---
title: S=1 Shastry–Sutherland Model
url: https://www.emergentmind.com/topics/s-1-shastry-sutherland-model
type: topic
---

# S=1 Shastry–Sutherland Model

The $S=1$ Shastry–Sutherland model denotes a family of frustrated quantum-spin systems defined on the two-dimensional Shastry–Sutherland lattice (SSL), a square lattice with an orthogonal covering of diagonal bonds. In its simplest isotropic Heisenberg form, it interpolates between a product-dimer regime and magnetically ordered states through intermediate phases that are absent in the classical limit. In anisotropic and field-dependent variants, the enlarged on-site Hilbert space of spin one further stabilizes phases not available for $S=1/2$, including a quantum paramagnet, two distinct spin supersolids, an “offspring” $m=1/6$ plateau, and chirality-split magnons in a collinear Néel phase. Closely related generalized constructions also admit exact trimer-singlet product ground states for integer spin [2605.24409][2507.08668][1406.5856][1202.5854].

## 1. Lattice, conventions, and model families

The SSL is obtained by placing alternating diagonal bonds on a square lattice so that orthogonal dimers live on two sets of diagonals that do not touch each other. The square plaquettes without a diagonal bond are the “empty plaquettes,” which form two checkerboard sublattices. In the isotropic $S=1$ Heisenberg study, the Hamiltonian is
\[
H = J\sum_{\langle i,j\rangle}\mathbf{S}_i\cdot\mathbf{S}_j
+ J'\sum_{\langle\!\langle i,j\rangle\!\rangle'}\mathbf{S}_i\cdot\mathbf{S}_j,
\qquad
g=\frac{J}{J'},
\]
with $J$ the nearest-neighbor inter-dimer coupling on square edges and $J'$ the diagonal intra-dimer coupling; energy units are set by $J'=1$ [2605.24409].

For $S=1$, the literature uses several closely related but not identical conventions. In the easy-axis magnon study, the model is a generalized SSL Heisenberg antiferromagnet with single-ion anisotropy,
\[
H=\sum_{(i,j)}J_{ij}\,\mathbf{S}_i\cdot\mathbf{S}_j + D\sum_i (S_i^z)^2,
\]
where $J$ labels intra-dimer bonds, $J'$ and $J''$ label two inequivalent inter-dimer directions, and $D<0$ is easy-axis. In the strongly anisotropic field-driven study, the Hamiltonian is an $S=1$ $XXZ$ model with single-ion anisotropy and longitudinal field,
\[
H = \sum_{\langle i,j\rangle}\!\left[-\Delta(S_i^xS_j^x+S_i^yS_j^y)+S_i^zS_j^z\right]
+ J' \sum_{\langle\!\langle i,j\rangle\!\rangle}\!\left[-\Delta(S_i^xS_j^x+S_i^yS_j^y)+S_i^zS_j^z\right]
+ D\sum_i (S_i^z)^2 - h\sum_i S_i^z.
\]
These differing notations are a persistent source of confusion; the diagonal-bond coupling is denoted by $J'$ in the isotropic Heisenberg work, but by $J$ in the easy-axis magnon work [2507.08668][1406.5856].

| Model variant | Hamiltonian ingredients | Distinctive $S=1$ phenomena |
|---|---|---|
| Isotropic Heisenberg SSL | $J$, $J'$, $g=J/J'$ | dimer, MPD, spiral, NAF |
| Easy-axis generalized SSL | $J$, $J'$, $J''$, $D<0$ | Néel (I), chirality-split magnons, SSE/SNE |
| Ising-like $XXZ$ SSL in field | $\Delta=0.2$, $J'$, $D>0$, $h$ | QP, SS1, SS2, $1/6$, $1/3$, $1/2$ plateaus |
| Modified trimer SSL | $J_1$, $J_2$ on decorated lattice | exact trimer-singlet product phase |

At the isotropic point $g=0$, each $J'$ bond hosts an exact $S=1$ singlet, and the bond expectation value is
\[
\langle \mathbf{S}_i\cdot\mathbf{S}_j\rangle=-2.
\]
The corresponding product state is the natural $S=1$ analogue of the dimer limit [2605.24409].

## 2. Zero-field isotropic Heisenberg phase diagram

For the isotropic $S=1$ SSL Heisenberg model, density matrix renormalization group (DMRG) and cluster mean-field theory (CMFT) identify the zero-field phase sequence
\[
\text{dimer} \;\to\; \text{mixed plaquette–dimer (MPD)} \;\to\; \text{spiral} \;\to\; \text{Néel antiferromagnet (NAF)}
\]
as $g=J/J'$ increases [2605.24409].

The transition from the dimer phase to the MPD phase occurs at $g_{c1}\approx 0.47$ and is first order. It is located from sharp features in $-\partial^2 e_0/\partial g^2$ and jumps in the bipartite entanglement entropy
\[
S_e=-\mathrm{Tr}\,\rho_\ell\ln\rho_\ell.
\]
The MPD-to-spiral transition is likewise first order at $g_{c2}\approx 0.58$, again diagnosed by discontinuities in $-\partial^2 e_0/\partial g^2$ and $S_e$. The spiral-to-NAF transition at $g_{c3}\approx 0.825$ is continuous and corresponds to commensurate locking of an incommensurate ordering wavevector to $(\pi,\pi)$; the energy derivative shows no clear singularity within finite-size resolution, while the magnetic Bragg peak evolves continuously and $S_e$ shows a sharp drop followed by a plateau [2605.24409].

The numerical evidence comes from SU(2)-symmetric DMRG on finite cylinders of widths $L_y=4,6$ and lengths up to $L_x=24$, retaining up to 8000 states with truncation errors below $5\times 10^{-6}$. Open clusters are also used to expose symmetry-broken bond patterns that are partially hidden on cylinders. CMFT calculations employ $L_y=4$ clusters with $L_x=8$ or $16$ and a DMRG impurity solver implemented via ITensor, with truncation errors below $10^{-6}$ [2605.24409].

Placed in a broader $S$-$g$ context, the $S=1$ sequence differs qualitatively from both the $S=1/2$ and classical limits. The global diagram assembled from $S=1/2$, $S=1$, $S=3/2$, and $S\to\infty$ shows that the spiral regime expands with increasing $S$, while the dimer and MPD windows shrink and ultimately vanish in the classical limit. This supports the interpretation that quantum-disordered phases are progressively suppressed as $S$ increases [2605.24409].

## 3. Microscopic characterization of the MPD, spiral, and Néel regimes

The primary local diagnostic is the bond energy
\[
B_{ij}=\langle \mathbf{S}_i\cdot\mathbf{S}_j\rangle.
\]
In the dimer phase, $B_{ij}$ is strongly negative on the diagonal $J'$ bonds and much weaker on the inter-dimer $J$ bonds. In the MPD phase, DMRG finds strong intradimer correlations together with weak tetramerization on the empty plaquettes. This tetramerization appears as a small but finite alternation of the four edge-bond energies around an empty plaquette, with two symmetry-related patterns residing on the two checkerboard plaquette sublattices [2605.24409].

A convenient MPD order parameter is
\[
T_{\rm MPD}=
\frac{1}{N_p}\sum_p \eta_p \sum_{\langle ij\rangle\in p}
\langle \mathbf{S}_i\cdot\mathbf{S}_j\rangle,
\qquad
\eta_p=\pm 1
\]
on the two empty-plaquette sublattices. On open clusters, $T_{\rm MPD}$ is small but finite and its sign selects one of the two degenerate plaquette patterns. On cylinders, the superposition of degenerate patterns strongly suppresses the apparent order parameter. This clarifies a common misconception: the MPD phase is not a pure plaquette state. It combines robust dimers with only weak empty-plaquette tetramerization, and is therefore distinct from the $S=1/2$ empty-plaquette phase [2605.24409].

Real-space spin correlations,
\[
C(\mathbf r)=\langle \mathbf S_{\mathbf 0}\cdot \mathbf S_{\mathbf r}\rangle,
\]
further separate the phases. In the MPD regime, correlations decay exponentially and display a sawtooth-like modulation along the $x$ direction, consistent with tetramerization on empty plaquettes. In the spiral regime, correlations show power-law decay with incommensurate oscillations. In the NAF phase, they exhibit power-law decay without oscillations, consistent with commensurate $(\pi,\pi)$ order on finite cylinders [2605.24409].

The static structure factor,
\[
S(\mathbf q)=\frac{1}{N_s}\sum_{i,j}
e^{i\mathbf q\cdot(\mathbf r_j-\mathbf r_i)}
\frac{\langle \mathbf S_i\cdot\mathbf S_j\rangle}{S(S+1)},
\]
is broad in the dimer and MPD phases, develops a sharp peak at $\mathbf q=(q_x,\pi)$ in the spiral regime, and locks to $(\pi,\pi)$ in the NAF phase. The spiral ordering wavevector
\[
\mathbf q(g)=(q_x(g),\pi), \qquad g_{c2}<g<g_{c3},
\]
moves continuously toward $(\pi,\pi)$ as $g$ increases. The pitch angle is $\theta(g)=\pi-q_x(g)$ and the real-space period along $x$ is $L_x(g)=2\pi/\theta(g)$. At $g=1/\sqrt{2}\approx 0.707$, CMFT finds an $8\times 2$ magnetic unit cell, consistent with a commensurate approximation to an incommensurate spiral with $\theta\approx \pi/4$ [2605.24409].

## 4. Easy-axis Néel (I) order, altermagnetic magnons, and spin transport

A distinct $S=1$ SSL regime arises when easy-axis single-ion anisotropy stabilizes a collinear Néel (I) state. In the generalized model with antiferromagnetic $J$, $J'$, $J''$ and $D<0$, the representative parameters
\[
\frac{J'}{J}=1.2,\qquad \frac{J''}{J}=1,\qquad \frac{D}{J}=-0.6
\]
place the system deep inside a collinear $z$-axis antiferromagnet with a four-site magnetic unit cell. Sites 1 and 3 are spin-up and sites 2 and 4 are spin-down, and the ordered state belongs to the plane group $p4g$ [2507.08668].

Linear spin-wave theory around this state uses a four-sublattice Holstein–Primakoff expansion and produces an $8\times 8$ bosonic Bogoliubov–de Gennes Hamiltonian. Because there are four spins per unit cell, there are four positive-energy magnon bands. For the representative parameters, the spectrum separates into a lower pair and an upper pair, with each pair consisting of two bands of opposite chirality. Magnon chirality is labeled by
\[
\vartheta=\frac{\langle S^z\rangle}{\hbar}=\pm 1,
\]
so that the bands are denoted $E_{\pm 1}^{l,u}(\mathbf k)$ [2507.08668].

The nontrivial result is that opposite-chirality magnon bands are nondegenerate throughout the Brillouin zone without spin–orbit coupling, Dzyaloshinskii–Moriya interactions, or an external field. The mechanism is symmetry-based. In the Néel (I) SSL, opposite-spin sublattices are related by $C_{4z}$ rather than by inversion centered between opposite-spin sites, and inversion centers lie at mid-edges and map each sublattice to itself. The chirality splitting alternates sign along orthogonal momentum directions, a magnonic realization of altermagnetic alternation [2507.08668].

Transport is treated within a relaxation-time Kubo formalism for the magnon spin current driven by a thermal gradient. The resulting longitudinal spin Seebeck and transverse spin Nernst responses are finite because the two circularly polarized magnons have different thermal populations and group velocities when $E_{+1}\neq E_{-1}$. Using $J/k_B=100\,\mathrm K$ and $T=25\,\mathrm K$, the study finds that $\sigma_\parallel(\theta)$ is maximal near $\theta=0,\pi$ and minimal near $\theta=\pi/2$, while $\sigma_\perp(\theta)$ vanishes when the thermal gradient is aligned with crystal axes and peaks near $\theta\approx \pi/4,\,3\pi/4$. When $J'=J''$, the restored $C_4$ symmetry enforces $\sigma_{xx}=\sigma_{yy}$ and the spin Nernst coefficient vanishes for all $\theta$. By contrast, the intrinsic thermal Hall conductivity remains negligible or zero because Berry curvature is symmetry-forbidden or cancels pairwise in this setting [2507.08668].

## 5. Uniaxial anisotropy, magnetic field, and plateau physics

In the strongly Ising-like $S=1$ $XXZ$ SSL with single-ion anisotropy, stochastic series expansion quantum Monte Carlo on $L\times L$ tori with $L=4$–$32$ reveals a much richer field-driven structure than in the isotropic Heisenberg problem. The study focuses on $\Delta=0.2$, with ground-state behavior accessed by low temperatures using $\beta=8L$ and, in some scans, $\beta=L$ [1406.5856].

At zero field, four phases appear as functions of $J'$ and $D$: an Ising antiferromagnet with ordering vector $(\pi,\pi)$, a transverse XY-ordered phase labeled “SF,” a large-$J'$ triplet-dimer phase, and a quantum paramagnet stabilized by sufficiently large $D>0$. The SF–QP transition is continuous in the $O(2)$ universality class, with representative critical values $D_c=1.71(1)$ at $J'=1.0$ and $D_c=0.31(1)$ at $J'=3.0$ [1406.5856].

Under a longitudinal field, the magnetization process exhibits several plateau and supersolid phases. For small $J'$, the sequence is
\[
\mathrm{AFM}\to \mathrm{SS1}\to \mathrm{PL1}\,(m=1/2)\to \mathrm{SF}\to \mathrm{FP}.
\]
With increasing $J'$, SS1 and PL1 disappear and a $1/3$ plateau PL2 emerges. For $J'\gtrsim 2.6$, a $1/6$ plateau PL3 appears below PL2, and a second supersolid SS2 is stabilized between them. For very large $J'\gtrsim 3.6$, where the zero-field state is triplet-dimer-like, a distinct high-field $1/2$ plateau PL4 appears; unlike PL1, PL4 is a quantum-disordered dimer plateau without Bragg peaks in either longitudinal or transverse structure factors [1406.5856].

The $1/3$ and $1/6$ plateaus are particularly characteristic of the $S=1$ case. In the Ising limit, PL2 is a period-3 stripe state with ordering wavevector
\[
Q=(2\pi/3,\pi)\quad \text{or}\quad (\pi,2\pi/3).
\]
PL3 is obtained from the same parent stripe pattern by replacing each $(\uparrow\uparrow)$ stripe by an $(\uparrow 0)$ stripe, preserving the period-3 order. This “offspring” mechanism is impossible for $S=1/2$ because the $S^z=0$ state does not exist. SS2 inherits the same period-3 longitudinal order while developing finite spin stiffness $\rho_s>0$, thereby combining diagonal and off-diagonal long-range order [1406.5856].

This variant also resolves another common ambiguity in the literature: the large-$J'$ “dimer” phase need not be a singlet-dimer phase. In the present sign convention, the transverse exchange is ferromagnetic, so the large-$J'$ state is described as a triplet-dimer analogue of the canonical $S=1/2$ singlet-dimer phase [1406.5856].

## 6. Exact generalized constructions, comparisons, and open directions

A related but distinct integer-spin extension replaces each SSL diagonal by an equilateral triangle, producing a decorated square lattice of coupled spin trimers. For integer spin, and in particular for $s=1$, the local trimer has a unique singlet ground state with energy
\[
E_{\text{trimer}}^{(0)}(1)=-3J_1.
\]
Because the inter-trimer couplings satisfy a balance condition, the product
\[
\Phi=\bigotimes_t |\phi_t\rangle
\]
of local trimer singlets is an exact eigenstate of the full lattice Hamiltonian and, within a finite parameter window, the exact ground state. For $s=1$, exact diagonalization with periodic boundary conditions gives $x_{c1}\approx -0.587$ and $x_{c2}\approx 0.454$ for $x=J_2/J_1$ on $N=24$ sites, while a rigorous inner interval is
\[
\frac{3-\sqrt{73}}{16}\approx -0.3465 < x < \frac14.
\]
Within this trimer-singlet product phase, the ground-state energy per site is
\[
E_0/N=-J_1,
\]
independent of $J_2$ [1202.5854].

Although this decorated-trimer model is not the standard SSL Heisenberg model, it is part of the broader $S=1$ Shastry–Sutherland landscape because it preserves the underlying orthogonal-dimer logic while changing the local building block from dimers to trimers. It also exhibits robust magnetization plateaus at $m=1/3$ and $m=2/3$, and a ferrimagnetic phase for sufficiently negative $J_2$ [1202.5854].

Across the literature, several comparisons are now clear. First, $S=1$ physics is not obtained by merely weakening quantum fluctuations relative to $S=1/2$. The isotropic model supports an MPD phase and an incommensurate spiral between the dimer and Néel limits; the anisotropic field-driven model supports a quantum paramagnet, two supersolids, and an offspring $1/6$ plateau; and the easy-axis ordered regime supports nonrelativistic chirality-split magnon bands without SOC or field [2605.24409][1406.5856][2507.08668]. Second, the role of anisotropy is decisive: easy-axis single-ion anisotropy stabilizes Néel (I) order and altermagnetic magnon transport, whereas easy-plane single-ion anisotropy stabilizes the quantum paramagnet and reshapes the field-induced plateau hierarchy [2507.08668][1406.5856].

The experimental context is correspondingly broad. Materials mentioned in this literature include rare-earth-based $\mathrm{RE}_2\mathrm{Be}_2\mathrm{GeO}_7$ ($\mathrm{RE}=\mathrm{Pr}$–$\mathrm{Yb}$), $\mathrm{ErB}_4$, $\mathrm{Eu}_2\mathrm{MgSi}_2\mathrm{O}_7$, and Pr-based melilites such as $\mathrm{Pr}_2\mathrm{Be}_2\mathrm{GeO}_7$ and $\mathrm{Pr}_2\mathrm{Ga}_2\mathrm{BeO}_7$. Proposed signatures include short-range scattering without sharp Bragg peaks in the MPD regime, a sharp incommensurate peak at $(q_x,\pi)$ in the spiral regime, conventional $(\pi,\pi)$ order in the NAF regime, angle-dependent spin Seebeck and spin Nernst responses in the Néel (I) regime, and plateau structures at $m=1/6$, $1/3$, and $1/2$ in the anisotropic field-driven problem [2605.24409][2507.08668][1406.5856].

Open questions identified in these works include the determination of precise exchange ratios in larger-$S$ SSL compounds, the role of exchange anisotropy and interlayer couplings, finite-temperature crossovers in MPD and spiral regimes, possible field-tuned versions of the MPD and spiral phases, interaction and disorder effects beyond linear spin-wave theory, and a quantitative mapping of the full $(J'/J,J''/J,D/J)$ phase diagram in the altermagnetic Néel (I) regime [2605.24409][2507.08668].

Source: https://www.emergentmind.com/topics/s-1-shastry-sutherland-model