Gapless and gapped spin liquids in frustrated spin-1/2 models

We present our recent numerical calculations on the Heisenberg model on various two-dimensional lattices, showing that gapped and gapless spin liquids may be stabilized in highly-frustrated regimes. The magnetically disordered phases can be described by considering Abrikosov fermions coupled to gauge fields. This approach gives a flexible and transparent representation of a myriad of states, ranging from valence-bond solids to spin liquids with different properties, including chiral order. For the Kagome lattice, we find that a gapless U(1) spin liquid with Dirac cones is competitive with previously proposed gapped spin liquids [1,2] also when small second- ($J_2$) and third-neighbor ($J_3$) antiferromagnetic couplings are considered on top of the nearest-neighbor ($J_1$) super-exchange [3-5]. For the $J_1{-}J_2$ model on finite clusters, a gapped $Z_2$ spin liquid can be stabilized in presence of a finite $J_2$ super-exchange, with a substantial energy gain with respect to the gapless U(1) Dirac spin liquid. However, this energy gain vanishes in the thermodynamic limit [4]. The presence of $J_3$ favors a chiral spin liquid with non-trivial magnetic fluxes and $C=1/2$ fractionalized Chern number [5].\\[4pt] [1] S. Yan, D. Huse, and S. White, Science 332, 1173 (2011).\\[0pt] [2] S. Depenbrock, I.P. McCulloch, and U.Schollwock, Phys. Rev. Lett. 109, 067201 (2012).\\[0pt] [3] Y. Iqbal, D. Poilblanc, and F. Becca, Phys. Rev. B 89, 020407(R) (2014).\\[0pt] [4] Y. Iqbal, D. Poilblanc, and F. Becca, arXiv:1410.7359.\\[0pt] [5] W.-J. Hu, W. Zhu, Y. Zhang, S. Gong, F. Becca, and D.N. Sheng, arXiv:1411.1327.