Spin-orbit-locked symmetry and the stability of the triangular Dirac spin liquid in the ytterbium delafossites

Abstract

The Yb delafossites NaYbSe$_2$, KYbSe$_2$, CsYbSe$_2$ and NaYbO$_2$ are candidates for the triangular U(1) Dirac spin liquid, QED$_3$ with $N_f=4$, but their exchange is spin-orbit coupled. Under their spin-orbit-locked space group we determine which operators the XXZ anisotropy and the bond-dependent couplings $J_{\pm\pm}$, $J_{z\pm}$ induce, and at what order. Symmetry forbids every Dirac mass, density and current at any order and, assuming adiabatic continuity, any new monopole term through charge three. At linear order the couplings reach only spin-2 members of the quartic multiplets $\mathbf{20}'$ and $\mathbf{84}$, which share their dimensions with Heisenberg-allowed operators. At second order $J_{\pm\pm}$ and $J_{z\pm}$ renormalize the anisotropy through $J_{z\pm}^2-2J_{\pm\pm}^2$, and nine spin-1 quartics without Heisenberg partners appear, irrelevant at $O(1/N)$. A single layer of a stable Heisenberg spin liquid thus acquires no relevant Lorentz-scalar operator from spin-orbit coupling. The Heisenberg liquid itself allows a Lorentz-vector $\mathbf{20}'$ quartic of large-$N$ dimension 2.92, with two-loop $ε$-expansion estimates 2.9-3.5. Rhombohedral stacking adds triplet-monopole tunneling between layers at first order. At the transition to 120$^\circ$ order the anisotropy is relevant, with crossover exponent $y_T\simeq0.74$-1, and exact diagonalization shows that easy-plane anisotropy selects the chiral-Ising class. The out-of-plane spectrum at the K point then stays gapless, and the spin-flip to non-spin-flip ratio grows as $ω^{-0.56}$ at leading order. Existing KYbSe$_2$ neutron data below $J$ do not fix the K-point exponent. In-plane fields induce a chiral mass and an out-of-plane moment proportional to $\sin3φ$, zero along the bonds and canceling between CsYbSe$_2$ layers, which the planar thermal Hall effect can test.