  We study the quantum order by disorder effect of the $S=1/2$ system, which is programmable in quantum simulator composed of Rydberg atoms in the triangular optical lattice with a controllable diagonal anisotropy. When the total magnetization is zero, a set of sub-extensive degenerate ground states is present in the classical limit, composed of continuous strings whose configuration enjoys a large degree of freedom. Among all possible configurations, we focus on the stripe (up and down spins align straightly) and kinked (up and down spins form a zigzag shape) patterns. Adopting the the real space perturbation theory (RSPT), we estimate the leading order energy correction when the nearest-neighbor ($nn$) spin-flip coupling $J$ term is considered, and the overall model becomes an effective XXZ model with spatial anisotropy. Our calculation demonstrates a lifting of the degeneracy, favoring the stripe configuration. When $J$ becomes larger, we adopt the infinite projected entangled-pair state (iPEPS) and numerically check the effect of degeneracy lifting. The iPEPS results show that even when the spin-flip coupling term is strong the stripe pattern is still favored. While the above system is realizable with the deformed optical lattice, at last, we numerically calculate the hard-core bosonic Hamiltonian with the $nn$ dipole-dipole interaction for demonstrating the possible underlying phases, using the cluster mean-field theory. We provide various phase diagrams with different tilted angles, showing the abundant phases including the supersolid. Our proposal indicates a realizable scenario through quantum simulator in studying the quantum effect as well as extraordinary phases. 
Programmable Order by Disorder Effect through Quantum Simulator