Unified gauge-theory description of quantum spin liquids on square-based frustrated lattices

Abstract

Quantum spin liquids are commonly thought to be highly sensitive to lattice geometry, symmetry, and microscopic exchange patterns, leading to a proliferation of seemingly distinct phases across frustrated magnets. Here, we provide a framework that unifies phases that appear distinct from the viewpoint of this intuition. We postulate that the spin-$\tfrac{1}{2}$ Heisenberg antiferromagnets on the square, Shastry-Sutherland, and checkerboard lattices can realize a single unified quantum phase: a gapless $\mathbb{Z}_2$ Dirac quantum spin liquid, despite their markedly different lattice symmetries. Using a systematic projective symmetry group analysis, we identify a checkerboard spin-liquid state that completes a closed set of adiabatically connected phases linking the well-established square-lattice and Shastry-Sutherland spin liquids. Crucially, we show that this lattice-level unification is mirrored exactly in the continuum description. In all three cases, the spin liquids descend from a common SU(2) $π$-flux parent state and are governed by the same gauge theory, QED$_3$ with two Dirac fermion flavors coupled to two adjoint Higgs fields. As a result, we postulate that the surrounding Néel and valence-bond-solid phases and their confinement transitions admit a unified interpretation within the framework of deconfined quantum criticality. More broadly, our results suggest that quantum spin liquids are most fundamentally classified not by lattice geometry or microscopic couplings, but by the emergent gauge theory and its Higgs structure. Distinct frustrated lattices can thus host the same quantum phase and exhibit the same confinement mechanisms, despite substantial differences in their microscopic symmetries.

Figures (6)

• Figure 1: Lattice geometries and symmetry hierarchy of the square, Shastry--Sutherland (SS), and checkerboard models. (a) Square lattice with minimal generators of the $p4mm$ wallpaper group. (b) Shastry--Sutherland lattice ($p4g$), obtained by symmetry reduction via diagonal bond decoration. (c) Checkerboard lattice ($p4gm$). (d) Heisenberg exchange pattern at the lowest common symmetry level, corresponding to the SS geometry. The limit $J_{\boxtimes'}=J_{\boxtimes}$ yields the checkerboard lattice, while $J_{\boxtimes'}=J_{\boxtimes}=J_{\square}$ yields the square lattice.
• Figure 2: Triangle of adiabatically connected $\mathbb{Z}_2$ Dirac spin liquids on the square (Z2A$zz$13), Shastry--Sutherland ($\mathrm{Z3000}$), and checkerboard ($\mathrm{Z05}$) lattices. Continuous interpolation of symmetry-allowed hopping and pairing parameters preserves both the projective symmetry group (PSG) and the invariant gauge group (IGG), showing that all three states belong to the same class of quantum order. Representative Dirac dispersions are shown for selected parameter values along the interpolation. In all cases, the low-energy structure remains that of the same Dirac spin liquid.
• Figure 3: Mean-field phase diagram of the Higgs potential Eq. \ref{['eq:Higgs_potential']} for the square, Shastry--Sutherland, and checkerboard lattices. Dashed (solid red) lines indicate second (first) order transitions. We choose the mean-field parameters to be $w=u=1$, $v_2=-1$, $\tilde{u}=0.75$, and $v_4=0.5$ and employ the Ansatz$\Phi_1^a = c_1 \delta_{a,x}$, $\Phi_2^a = c_1 \delta_{a,y}$, and $\Phi_3^a = c_2 \delta_{a,z}$. Independent of the stable, gapless $\mathbb{Z}_2$ spin liquid---Z2A$zz$13 (square) Wen-2002Senthil-2000, Z3000 (Shastry--Sutherland) or Z05 (checkerboard)---the phase diagram is qualitatively identical. We assume the $\mathrm{SU}(2)$$\pi$-flux state confines to a Néel state whilst the $\mathrm{U}(1)$ staggered flux state confines to a columnar or plaquette VBS state, consistent with earlier checkerboard studies finding plaquette, staggered spin-Peierls, and crossed-dimer valence-bond orders in different regimes Bernier-2004Starykh-2005Bishop-2012. If confinements are reversed, the arrow direction (increasing $J_2$ for square, $J_d$ for Shastry--Sutherland, $J_\boxtimes$ for checkerboard) reverses. As the confinement mechanisms are the same across lattices, we postulate that also the ordered phases are adiabatically connected between the models.
• Figure 4: (a) SU(2) fluxes characterizing the U(1) Ansätze, defined with respect to a base site indicated by the black circle. (b)–(d) Illustrations of the 2-, 4-, and 8-site unit cells required to realize different classes of QSL Ansätze, respectively.
• Figure 5: Sign structure of all U(1) Ansätze. Gray (black) dots on the lattice sites denote positive (negative) signs of the chemical potential, while solid and dashed lines or arrows represent positive and negative hopping amplitudes, respectively. Among these Ansätze, U01--U16 consist only of real hoppings, whereas the remaining states involve complex hoppings. The directions of the complex link fields $u^{\space}_{ij}=\dot\iota t^{0}_{ij}\tau^0+t^{z}_{ij}\tau^z$ are indicated by arrows. The fluxes $(\varphi^{}_{\boxtimes},\varphi^{}_{\square},\varphi^{}_{\triangle,1},\varphi^{}_{\triangle,2},\varphi^{}_{\triangle,3},\varphi^{}_{\triangle,4})$ are also shown for each Ansatz.