Spiral Phase and Phase Diagram of the $S$=1/2 XXZ Model on the Shastry-Sutherland Lattice

TL;DR

This work maps the ground-state phase diagram of the $S=1/2$ XXZ model on the Shastry–Sutherland lattice as a function of the coupling ratio $g=J/J'$ and anisotropy $\Delta$, using ED, CMFT, and DMRG to combine complementary strengths. A key finding is the emergence of a coplanar spiral phase at small $\Delta$, alongside a robust intermediate empty plaquette (EP) phase that narrows away from the isotropic point; the full phase diagram also includes a dimer, full plaquette (FP), and $z$-AFM/$xy$-AFM phases. ED reveals sharp first-order transitions at certain boundaries, CMFT confirms EP as the stable intermediate phase but exhibits mean-field artifacts, and DMRG on long cylinders provides the most reliable boundaries for $\Delta<1$, resolving discrepancies and revealing EP–ICM–xy‑AFM competition. The results offer a plausible explanation for spin-liquid-like behavior in related SSL materials and highlight the crucial role of XXZ anisotropy in shaping the SSL phase diagram.

Abstract

We investigate the ground-state phase diagram of the $S$=1/2 XXZ model on the two-dimensional Shastry-Sutherland lattice using exact diagonalization (ED), density-matrix renormalization group (DMRG), and cluster mean-field theory (CMFT) with DMRG as a solver. In the isotropic case ($Δ=1$), CMFT results reveal an intermediate empty plaquette (EP) phase that has a lower energy than the full plaquette (FP) phase. However, due to mean-field artifacts, CMFT alone is not suitable for accurately determining phase boundaries. Therefore, we combined three methods to map out the reliable phase diagram. Our calculations show that the EP phase narrows as $Δ$ deviates from unity and eventually vanishes. More importantly, we identify a spiral phase at small $Δ$, which has not been reported in previous studies. This phase is clearly captured by DMRG simulations on long cylindrical geometries. The competition between the EP, spiral, and $xy$-AFM phases near their boundaries provides a plausible explanation for the emergent spin-liquid-like behavior in RE$_2$Be$_2$GeO$_2$, while shedding new light on the role of XXZ anisotropy in the Shastry-Sutherland XXZ model.

Figures (7)

• Figure 1: (a) Shastry-Sutherland lattice and three clusters used in the numerical calculations. Blue and green boxes represent $6\times6$ E- and F- type clusters, respectively, with stars of the same colors indicating their centers. The red box denotes the 32-site cluster used in ED calculations. (b) Phase diagram of the XXZ model on the Shastry-Sutherland lattice constructed from combined ED, CMFT, and DMRG results. Solid and dashed lines correspond to first-order and (possible) continuous phase transitions, respectively.(c,d) Schematic of empty plaquette (EP) and full plaquette (FP) states. Here, orange-shaded $2\times2$ plaquettes indicate plaquette-singlet patterns.
• Figure 2: (a-e) Exact diagonalization (ED) energy spectra for a 32-site system at $\Delta=1, 0, 0.5, 1.5, 2$. The ED calculation uses symmetries including the total spin $M_z$ and spin-flip quantum number $Z$. Different colors represent the energy spectra from different symmetry sectors. Dashed vertical lines highlight the level crossing points indicating the quantum phase transitions. The phase diagram obtained by finite-size ED calculations is shown in (f).
• Figure 3: (a) and (d) illustrate the bond energy and magnetic order strength on each site for the E-type and F-type $8\times8$ clusters at $g=0.72$ and $g=0.66$, respectively. The evolution of dimer, plaquette, and AFM order parameters with $g$ is presented for the E-type cluster in (b) and for the F-type cluster in (e). Extrapolations of the order parameters are shown in (c) and (f): linear extrapolations of the plaquette order from the E-type cluster at $g=0.7$ and $0.82$ are displayed in (c), while second-order polynomial extrapolations of the AFM order obtained from both E-type and F-type clusters at $g=1$ are shown in (f).
• Figure 4: (a-c) The order parameter as a function of g for E-type structures when $\Delta = 0, 0.5, 2$ in sizes of 4$\times$4 and 6$\times$6. (d) Phase diagram obtained by collecting the phase boundaries calculated via CMFT for $\Delta = 0, 0.5, 1, 1.5, 2$ in size of 6$\times$6. Solid symbols and solid lines represent first-order phase transitions, while open symbols and dashed lines represent continuous phase transitions.
• Figure 5: (a) Logarithm of the absolute second derivative of the ground-state energy as a function of $g$. The inset shows the corresponding first derivative. (b) Spin structure factor at $\Delta=0$ and $g=0.65$, exhibiting two magnetic Bragg peaks near $(\pi,\pi)$. (c) Evolution of the magnetic Bragg peak positions with $g$. While the $L_y=4$ cylinder exhibits finite-size effects, the $L_y=8$ result at $g=0.68$ (open square) is consistent with the $L_y=6$ data, indicating weaker finite-size effects for $L_y=6$. The inset shows a negligible plaquette VBS order. (d) Real-space spin-spin correlation functions along the $x$ direction for various values of $g$. (e) Schematic representation of spin-spin correlations: between the reference site and all other sites, and the nearest-neighbor spin-spin correlations (or bond energies). The thickness of each bond represents the strength of the nearest-neighbor antiferromagnetic spin correlation. Circles of different colors indicate whether the spin correlation between a lattice site and the reference point (the largest light-blue circle) is positive or negative, while the size of each circle reflects the magnitude of the correlation.