Fractionalization on the Surface: Is Type-II Terminated $1T$-TaS$_2$ Surface an Anomalously Realized Spin Liquid?

The type-II terminated $1T$-TaS$_2$ surface of a three-dimensional $1T$-TaS$_2$ bulk material realizes the effective spin-$1/2$ degree of freedom on each David star cluster with ${\mathcal{T}^2=-1}$ such that the time-reversal symmetry is realized anomalously, despite the fact that bulk three-dimensional $1T$-TaS$_2$ material has an even number of electrons per unit cell with ${\mathcal{T}^2=+1}$. This surface is effectively viewed as a spin-$1/2$ triangular lattice magnet, except with a fully gapped topological bulk. We further propose this surface termination realizes a spinon Fermi surface spin liquid with the surface fractionalization but with a nonexotic three-dimensional bulk. We analyze possible experimental consequences, especially the surface spectroscopic measurements, of the type-II terminated surface spin liquid.