fixed as long as the gap stays open. The Qi–Wu–Zhang square-lattice two-band Hamiltonian gives a clear example of phases C=0,+1,−1,0 separated by three critical masses.

Make a prediction. Start with M=−1. Predict the lower-band Chern number and then change M through 0 to +1. Which momentum-space points become gapless at the change?

Prerequisites: Bloch vectors, Pauli matrices, Berry curvature, the Brillouin zone and numerical integration.

Physics readout

Numerical and exact predictions

Momentum-space curvature

Berry curvature Ω−(kx,ky)

The signed curvature density has peaks concentrated near gapped Dirac points. Its integral—not the sign at a single point—determines the Chern number.

Blue and orange depict opposite signs of the lower-band curvature under A=i〈u|∂u〉. The plotted range kx,ky∈[−π,π]. At exact band touchings the curvature is singular and the Chern number is undefined.

Numerical global invariant

Integrated Chern number versus mesh resolution

A midpoint grid approximates the continuous Brillouin-zone integral. Near a phase boundary much finer grids are required to resolve sharply concentrated curvature.

C(N)=ΣΩ(kij)(2π/N)²/(2π). The analytic gapped-phase integer is provided independently using the four Dirac masses; numerical convergence is a required reliability check.

Reconstruct the result

The topological index is a sphere-covering integral

Consider h(k)=sin kx σx+sin ky σy+[M+coskx+cosky]σz. Its Bloch vector d=(sin kx,sin ky,M+coskx+cosky) sets energies E±=±|d|. For the lower band and Berry convention A=i〈u|∇u〉,

Ω₋(k)=½ d·(∂kx d×∂ky d)/|d|³, C₋=(1/2π)∫BZ Ω₋ d²k.

Explicit derivatives yield ∂kx d=(coskx,0,−sinkx) and ∂ky d=(0,cosky,−sinky). Their cross product and the dot product with d give a direct curvature formula in terms of elementary sines and cosines. The integration is over the periodic two-dimensional Brillouin torus.

Gaps can close only when both sin kx and sin ky vanish and the remaining mass term is zero: M=−2 at Γ, M=0 at two inequivalent X points, and M=2 at the corner. The oriented contributions of these Dirac masses give

C₋=½[sgn(M+2)−2sgn(M)+sgn(M−2)], M∉{−2,0,2}.

The predicted values are 0 for M<−2, +1 for −2<M<0, −1 for 0<M<2, and 0 for M>2. A finite-grid numerical integral should approach the corresponding integer as the mesh is refined sufficiently away from gap closings.

An exactly soluble reference limit

For |M|≫2 the d vector points predominantly along the z direction and the Chern number is zero. At M=−1 it is +1; at M=+1 it is −1. At M=0 the two occupied and empty bands touch at (π,0) and (0,π), and the band Chern number is undefined. A finite discretized integral at such a point should not be used to assign topology.

Assumptions and numerical reliability

Ideal clean noninteracting two-band lattice Hamiltonian with lattice spacing one. The continuous Chern number is mathematically defined for an isolated band only; the midpoint-rule calculation is an approximation with a mesh-dependent error. At critical masses the UI intentionally suppresses the invariant and identifies the gap closure.

Independent investigations

Three experiments to reason through

Map a topological phase. Choose M=−1, vary the mesh and compare the integrated C with the analytic +1 prediction. Inspect the sign structure of curvature throughout the Brillouin zone.
Cross two gaps. Change M from −1 to +1 through zero. Explain why the Chern number jumps by −2, accounting for the two gapless points at the transition.
Challenge numerical convergence. Set M=.2, compare 16 and 96 points per axis. Explain why a small minimum gap produces narrow curvature peaks and a slower convergence of the integral.

Self-check: For the specified lower-band convention, what is the Chern number when 0 +1 −1 0

Extension: derive the signed Dirac-cone contributions at Γ, X and the zone corner, then connect the net chirality of open-boundary modes to C via bulk-boundary correspondence.

Physical scope

What the experiment means

A topological phase is characterized by a global band property with quantized stability, even as local Berry curvature changes smoothly. Here the critical gaps and analytic Chern number independently benchmark the finite numerical integration. This is a starting point for quantum Hall response and topological pumping.

Compare with the manuscript: Berry curvature, QWZ phase diagram · Berry connection foundations.