Numerical transfer matrix study of frustrated next-nearest-neighbor Ising models on square lattices

Ising models with frustrated next-nearest-neighbor interactions present a rich array of modulated phases. These phases, however, assemble and relax slowly, which hinders their computational study. In two dimensions, strong fluctuations further hamper determining their equilibrium phase behavior from theoretical approximations. The exact numerical transfer matrix (TM) method, which bypasses both difficulties, can serve as a benchmark method once its own numerical challenges are surmounted. Building on our recent study [Hu and Charbonneau, Phys. Rev. B 103, 094441 (2021)], in which we evaluated the two-dimensional axial next-nearest-neighbor Ising model with transfer matrices, we here extend the effective usage of the TM method to Ising models with biaxial, diagonal, and third-nearest-neighbor frustration models. The high-accuracy TM numerics help resolve various physical ambiguities about these reference models, thus providing a clearer overview of modulated phase formation in two dimensions.