Dzyaloshinskii-Moriya interactions in the SU(2) gauge theory of the Shastry-Sutherland Dirac spin liquids

We classify the fermion bilinears of the SU(2) gauge theory of the Shastry-Sutherland Dirac spin liquids under the spin-orbit-locked space group and treat the Dzyaloshinskii-Moriya (DM) interactions of SrCu${}_2$(BO${}_3$)${}_2$ at leading order in $1\mathrm{/}N$ and at one loop. At linear order DM can neither gap the spinons nor destabilize the SO(5) point above a scale $T^{\ast }$. Symmetry forbids every local gauge-singlet fermion mass at any order. Among the singlet bilinears, the in-plane couplings induce only a conserved spin current of dimension exactly 2, which sets $T^{\ast }=4\lambda \simeq 0.03$-$0.17$ K, and the out-of-plane coupling only a flavored spin-2 current, irrelevant at a unitary critical point. DM-induced Yukawa couplings start below $2\mathrm{\%}$ of the one that bounds the pseudocritical window. In zero field, measured samples order above $1$ K, at or above $T^{\ast }$. Below $T^{\ast }$, in a two-dimensional sample at the critical pressure, the Dirac cones reconstruct into a chirally protected $d$-wave quadratic band touching whose only gauge-singlet gap channels are the two valence-bond components and the scalar chirality. A one-loop flow, with the gauge field screened by the band touching, ends at $T^{\ast \ast }\sim (0.03$-$0.1)\lambda$ on a ray containing valence-bond order for the material’s exchange and for most random couplings, and otherwise on a gapped chiral ${\mathbb{Z}}_2$ spin liquid. The DM bias between the two plaquette patterns is at most of order ${10}^{-4}$, with a direction not fixed at $O(1\mathrm{/}N)$, whereas non-magnetic quenched disorder couples a relevant random field to the plaquette doublet and none to the chirality. A field along $[100]$ induces a $(\pi ,\pi )$ Néel moment linear in the field; a field along $[001]$ does not. This selection rule also holds in the dimer phase and is testable by ${}^{11}$B NMR under pressure.