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arXiv:2609.22907

On the precise boundary of the Néel antiferromagnetic phase in the Shastry-Sutherland model

Rongyi Lv, Xiangjian Qian, Mingpu Qin

Published Sep 19, 2026• 0 citations
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Abstract

The ground state phase diagram of the Shastry-Sutherland model remains controversial. In particular, the nature of the transition between the plaquette valence bond solid (PVBS) phase and the Néel antiferromagnetic (AFM) phase is still under debate. While some studies suggest a direct transition between the two phases, others propose the existence of a narrow intermediate quantum spin liquid regime. To address this issue, we perform large-scale Density Matrix Renormalization Group calculations to determine the Néel AFM phase boundary in the Shastry-Sutherland model. By applying staggered boundary pinning fields and directly measuring the bulk staggered magnetization, and through systematic extrapolations with respect to truncation error as well as careful finite-size scaling, we find that the staggered magnetization remains finite across the controversial region $0.78 \lesssim J_{1}/J_{2} \leq 0.81$, which was suggested to host a quantum spin liquid phase in some studies. Together with the previously established consensus placing the PVBS phase boundary at $J_{1}/J_{2} \approx 0.78$, these results provide evidence against the existence of an intermediate quantum spin liquid phase and support a direct PVBS--Néel AFM transition in the Shastry-Sutherland model.

DOI
10.48550/arxiv.2609.22907

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The ground state phase diagram of the Shastry-Sutherland model remains controversial. In particular, the nature of the transition between the plaquette valence bond solid (PVBS) phase and the Néel antiferromagnetic (AFM) phase is still under debate. While some studies suggest a direct transition between the two phases, others propose the existence of a narrow intermediate quantum spin liquid regime. To address this issue, we perform large-scale Density Matrix Renormalization Group calculations to determine the Néel AFM phase boundary in the Shastry-Sutherland model. By applying staggered boundary pinning fields and directly measuring the bulk staggered magnetization, and through systematic extrapolations with respect to truncation error as well as careful finite-size scaling, we find that the staggered magnetization remains finite across the controversial region 0.78≲J1/J2≤0.810.78 \lesssim J_{1}/J_{2} \leq 0.810.78≲J1​/J2​≤0.81, which was suggested to host a quantum spin liquid phase in some studies. Together with the previously established consensus placing the PVBS phase boundary at J1/J2≈0.78J_{1}/J_{2} \approx 0.78J1​/J2​≈0.78, these results provide evidence against the existence of an intermediate quantum spin liquid phase and support a direct PVBS--Néel AFM transition in the Shastry-Sutherland model.
We propose a generalized Shastry-Sutherland model which bridges the Shastry-Sutherland model and the J1J_1J1​-J2J_2J2​ Heisenberg model. By employing large scale Density Matrix Renormalization Group and Fully Augmented Matrix Product State calculations, combined with careful finite-size scaling, we find the phase transition between the plaquette valence bond state (PVBS) and Neel anti-ferromagnetic (AFM) phase in the pure Shastry-Sutherland model is a weak first one. This result indicates the existence of an exotic tri-critical point in the PVBS to AFM transition line in the phase diagram, as the transition in the J1J_1J1​-J2J_2J2​ Heisenberg model was previously determined to be continuous. We determine the location of the tri-critical point in the phase diagram at which first-order transition turns to continuous. Our generalized Shastry-Sutherland model provides not only a valuable platform to explore exotic phases and phase transitions but also more realistic description of Shastry-Sutherland materials like SrCu2_22​(BO3_33​)2_22​.
The Shastry-Sutherland model as a canonical example of frustrated magnetism has been extensively studied. The conventional wisdom has been that the transition from the plaquette valence bond order to the Neel order is direct and potentially realizes a deconfined quantum critical point beyond the Ginzburg-Landau paradigm. This scenario however was challenged recently by improved numerics from density matrix renormalization group which offers evidence for a narrow gapless spin liquid between the two phases. Prompted by this controversy and to shed light on this intricate parameter regime from a fresh perspective, we report high-resolution functional renormalization group analysis of the generalized Shastry-Sutherland model. The flows of over 50 million running couplings provide a detailed picture for the evolution of spin correlations as the frequency/energy scale is dialed from the ultraviolet to the infrared to yield the zero temperature phase diagram. The singlet dimer phase emerges as a fixed point, the Neel order is characterized by divergence in the vertex function, while the transition into and out of the plaquette order is accompanied by pronounced peaks in the plaquette susceptibility. The plaquette order is suppressed before the onset of the Neel order, lending evidence for a finite spin liquid region for J1/J2∈(0.77,0.82)J_1/J_2\in (0.77,0.82)J1​/J2​∈(0.77,0.82), where the flow is continuous without any indication of divergence.
The Shastry-Sutherland (SS) model as a canonical example of frustrated magnetism has been extensively studied. The conventional wisdom has been that the transition from the plaquette valence bond order to the Neel order is direct and potentially realizes a deconfined quantum critical point beyond the Ginzburg-Landau paradigm. This scenario, however, was challenged recently by improved numerics from density matrix renormalization group which offers evidence for a narrow gapless spin liquid between the two phases. Prompted by this controversy and to shed light on this intricate parameter regime from a fresh perspective, we report high-resolution functional renormalization group analysis of the generalized SS model. The flows of over 50 million running couplings provide a detailed picture for the evolution of spin correlations as the frequency/energy scale is dialed from the ultraviolet to the infrared to yield the zero-temperature phase diagram. The singlet dimer phase emerges as a fixed point, the Neel order is characterized by divergence in the vertex function, while the transition into and out of the plaquette order is accompanied by pronounced peaks in the plaquette susceptibility. The plaquette order is suppressed before the onset of the Neel order, lending evidence for a finite spin liquid region for ${J}_{1}/{J}_{2}\ensuremath{\in}(0.77,0.82)$, where the flow is continuous without any indication of divergence.
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 Néel antiferromagnetic (AFM) order with staggered magnetization and a nonmagnetic regime lacking conventional order, consistent with a nonmagnetic insulating phase that may be related to the spin-liquid-like regimes reported in previous studies. In contrast, the ( U / t , J / t ) phase diagram exhibits a competition between the Néel antiferromagnet 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 Néel AFM phase 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 SrCu 2 ( BO 3 ) 2 [J. Guo , ]. 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.
We study the half-filled single-band Hubbard model on the Shastry–Sutherland lattice with nearest-neighbor hopping ttt, incorporating (i) a frustrating diagonal hopping t′t't′ to probe the emergence of spin-liquid behavior, and (ii) an additional Heisenberg exchange JJJ 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)(U/t,t'/t)(U/t,t′/t) and (U/t,J/t)(U/t,J/t)(U/t,J/t) parameter planes. In the (U/t,t′/t)(U/t,t'/t)(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)(U/t,J/t)(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_22​(BO3_33​)2_22​ [Guo \emph{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.
We perform an in-depth investigation of the phase diagram of the ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model on the square lattice. We take advantage of density matrix renormalization group and fully augmented matrix product states methods and reach unprecedented accuracy with large bond dimensions. We utilize excited-level crossing analysis to pinpoint the phase transition points. It was believed before that there exists a narrow spin liquid phase sandwiched by the N\'eel antiferromagnetic (AFM) and valence bond solid (VBS) phases. Through careful finite-size scaling of the level crossing points, we find a direct phase transition between the N\'eel AFM and VBS phases at J2/J1=0.535(3){J}_{2}/{J}_{1}=0.535(3)J2​/J1​=0.535(3), suggesting the absence of an intermediate spin liquid phase. We also provide accurate results for ground-state energies for a variety of sizes, from which we find that the transition between the N\'eel AFM and VBS phases is continuous. These results indicate the existence of a deconfined quantum critical point at J2/J1=0.535(3){J}_{2}/{J}_{1}=0.535(3)J2​/J1​=0.535(3) in the model. From the crossing of the first derivative of the energies with J2{J}_{2}J2​ for different sizes, we also determine the precise location of the first-order phase transition between the VBS and stripe AFM phases at J2/J1=0.610(5){J}_{2}/{J}_{1}=0.610(5)J2​/J1​=0.610(5).