Two-Dimensional J_{1}-J_{2} Clock Model: Enhanced Symmetries, Emergent Orders, and Landau-Incompatible Transitions

We present a comprehensive study on the frustrated J_{1}-J_{2} classical q-state clock model with even q>4 on a two-dimensional square lattice, revealing a rich ensemble of phases driven by competing interactions. In the unfrustrated regime (J_{1}>2J_{2}), the model reproduces the standard clock model phenomenology: a low-temperature Z_{q}-broken ferromagnet, an intermediate XY-like critical quasi-long-range-ordered phase with emergent U(1) symmetry, and a high-temperature paramagnet. For J_{1}<2J_{2}, frustration stabilizes five distinct regimes: the disordered paramagnet, a stripe-ordered phase breaking Z_{q}×Z_{2} symmetry, two Z_{2}-broken nematic phases (one with and one without quasi-long-range order), and an exotic stripe phase with emergent discrete Z_{q} spin degrees of freedom prohibited in the microscopic Hamiltonian. Remarkably, this seemingly forbidden Z_{q} order emerges via a relevant operator in the infrared long-wavelength limit, rather than from an irrelevant perturbation, highlighting a nonstandard route to emergence. Using large-scale corner transfer matrix renormalization group calculations, complemented by classical Monte Carlo simulations, we map the complete phase diagram and identify Berezinskii-Kosterlitz-Thouless, Ising, first-order, and unconventional Landau-incompatible transitions between different phases. Finally, we propose an effective field-theoretic framework that encompasses these emergent orders and their interwoven transitions.