Pairing susceptibility in the multi-layer Hubbard model

We present a systematic study of the interaction, doping, and layer dependence of the dx²-y²-wave pairing susceptibility of the Hubbard model for a stacked 2D square lattice. We perform a multi-index perturbative expansion up to fourth-order to obtain coefficients in powers of the Hubbard U, the inter-layer V, and the pair-hopping J interactions. We evaluate the vertex diagrams that contribute to the pairing susceptibility for = 2,3, 4 layered models in the U-V-J interaction space. This provides unprecedented access to the pairing amplitudes, allowing us to identify the processes that enhance or reduce pairing. We distinguish pairing within the diagonal channel, P∥d, and off-diagonal channel, P⊥d, and find that, in the absence of J, the qualitative behavior of the layered system is equivalent to the single-layer model. In the presence of J, we show that pairing is enhanced sublinearly with increasing and is primarily mediated by the P⊥d component and find which coefficients and diagram sets are responsible. Finally, we construct a generalized -dependent equation for the d-wave pair susceptibility to speculate pairing beyond =4.