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 morphology 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 these 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 (ANNNI) model with transfer matrices, we here extend the effective usage of the TM method into the Ising models with biaxial, diagonal, and third-nearest-neighbor frustrations (BNNNI, DNNI, and 3NNI models). Thanks to the high-accuracy numerics provided by the TM results, various physical ambiguities about these reference models are resolved and an overview of modulated phase formation is obtained.