Uniform separation of the first two bands away from odd multiples of the Berry-curvature parameter

Prove that there exists a constant h_0>0, independent of B 3[0,c0), such that for every 0<h<h_0 and every quasi-momentum k 3^2/^2, the first two eigenvalues of the infinite Chern band Hamiltonian H_B(k) satisfy E_2(k;B,h)>E_1(k;B,h).

Background

The paper analyzes the semiclassical splitting of the two lowest Bloch bands of the infinite Chern band model. For Berry-curvature parameter B equal to an odd multiple of \pi, the authors prove an exponentially small tunneling formula for the band splitting and show that symmetry enforces an exact degeneracy at a special quasi-momentum. The behavior for other values of B is not established analytically in the paper.

In the numerical appendix, the family H_B(k) is considered for arbitrary B, with the gap E_2(k;B,h)-E_1(k;B,h) examined numerically. The computations show a positive gap at all sampled B<\pi and an exact closing at B=\pi, motivating the conjecture that, uniformly for B in [0,\pi), the first two bands remain separated throughout the Brillouin torus for sufficiently small semiclassical parameter h.

References

Figure~\ref{app:fig-closing} shows a positive gap at every sampled $B<\pi$, closing at $\pi$. Together with Figure~\ref{app:fig-surfaces}, this provides numerical evidence for the following conjecture. There exists $h_0>0$, independent of $B\in[0,\pi)$, such that, for every $0<h<h_0$ and every $B\in[0,\pi)$,

E_2(k;B,h)>E_1(k;B,h) \qquad\text{for all }k\in\mathbb R2/\mathbb Z2.

— Semiclassical Band Splitting in the Infinite Chern Band Model  (2610.21462 - Guo, 15 Oct 2026) in Conjecture (Appendix, Numerical Results), labeled Conjecture \ref{app:conjecture}