We present a comprehensive study on the frustrated $J_1\text{−}{J}_2$ 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 ${\mathbb{Z}}_q$-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 ${\mathbb{Z}}_q\times {\mathbb{Z}}_2$ symmetry, two ${\mathbb{Z}}_2$-broken nematic phases (one with and one without quasi-long-range order), and an exotic stripe phase with emergent discrete ${\mathbb{Z}}_q$ spin degrees of freedom prohibited in the microscopic Hamiltonian. Remarkably, this seemingly forbidden ${\mathbb{Z}}_q$ 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.

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• Critical phenomena
• Frustrated magnetism
• Phase diagrams
• Phase transitions
• Monte Carlo methods
• Tensor network methods
• XY model