When the change in binding energy is not negligible for the most weakly bound visible exciton, the band gap can be estimated by extrapolating from the exciton Rydberg series.

Since $W$ depends in a coordinate representation on the positions of the two interacting particles in the semiconductor layer, its Fourier representation is a matrix in momentum space for every $\mathit{q}$ in the Brillouin zone. If atomic scale spatial variations were relevant in $\mathrm{\Delta }W$, the fact that the semiconductor layer and the remote layer have different lattices would complicate its description. $\mathrm{\Delta }W$ would have the lattice periodicity of the remote crystal, and this would complicate its matrix elements in the representation of semiconductor layer Bloch states. Fortunately, the $\exp (-|\mathit{q}+\mathit{G}|d)$ factors in the interactions between layers make these complications irrelevant.

When the TMD is directly adjacent to the correlated fluid layer, we use the dielectric constants of the TMD (in case of ${\mathrm{WSe}}_2$, ${\epsilon }_{\parallel }\simeq 15$ and ${\epsilon }_{zz}\simeq 8$), and in presence of a hBN spacer, we use ${\epsilon }_{\parallel }\simeq 7$ and ${\epsilon }_{zz}\simeq 3$.

We use a phenomenological dielectric constant ${\epsilon }_r=35$ in the Hartree-Fock calculation to obtain a valley polarization phase diagram that is approximately consistent. Otherwise, we use realistic dielectric functions. We also account for gate-screening by using interaction $V_{\mathrm{eff}}(\mathit{q})=2\pi e^2\tanh (d_{\mathrm{gate}}q)/({\epsilon }_{r}q)$ with $d_{\mathrm{gate}}=50\mathrm{nm}$.