This paper finds clear limits on how large a Chern number can be in a two-orbital square-lattice model when electrons are allowed only finite-range hopping. The authors extend the well-known Qi–Wu–Zhang model to include hopping up to the thirteenth neighbor. Using algebraic zero counting and “mass-weighted” sum rules, they give exact upper bounds and build explicit Hamiltonians that reach those bounds. For eleventh-, twelfth-, and thirteenth-neighbor hopping they report maximal absolute Chern numbers |C|_max = 36, 40, and 51, respectively.

The Chern number is an integer that controls the quantized Hall conductance of a two-dimensional band insulator and counts net chiral edge channels. In two-band models like the one studied here, the Chern number can be read from the pattern of Dirac points (zeros where the two bands touch before a mass term opens a gap) and from the signs of the masses that gap them. Each gapped Dirac point contributes plus or minus one half, so the total Chern number is one half times the sum of vortex charges times the sign of the mass at each Dirac point.

What the authors did was twofold. First, they counted how many Dirac zeros the hybridization term can produce when hopping is limited to a finite radius. They used algebraic tools that relate the allowed trigonometric harmonics to the number of complex zeros and then to physical Dirac points. Second, they studied how the mass term — which is itself a finite trigonometric polynomial when hopping is finite range — can assign signs to those zeros. Because all masses are values of the same finite harmonic expansion, their signs cannot be chosen independently at each zero. Combining those two ideas yields global bounds on the possible Chern numbers. The paper also supplies explicit model coefficients and supplemental proofs that show the bounds are tight for the family they consider.