Local Density of States of Impurities in Mott Insulators

Motivated by recent scanning tunneling microscope measurements of local density of states on impurities in parent compounds and very lightly doped, insulating compounds of the cuprates high-$T_C$ superconductors (C. Ye et. al. Nat. Commun. {\bf 4}, 1365 (2013), and P. Cai et. al. Nat. Phys. {\bf 12}, 1047 (2016)), we start with the slave rotor description of the Mott insulating phase, and solve the local density of states of the impurities sites using a $T$-matrix approach. Excellent agreement with the experimental results is found. Our calculations account for all the following key features of the observed local density of states: i) positions and amplitudes of the in-gap spectral weights of a single impurity; ii) the spectral weight transfer from the upper Hubbard band to the lower Hubbard band; iii) the difference between two different types of impurities. By generalizing the $T$-matrix solutions to the finite-doping, insulating case, we suggest that the observed suppression of spectral weight at the Fermi energy can be attributed to the protection from the bulk insulating phase.