Frustrated spin-systems have traditionally proven challenging to understand, owing to scarcity of controlled methods for their analyses. By contrast, under strong magnetic fields, certain aspects of spin systems admit simpler description. For instance, it is well known that the saturation transition into the field-polarized phase maps onto the BEC condensation of hardcore bosons. However, frustrations provide an additional tuning parameter allowing a scenario where this conventional outcome breaks down. In particular, by studying a representative J1-J2-J3 model on the square lattice, we establish (i) transitions that go beyond the BEC universality class, with new critical exponents, and (ii) find analytical and numerical evidence of exotic Bose metal – a compressible state with power-law correlations, which in the language of spins translates into an algebraic spin liquid [1]. Another example of exotic physics in an applied field is supplied by the supersolid – an emergent state of matter originally conjectured to exist in 4He, where both the global U(1) symmetry of the order parameter and the translation symmetry are spontaneously broken. Motivated by the very recent experimental evidence for the supersolid state in a triangular lattice magnet from the inelastic neutron scattering data on K2Co(SeO3)2, we perform projected entangled-pair state (iPEPS) simulations of the XXZ spin-1/2 model with strong easy-axis exchange anisotropy [2]. We find two phases with supersolid characteristics: the low-field “Y”-phase and the high-field (B ≥ 20T) “Ψ”-phase. While many methods have been developed to target the ground states of frustrated magnets, few methods capable of computing the excitation spectra exist. We have developed the iPEPS-bases excitation ansatz and computed the dynamical spin structure factor which agrees very well with the experimental neutron scattering data in the “Y” phase, corroborating its interpretations as a supersolid. [1] S. Sur, Y. Xu, S. Li, S.-S. Gong, and A.H. Nevidomskyy, Phys. Rev. Lett. 132, 066701 (2024). [2] Yi Xu, J. Hasik, B. Ponsioen and A. H. Nevidomskyy, Phys. Rev. B 111, L060402 (2025).