Pairing susceptibility in the weakly interacting multilayer Hubbard model evaluated by direct perturbative expansion

We present a systematic study of the interaction, doping, and layer dependence of the $d_{x^2-y^2}$-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 $\ell= 2,3, 4$ 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, $P^{\parallel}_{d}$, and off-diagonal channel, $P^{\perp}_{d}$, 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 $\ell$ and is primarily mediated by the $P^{\perp}_{d}$ component and find which coefficients and diagram sets are responsible. Finally, we construct a generalized $\ell$-dependent equation for $ P^{\perp}_{d}$ to speculate pairing beyond $\ell=4$.
Comment: 21 pages, 9 figures in the main text, 10 figures in the supplemental material (appended)