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 ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-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 interlayer $V$, and the pair-hopping $J$ interactions. We evaluate the vertex diagrams that contribute to the pairing susceptibility for $\ensuremath{\ell}=2,\phantom{\rule{4pt}{0ex}}3,\phantom{\rule{4pt}{0ex}}4$ layered models in the $U\text{\ensuremath{-}}V\text{\ensuremath{-}}J$ 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, ${P}_{d}^{\ensuremath{\parallel}}$, and off-diagonal channel, ${P}_{d}^{\ensuremath{\perp}}$, 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 $\ensuremath{\ell}$ and is primarily mediated by the ${P}_{d}^{\ensuremath{\perp}}$ component and find which coefficients and diagram sets are responsible. Finally, we construct a generalized $\ensuremath{\ell}$-dependent equation for ${P}_{d}^{\ensuremath{\perp}}$ to speculate pairing beyond $\ensuremath{\ell}=4$.