The Shastry-Sutherland model — a spin-1/2 Heisenberg antiferromagnet on a two-dimensional orthogonal-dimer lattice — admits an exact product-of-dimer-singlets ground state and exhibits, between that phase and Néel order, a square-plaquette valence-bond solid and a narrow candidate quantum spin liquid. Salou-Smith, Ralko and Jaubert (arXiv:2507.13877, 2025) introduced a three-dimensional analogue, built by removing the top bond of every tetrahedron of the pyrochlore lattice (the three-dimensional network of corner-sharing tetrahedra) so that the local connectivity is indistinguishable from the 2D Shastry-Sutherland lattice, and proved that the dimer singlet remains an exact ground state for diagonal coupling J2≥2J1J_2\ge 2J_1. The zero-field quantum phase diagram below this exact-solvability boundary is undetermined in three dimensions: the true lower edge of the dimer phase, the existence and nature of any intermediate phases (the 2D square-plaquette solid is geometrically impossible because, beyond the four-site unit cell, the minimal 3D loop has seven sites and no empty four-site squares exist), and the character of the transition(s) to the Néel state — including the possibility of a 3D quantum spin liquid or deconfined quantum critical point — are all open. Exact diagonalization up to 32 sites suggests enhanced dimer stability in 3D.

Background

The Shastry-Sutherland (SS) model is a spin-1/2 Heisenberg antiferromagnet on a two-dimensional lattice of corner-sharing orthogonal dimer units: each site has four nearest-neighbor J1J_1 bonds forming a modified square lattice, plus one diagonal J2J_2 bond, and Shastry and Sutherland (1981) proved that for large enough J2J_2 (with J1=1J_1=1, the exact window is J2≥2J_2\ge 2) the ground state is exactly the product of singlets on the J2J_2 dimers. (A singlet is the two-spin entangled state (∣↑↓⟩−∣↓↑⟩)/2(|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle)/\sqrt 2; a Néel state is a state with two-sublattice staggered antiferromagnetic order; a valence-bond solid, VBS, is a non-magnetic state in which singlets form in a regular pattern that breaks lattice symmetries; a quantum spin liquid, QSL, is a non-magnetic state without any broken lattice symmetry, typically with fractionalized excitations; a triplon is the collective S=1S=1 excitation obtained by breaking a dimer singlet.) The model is realized in the layered compound SrCu2_2(BO3_3)2_2 (Kageyama et al., 1999). Its two-dimensional zero-field phase diagram is well mapped: the dimer phase survives below the exact boundary down to J2≈1.48J_2\approx 1.48 (Koga and Kawakami, 2000; Corboz and Mila, 2013, using infinite projected entangled-pair states, iPEPS — a two-dimensional tensor-network ansatz whose bond dimension, the variational parameter controlling how much entanglement the state can encode, is extrapolated to reach the thermodynamic limit); for 1.28≲J2≲1.481.28\lesssim J_2\lesssim 1.48 the ground state is a square-plaquette VBS (singlets resonating on empty four-site squares); recent density-matrix renormalization-group and neural-quantum-state studies (Yang, Sandvik and Wang, 2022; Viteritti et al., 2025; Nutakki et al., 2025) and high-precision iPEPS (Corboz, Zhang, Ponsioen and Mila, 2026) find a narrow QSL candidate for 1.22≲J2≲1.281.22\lesssim J_2\lesssim 1.28; below it the system has Néel order. The plaquette-to-Néel and dimer-to-plaquette boundaries have also been discussed as possible deconfined quantum critical points — continuous transitions between phases with unrelated order parameters (Lee, You, Sachdev and Vishwanath, 2019). Several three-dimensional SS-type models with exact dimer ground states existed before (Koga's stacked-layer orthogonal-dimer system for SrCu2_2(BO3_3)2_2, 2000; cubic-lattice generalizations by Chen and Büttner, 2002), but none reproduces the local SS connectivity. Salou-Smith, Ralko and Jaubert (2025) constructed such a lattice: starting from the pyrochlore lattice (the 3D network of corner-sharing tetrahedra), remove the top bond of every tetrahedron and let the opposite bottom bond be the J2J_2 diagonal, leaving corner-sharing four-site square units with four J1J_1 bonds each and coordination five. The Hamiltonian is H=J1∑⟨i,j⟩Si⋅Sj+J2∑⟨⟨i,j⟩⟩Si⋅Sj\mathcal{H}=J_1\sum_{\langle i,j\rangle}\mathbf S_i\cdot\mathbf S_j+J_2\sum_{\langle\langle i,j\rangle\rangle}\mathbf S_i\cdot\mathbf S_j with J1=1J_1=1 and antiferromagnetic J2>0J_2>0. Classically the model is exactly solvable (Ising and Heisenberg phase diagrams coincide with the 2D ones, including a 1/3 magnetization plateau and umbrella states), and for S=1/2S=1/2 the dimer singlet is proven to be the exact ground state for J2≥2J_2\ge 2 by a local triangle-unit argument. The crucial geometric difference is that beyond the four-site unit cell the smallest closed loop contains seven spins in 3D (four in 2D), so the square-plaquette mechanism that stabilizes the 2D intermediate phase is absent.

Problem Statement

For the spin-1/2 Heisenberg antiferromagnet H=J1∑⟨i,j⟩Si⋅Sj+J2∑⟨⟨i,j⟩⟩Si⋅Sj\mathcal{H}=J_1\sum_{\langle i,j\rangle}\mathbf S_i\cdot\mathbf S_j+J_2\sum_{\langle\langle i,j\rangle\rangle}\mathbf S_i\cdot\mathbf S_j on the three-dimensional Shastry-Sutherland lattice of Salou-Smith, Ralko and Jaubert (pyrochlore lattice with the top bond of every tetrahedron removed; four J1J_1 bonds and one J2J_2 diagonal per four-site unit cell), at zero magnetic field with J1=1J_1=1 and antiferromagnetic J2>0J_2>0: determine the ground-state phase diagram for 0<J2<20<J_2<2, the region below the exactly solved dimer boundary. This comprises (i) the true lower boundary J2cJ_2^c of the dimer-singlet phase (exact for J2≥2J_2\ge 2; exact diagonalization up to 32 sites suggests it persists below, but the thermodynamic-limit value is unknown); (ii) the nature of whatever phase(s) occupy the region between the dimer phase and the small-J2J_2 Néel state — whether analogues of the 2D intermediate phases exist (the 2D square-plaquette VBS is excluded by the seven-site minimal loop, but valence-bond solids on longer loops or on J2J_2-bonded squares, a quantum spin liquid, or no intermediate phase at all are all possibilities); and (iii) the character of the phase transition(s), in particular whether any of them can be a deconfined quantum critical point — a continuous transition between phases with unrelated order parameters — in this three-dimensional geometry. Only the zero-field phase diagram below J2=2J_2=2 is in scope; the paper's companion questions on this lattice — magnetization plateaux in a field, Dzyaloshinskii-Moriya-interaction (the antisymmetric exchange term allowed by the lattice's lack of inversion symmetry) effects on triplon-band topology, and the J3J_3 path back to the pyrochlore — are not part of the present problem.

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