Which Kitaev-like honeycomb model has the lowest energy?

Unlike the Kitaev honeycomb model, where each spin interacts with three nearest neighbors via XX, YY, or ZZ interactions depending on the bond orientation, the Kitaev-like models considered in this talk relax the strict dependence on bond orientation while still requiring each spin to participate in all three types of interactions. Despite this relaxation, all Kitaev-like models remain exactly solvable. By applying the Jordan--Wigner transformation, we can map any of these models onto a bilinear form of Majorana operators within each flux sector of the Hamiltonian labeled by conserved Z₂ fluxes. A natural question arises: Which Kitaev-like model has the lowest ground state energy? Using arguments based on reflection positivity, we show that the so-called Kekulé--Kitaev model, characterized by the highest degree of reflection symmetry, is energetically favorable across the entire parameter space. (Its GS energy is degenerate with that of other Kitaev-like models at the isotropic and extreme Toric Code points of the parameter space; everywhere else it is a winner.) Furthermore, reflection positivity helps identify the ground-state flux sector, where the spectrum is obtained and compared with the original Kitaev honeycomb model.