Emergent gauge flux and spin ordering in magnetized triangular spin liquids: applications to Hofstadter-Hubbard model

Motivated by the recent progress in the moiré superlattice systems and spin-1/2 triangular lattice antiferromagnets, we revisit the triangular-lattice spin liquids and study their magnetic responses. While the magnetic responses on the ordered phases can be mundane, the orbital magnetic flux and the Zeeman coupling have synergetic effects on the internal gauge flux generations in the relevant spin liquid phases. The former was known to induce an internal U(1) gauge flux indirectly through the charge fluctuations and ring exchange, and thus could lead to the formation of a chiral spin liquid. The latter could spontaneously generate a uniform field-dependent internal gauge flux, driving a conically-ordered state. The competition and interplay between these two field effects are discussed through a generic spin-1/2 $J_1$-$J_2$-$J_χ$ model and with the experimental consequences. Our results could find applications in the moiré superlattice systems with the Hofstadter-Hubbard model as well as the triangular lattice antiferromagnets.