Proximate Dirac spin liquid in honeycomb lattice $J_1$-$J_3$ XXZ model: Numerical study and application to cobaltates

Recent theoretical and experimental work suggest that the honeycomb cobaltates, initially proposed as candidate Kitaev quantum magnets, are in fact described by a pseudospin-$1\mathrm{/}2$ easy-plane spin Hamiltonian with nearest neighbor ferromagnetic (FM) exchange $J_1$ being frustrated by antiferromagnetic third-neighbor exchange $J_3$ and weaker compass anisotropies. Using exact diagonalization and density-matrix renormalization group (DMRG) calculations, we show that this model exhibits FM order at small $J_3\mathrm{/}{J}_1$ and zig-zag (ZZ) order at large $J_3\mathrm{/}{J}_1$, separated by an intermediate phase, which we label as $\widetilde{\mathrm{S}\mathrm{L}}$. This $\widetilde{\mathrm{S}\mathrm{L}}$ phase is shown to exhibit spin-liquid-like correlations in DMRG, although we cannot preclude weak broken symmetries, e.g. weak Ising type Néel order, given the limits on our explored system sizes. Using a modified parton mean field theory and variational Monte Carlo on Gutzwiller projected wavefunctions, we show that the optimal FM and ZZ orders as well as the intermediate $\widetilde{\mathrm{S}\mathrm{L}}$ state are proximate to a `parent’ Dirac spin liquid (SL). This Dirac SL is shown to capture the broad continuum in the temperature and magnetic field dependent terahertz spectroscopy of BaCo${}_2$(AsO${}_4$)${}_2$, and the reported low temperature metallic thermal conductivity in Na${}_2$Co${}_2$TeO${}_6$ and BaCo${}_2$(AsO${}_4$)${}_2$ upon incorporating disorder induced broadening.