Emergent Intermediate Phase in the $J_1$-$J_2$ XY model from Tensor Network Approaches

We investigate the finite-temperature phase diagram of the classical $J_1$-$J_2$ XY model on a square lattice using a tensor network approach designed for frustrated spin systems. This model, characterized by competing nearest-neighbor and next-to-nearest-neighbor interactions, exhibits a complex interplay between $U(1)$ and $Z_2$ symmetries. Our study reveals an emergent intermediate phase around $J_2/J_1 \sim 0.505$, which is characterized by a $Z_2$ long-range stripe order without phase coherence in the XY spins. The intermediate phase features two well-separated phase transitions: a higher-temperature Ising transition and a lower-temperature Berezinskii-Kosterlitz-Thouless transition. The relative separation between these transitions is significantly larger than previously reported, enabling a clearer investigation of their distinct thermodynamic properties. For $0.5<J_2/J_1 < 0.501$, two transitions merge into a single first-order phase transition, a phenomenon that cannot be explained solely by mapping to the Ising-XY model. As $J_2/J_1 \to \infty$, the transition evolves continuously into the BKT universality class. These findings advance the understanding of the mechanisms driving phase transitions in frustrated spin systems and suggest potential experimental realizations in platforms such as ultracold atoms, Josephson junction arrays, and optical lattices.