Single-particle Bloch bundles and band topology — Examples

Write $d_M=(\sin k_1,\sin k_2,M+\cos k_1+\cos k_2)$. The band energies are $\pm \mathrm{\mid }{d}_M\mathrm{\mid }$, so the gap is $2\mathrm{\mid }{d}_M\mathrm{\mid }$ and closes exactly if all three components vanish. The first two vanish only when each $k_j$ is $0$ or $\pi$ modulo $2\pi$; the third component at $(0,0),(\pi ,0),(0,\pi ),(\pi ,\pi )$ is respectively $M+2,M,M,M-2$. Thus a zero exists exactly at $M=-2,0,2$. For every other fixed $M$, compactness of the torus and continuity give ${\min }_k\mathrm{\mid }{d}_M(k)\mathrm{\mid }>0$, so the two bands have a uniform gap. The sine and cosine terms are finite Fourier sums with nearest-neighbor support, hence this is a finite-range Hermitian Bloch matrix.

Fix $M$ outside the closing set and let $N=(0,0,1)\in S^2$. The preimages of $N$ under $n_M=d_M\mathrm{/}\mathrm{\mid }{d}_M\mathrm{\mid }$ are exactly the four candidate points above for which the third component is positive. Near such a point, use the oriented coordinates $(n_1,n_2)$ at $N$. The derivative matrix is diagonal with entries $\cos k_1\mathrm{/}{d}_3$ and $\cos k_2\mathrm{/}{d}_3$, where $d_3=M+\cos k_1+\cos k_2>0$; its determinant is nonzero. Therefore $N$ is a regular value, and the local orientation signs at $(0,0),(\pi ,0),(0,\pi ),(\pi ,\pi )$, whenever present, are respectively $+1,-1,-1,+1$. By [F2], degree is the sum of the signs for those points with $d_3>0$.

If $M<-2$, no candidate has $d_3>0$ and the sum is $0$. If $-2<M<0$, only $(0,0)$ contributes, giving $+1$. If $0<M<2$, $(0,0),(\pi ,0),(0,\pi )$ contribute $+1-1-1=-1$. If $M>2$, all four contribute $+1-1-1+1=0$. Substitution into [F1] gives every table entry. For any two values in one of the four intervals, the closed segment between them remains in that interval; the joint function $\mathrm{\mid }{d}_M(k)\mathrm{\mid }$ is positive on the compact torus times that segment, so it has a positive minimum and supplies a gap-preserving deformation.

By [F1], this is a smooth rank-one orthogonal band projector with $C(P_{-})=-1\mathrm{\ne }0$. Suppose it had a global smooth periodic normalized eigenvector $u(k)$. Since its range is one-dimensional, $u(k)$ would be a global orthonormal frame of the line bundle.

Apply [F2] to that frame: it would force $C(P_{-})=0$, contradicting step 1.1's value $-1$. Thus this smooth isolated band has no global smooth periodic normalized eigenvector, refuting the universal statement.