The frustrated XY model on the honeycomb lattice has drawn lots of attentions because of the potential emergence of chiral spin liquid (CSL) with the increasing of frustrations or competing interactions. In this work, we study the extended spin-$frac
{1}{2}$ XY model with nearest-neighbor ($J_1$), and next-nearest-neighbor ($J_2$) interactions in the presence of a three-spins chiral ($J_{chi}$) term using density matrix renormalization group methods. We obtain a quantum phase diagram with both conventionally ordered and topologically ordered phases. In particular, the long-sought Kalmeyer-Laughlin CSL is shown to emerge under a small $J_{chi}$ perturbation due to the interplay of the magnetic frustration and chiral interactions. The CSL, which is a non-magnetic phase, is identified by the scalar chiral order, the finite spin gap on a torus, and the chiral entanglement spectrum described by chiral $SU(2)_{1}$ conformal field theory.
With the published data of apparent axis ratios for 1109 ultra-diffuse galaxies (UDGs) located in 17 low-redshift (z~ 0.020 - 0.063) galaxy clusters and 84 UDGs in 2 intermediate-redshift (z~ 0.308 - 0.348) clusters, we take advantage of a Markov Cha
in Monte Carlo approach and assume a ubiquitous triaxial model to investigate the intrinsic morphologies of UDGs. In contrast to the conclusion of Burkert (2017), i.e., the underlying shapes of UDGs are purely prolate ($C=B<A$), we find that the data favor the oblate-triaxial models ($C<Blesssim A$) over the nearly prolate ones. We also find that the intrinsic morphologies of UDGs are relevant to their stellar masses/luminosities, environments, and redshifts. First, for the low-redshift UDGs in the same environment, the more-luminous ones are always thicker than the less-luminous counterparts, possibly due to the more voilent internal supernovae feedback or external tidal interactions for the progenitors of the more-luminous UDGs. The UDG thickness dependence on luminosity is distinct from that of the typical quiescent dwarf ellipticals (dEs) and dwarf spheroidals (dSphs) in the local clusters and groups, but resembles that of massive galaxies; in this sense, UDGs may not be simply treated as an extension of the dE/dSph class with similar evolutionary histories. Second, for the low-redshift UDGs within the same luminosity range, the ones with smaller cluster-centric distances are more puffed-up, probably attributed to tidal interactions. Finally, the intermediate-redshift cluster UDGs are more flattened, which plausibly suggests a `disky origin for high-redshift, initial UDGs.
