Title: Identifying the ground state phases by spin-patterns in the Shastry-Sutherland model

Abstract: Exploring the influence of frustration on the phases and related phase transitions in condensed matter physics is of fundamental importance in uncovering the role played by frustration. In the two-dimensional square lattice, a minimal frustration has been formulated in 1981 as the Shastry-Sutherland (SS) model described by competitions between the nearest-neighbor bond ($J_1$) and the next-nearest-neighbor one ($J_2$). In the two limits of $\alpha=J_2/J_1$, i.e. $\alpha \ll 1$ and $\alpha \gg 1$, the corresponding phases are the N{\'e}el antiferromagnet (AFM) and the dimer-singlet(DS). Unfortunately, the intermediate regime remains controversial, and the nature of transition from the N{\'e}el AFM to the intermediate state is also unclear. Here we provide a pattern language to explore the SS model and take the lattice size $L=4 \times4$ with periodic boundary condition. We firstly diagonalize the Hamiltonian in an operator space to obtain all fundamental spin-patterns and then analyze their energy and occupancy evolutions with the frustration parameter $\kappa=\alpha / (1+\alpha)$. Our results indicate that the intermediate regime is characterized by diagonal two-domain spin-pattern while the N{\'e}el AFM state has a diagonal single-domain and the DS has mixings of diagonal single- and four-domain. While the transition from the DS to the intermediate phase occurred around $\alpha_c = 1.5$ is the first-order in nature, consistent with that in literature, the one from the intermediate phase to the AFM is clearly seen around $\alpha_c = 1.277$, where it has a reversal of the contributions from the single- and two-domain patterns to the ground state. The result indicates that the pattern language is powerful in identifying the possible phases in frustrated models.