Vanishing-gap conjecture at zero and pi magnetic flux

Prove that, for the two-leg Creutz model with long-range hopping and equal hopping amplitudes t_v=t_h=t_d=1, the energy gap vanishes at the equilibrium critical phases theta_c^eq=0 and theta_c^eq=±pi modulo 2pi for arbitrary hopping range D and decay exponent nu.

Background

The paper studies a two-leg Creutz ladder in which horizontal and diagonal hopping amplitudes extend over a finite range D and decay with distance as an inverse power law characterized by the exponent nu. The single-particle dispersion is expressed through the function C_k(nu,D), and an equilibrium gap closing requires simultaneous solutions of equations involving C_k and the magnetic-flux parameter theta.

For theta_c^eq=0 and ±pi, symmetry guarantees one of the conditions for gap closing, but establishing the existence of a momentum k_c^eq satisfying the remaining condition for every D and nu is not analytically demonstrated. The authors report numerical verification for several values of these parameters and formulate the general statement as a conjecture.