Exact diagonalization study of the anisotropic Heisenberg model related to YbMgGaO$_4$

Employing exact diagonalization, we systematically study the anisotropic Heisenberg model which is related to rare-earth triangular-lattice materials. We probe its full 3D phase diagram afresh and identify a large region of quantum spin liquid (QSL) phase which can extend to the QSL region of the $J_1$--$J_2$ triangular Heisenberg model. Furthermore, we explore the magnetization curves of different phases and reproduce the 1/3-magnetization plateau in the quantum spin liquid phase region. More importantly, to study the possible chemical disorders in real materials, we consider the randomness of exchange interactions and find no spin glass order. And there is a large region of random-singlet phase which contains strongly random spin networks, dominated by two-spin singlets, four-spin singlets and other singlet domains. Our comprehensive ED study can give detailed insightful understanding of the microscopic Hamiltonian related to the YbMgGaO$_4$ and some other related rare-earth triangular-lattice materials.