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RVB States in doped Band Insulators from Coloumb forces: Theory and a case study of Superconductivity in BiS$_2$ Layers

Doped band insulators, HfNCl, WO$_3$, diamond, Bi$_2$Se$_3$, \bis2 families, STO/LAO interface, gate doped SrTiO$_3$ and MoS$_2$ etc. are unusual superconductors. With an aim to build a general theory for superconductivity in doped band insulators we focuss on \bis2 family, discovered by Mizuguchi et al. in 2012. While maximum Tc is only $\sim$ 11 K in \laofx, a number of experimental results are puzzling and anomalous; they resemble high Tc and unconventional superconductors. Using a two orbital model of Usui, Suzuki and Kuroki we show that the uniform low density free fermi sea in \laofh is \textit{unstable towards formation of next nearest neighbor Bi-S-Bi diagonal valence bond} (charge -2e Cooper pair) and their \textit{Wigner crystallization}. Instability to this novel state of matter is caused by unscreened nearest neighbor coulomb repulsions (V $\sim$ 1 eV) and a hopping pattern with sulfur mediated diagonal next nearest neighbor Bi-S-Bi hopping t' $\sim$ 0.88 eV, larger than nearest neighbor Bi-Bi hopping, t $\sim$ 0.16 eV. Wigner crystal of Cooper pairs quantum melt for doping around x = 0.5 and stabilize certain resonating valence bond states and superconductivity. We study few variational RVB states and suggest that \bis2 family members are latent high Tc superconductors, but challenged by competing orders and fragile nature of manybody states sustained by unscreened Coulomb forces. One of our superconducting state has d$_{xy}$ symmetry and a gap. We also predict 2d \textit{Bose metal or vortex liquid} normal state, as charge -2e valence bonds survive in the normal state.