Antiferromagnetism, Spin Liquid, and Plaquette Singlet Phases in the Shastry–Sutherland Model: A Variational Cluster Approximation Approach

We study the half-filled single-band Hubbard model on the Shastry–Sutherland lattice with nearest-neighbor hopping t, incorporating (i) a frustrating diagonal hopping $t'$ to probe the emergence of spin-liquid behavior, and (ii) an additional Heisenberg exchange J that explicitly stabilizes plaquette-singlet correlations. Using the variational cluster approximation (VCA) with exact diagonalization at zero temperature on an eight-site cluster, we map the phase diagrams in the $(U/t,t'/t)$ and $(U/t,J/t)$ parameter planes. In the $(U/t,t'/t)$ plane, our analysis uncovers a competition between N\'eel antiferromagnetic (AFM) order with staggered magnetization and a nonmagnetic regime lacking conventional order, consistent with an algebraic spin-liquid (ASL) phase characterized by a gapped excitation spectrum and vanishing staggered magnetization. In contrast, the $(U/t,J/t)$ phase diagram exhibits a competition between the N\'eel AFM and a solid valence-bonding valence bond plaquette (PS), where spins form localized singlet plaquettes. Most notably, we observe that the transition from the PS phase to the N\'eel AFM is a first-order phase transition, with a discontinuous jump of the order parameters at the phase boundary, in quantitative agreement with high-pressure heat-capacity measurements on SrCu₂(BO₃)₂ [Guo et al., Commun. Phys. 8, 75 (2025)]. Thus, our VCA analysis underscores the subtle interplay of geometric frustration, quantum fluctuations, and dimensionality, and elucidates the crucial role of short-range correlations in stabilizing and competing ground states of frustrated quantum magnets.