Renormalized Mean Field Theory

Abstract

To deal with the issues mentioned in the previous chapter, we chose to follow certain numerical approaches. Like the protocol most of us would follow when encountering a physical scenario, one selects a proper Hamiltonian that we conjecture to be most likely capable of reflecting the system of interest. Here, we chose the strongly correlative $t-J$ Hamiltonian as our model Hamiltonian to investigate. Normally, in solving for such a strongly correlative system, a heavy amount of calculation is involved since there is no easy way to construct its low energy states. But thanks to M. Gutzwiller, he came up with an approximation based on the idea that one can merge the effect of strong correlation into a projection operator. This brilliant proposal will help alleviate the burden of calculation. In this chapter, we will go through the main method of ours, the renormalized mean field theory, which is a mean-field extension of Gutzwiller approximation, in detail starting from the $t-J$ Hamiltonian. We will also demonstrate how we calculate some of the key properties such as local density of states and spectra weight with our Bogoliubov-deGenne wavefunctions. Moreover, after introducing an additional phase which accounts for the external magnetic field, we will show the way of calculating the topological invariants, which is the Chern number in our case, for each mean-field band that we are interested in.