Bond disordered spin liquid and the honeycomb iridate H$_3$LiIr$_2$O$_6$ $-$ abundant low energy density of states from random Majorana hopping

The 5d-electron honeycomb compound H$_3$LiIr$_2$O$_3$ [K. Kitagawa $et$ $al.$, Nature 554, 341-345 (2018)] exhibits an apparent quantum spin liquid (QSL) state. In this intercalated spin-orbital compound, a remarkable pile up of low-energy states was experimentally observed in specific heat and nuclear magnetic (NMR) spin relaxation. We show that a bond disordered Kitaev model can naturally account for this phenomenon, suggesting that disorder plays an essential role in its theoretical description. In the exactly soluble Kitaev model, we obtain, via spin fractionalization, a random bipartite hopping problem of Majorana fermions in a random flux background. This has a divergent low-energy density of states (DOS) of the required power-law form $N(E)\propto E^{-\nu}$ with a drifting exponent which takes on the value $\nu \approx 1/2$ for relatively strong bond disorder. Breaking time reversal symmetry (TRS) removes the divergence of the DOS, as does applying a magnetic field in experiment. We discuss the implication of our scenario for future experiments and its broader implications