Unifying Dirac Spin Liquids on Square and Shastry-Sutherland Lattices via Fermionic Deconfined Criticality

TL;DR

This work develops a fermionic SU(2) gauge theory to describe deconfined quantum criticality on the Shastry–Sutherland lattice, showing that its low-energy description matches the square-lattice theory at leading order. Starting from an $ ext{SU}(2)$ π-flux parent state, Dirac spinons couple to an SU(2) gauge field and three adjoint Higgs fields, whose condensates realize transitions to a $U(1)$ staggered-flux spin liquid and a gapless $Z_2$ Dirac spin liquid (Z3000). A controlled large-$N_f$, $N_b$ expansion yields critical exponents and identifies the Yukawa coupling as the relevant perturbation that destabilizes the exact SO(5) fixed point, consistent with pseudocritical scaling seen in Monte Carlo studies. The analysis predicts large anomalous dimensions for the Néel and VBS order parameters, aligning with numerical indications of strong fluctuations near deconfined criticality, and demonstrates that fermionic deconfined criticality extends from square to frustrated lattices with reduced symmetry. Overall, the paper provides a coherent field-theoretic framework for understanding a gapless Z2 spin liquid on SS lattices within the broader SO(5) deconfined-criticality paradigm, and it highlights extended pseudocritical behavior as a robust feature of these systems.

Abstract

We present a fermionic gauge theory for deconfined quantum criticality on the Shastry-Sutherland lattice and reveal its shared low-energy field-theoretic structure with the square lattice. Starting from an SU(2) $π$-flux parent state, we construct a continuum theory of Dirac spinons coupled to an SU(2) gauge field and adjoint Higgs fields whose condensates drive transitions to a staggered-flux U(1) spin liquid and a gapless $\mathbb{Z}_{2}$ Dirac spin liquid. While the Shastry-Sutherland lattice permits additional symmetry-allowed fermion bilinears compared to the square lattice, the quantum field theories are identical up to additional irrelevant terms. Consequently, the Higgs potential structure and the leading low-energy theory coincide with the square-lattice case at the quantum critical point. The SO(5) critical point is expected to realize conformal deconfined criticality: we analyze it in a large flavor expansion, calculate its critical exponents, and identify the Yukawa coupling between the fermions and Higgs fields as the relevant perturbation that destabilizes it, consistent with pseudocritical behavior observed in recent Monte Carlo studies. We show that the emergent SO(5) order parameter acquires a large anomalous dimension at the critical point, leading to strongly enhanced Néel and VBS susceptibilities-a hallmark of fermionic deconfined quantum criticality consistent with numerical studies. Our results place recent numerical evidence for a gapless $\mathbb{Z}_{2}$ Dirac spin liquid on the Shastry-Sutherland lattice within a controlled field-theoretic framework and demonstrate that fermionic deconfined criticality on the square lattice-including critical exponents and stability-extends to frustrated lattices with reduced symmetry.

Figures (11)

• Figure 1: (a) Illustration of the Shastry--Sutherland lattice. The four orbitals within the unit cell are denoted by two sublattice degrees of freedom $(m_x,m_y)=(A,A), (B,A), (B,B), (A,B)$, respectively. (b) Shown are additional diagonal links with the Heisenberg couplings needed to interpolate to the square lattice limit.
• Figure 2: Mean field phase diagram of the Higgs potential in Eq. \ref{['eq:Higgs_potential']}. The dashed (solid red) lines indicate second (first) order transitions. Following Shackleton2021, we choose the mean-field parameters to be $w=u=1$, $v_2=-1$, $\tilde{u}=0.75$, and $v_4=0.5$. We employ the ansatz $\Phi_1^a = c_1 \delta_{a,x}$, $\Phi_2^a = c_1 \delta_{a,y}$, and $\Phi_3^a = c_2 \delta_{a,z}$. We assume that 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. In case the confinements are opposite to our assumption, the direction of the arrow, which indicates a possible trajectory of the Shastry--Sutherland model with increasing $J_d/J_s$, has to be reversed. The $\mathbb{Z}_2$ spin liquid $\mathrm{Z3000}$ is a stable Dirac spin liquid Wen2002Senthil-2000. Our field theoretic analysis shows that it is connected to Wen's Wen2002 gapless spin liquid Z2A$zz$13, consistent with an infinite density matrix renormalization group, exact diagonalization and PSG/Variational Monte Carlo study Maity2024.
• Figure 3: Definition of diagrammatic symbols for the propagators $D_{\mu\nu},D_\lambda,G_\psi,$ and $G_\Phi$.
• Figure 4: Self-energy correction to the fermion propagator due to the gauge field.
• Figure 5: Self-energy corrections to the Higgs propagator due to the gauge field and the Lagrange multiplier field.