#### 4.2. Method

Because of the large band gaps, we need only to consider conduction band processes [

16,

29] (see

Figure 1). For each of the monolayers, we label the bottom and top conduction subbands

$\beta =b$ and

$\beta =t$. Due to the large valley separation in momentum space, inter-valley scattering is negligible, so the effect of the two valleys appears only in a valley degeneracy factor,

${g}_{v}=2$.

The multiband electron-hole Hamiltonian is,

For the

n-doped monolayer,

${c}_{\beta ,k}^{\u2020}$ and

${c}_{\beta ,k}$ are the creation and annihilation operators for electrons in conduction subband

$\beta $. For the

p-doped monolayer,

${d}_{\beta ,k}^{\u2020}$ and

${d}_{\beta ,k}$ are the corresponding operators for holes. The kinetic energy terms are

${\xi}_{\beta}^{\left(i\right)}\left(k\right)={\epsilon}_{\beta}^{\left(i\right)}\left(k\right)-{\mu}^{\left(i\right)}$ where

${\epsilon}_{\beta}^{\left(i\right)}\left(k\right)$ is the energy dispersion for the

$i=e,h$ monolayer, as in Equation (

2). We consider only equal carrier densities

${n}^{e}={n}^{h}=n$, so the chemical potentials

$\mu =\frac{1}{2}({\mu}^{\left(e\right)}+{\mu}^{\left(h\right)})$.

${V}_{k\phantom{\rule{0.166667em}{0ex}}{k}^{\prime}}^{D}$ is the bare attractive Coulomb interaction between electrons and holes in the opposite monolayers,

where

d is the thickness of the barrier.

${V}_{k\phantom{\rule{0.166667em}{0ex}}{k}^{\prime}}^{S}$ is the bare repulsive Coulomb interaction between carriers in the same monolayer.

In contrast with conventional BCS pairing, the Coulomb pairing interaction is independent of the electron and hole spins. There are four possible electron-hole pairings, corresponding to four superfluid condensates [

37,

38]. We introduce the temperature dependent normal and anomalous multiband Matsubara Green functions, with subband indices

$\alpha $ and

$\beta $,

where

${T}_{\tau}$ is the time-ordering operator. These result in mean field equations for the gaps and the densities [

37,

38]:

where

${F}_{k{k}^{\prime}}^{\alpha \beta {\alpha}^{\prime}{\beta}^{\prime}}=\u2329{\alpha}^{\prime}{k}^{\prime}\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}\alpha k\u232a\u2329\beta k\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}{\beta}^{\prime}{k}^{\prime}\u232a$ is the form factor that represents the overlap of the single particle wave functions [

39]. Because of the spin polarization in the bands, there is no spin degeneracy,

${g}_{s}=1$.

We determined that the

$\left\{bt\right\}$ and

$\left\{tb\right\}$ cross-pairing terms are negligible. Thus we can focus on intraband pairing, retaining the Green functions and form factors only with

$\alpha =\beta $(

${\alpha}^{\prime}={\beta}^{\prime}$). Since the Coulomb interaction

${V}_{k\phantom{\rule{0.166667em}{0ex}}{k}^{\prime}}^{eh}$ conserves the spin of the electron-hole pair, there are no spin-flip scattering processes. This implies that

${F}_{k{k}^{\prime}}^{\beta \beta {\beta}^{\prime}{\beta}^{\prime}}=0$ for

$\beta \ne {\beta}^{\prime}$, so there is no Josephson-like pair transfer between the bands [

29].

At zero temperature the gap equations are decoupled,

where we abbreviate the notation

${F}_{k{k}^{\prime}}^{\beta \beta {\beta}^{\prime}{\beta}^{\prime}}\equiv {F}_{k{k}^{\prime}}^{\beta {\beta}^{\prime}}$.

${E}_{\beta}\left(k\right)=\sqrt{{\xi}_{\beta}{\left(k\right)}^{2}+{\Delta}_{\beta \beta}^{2}\left(k\right)}$ is the quasi-particle excitation energy for subband

$\beta $, with

${\xi}_{\beta}\left(k\right)=({\xi}_{\beta}^{\left(e\right)}+{\xi}_{\beta}^{\left(h\right)})/2$.

$\theta \left[{E}_{t}^{-}\left(k\right)\right]=1-\mathit{f}[{E}_{t}^{-}\left(k\right),0]$ is a step function, with

$\mathit{f}[{E}_{t}^{-}\left(k\right),0]$ the zero temperature Fermi-Dirac distribution.

${E}_{t}^{\pm}\left(k\right)={E}_{t}\left(k\right)\pm \delta \lambda /2$, with

$\delta \lambda =|{\lambda}_{v}|-|{\lambda}_{c}|$ the energy misalignment of the top bands.

Equation (

8) has the same form as for a decoupled one-band system, because the two bottom bands are aligned [

40]. In contrast, Equation (9) contains the effect of misalignment of the top bands through the term

$\theta \left[{E}_{t}^{-}\left({k}^{\prime}\right)\right]\equiv \theta [\sqrt{{\xi}_{t}{\left(k\right)}^{2}+{\Delta}_{tt}^{2}\left(k\right)}-\delta \lambda /2]$. This term drops below unity only at higher densities where the pair coupling strength is weak compared with the misalignment.

${V}_{k{k}^{\prime}}^{eh}$ in Equations (8) and (9) is the screened electron-hole interaction. We use the linear-response Random Phase Approximation for static screening in the superfluid state [

16],

with

$q=|\mathbf{k}-{\mathbf{k}}^{\prime}|$.

${\Pi}_{n}\left(q\right)$ is the normal static polarizability in the superfluid state and

${\Pi}_{a}\left(q\right)$ is the anomalous static polarizability [

13,

14]. The polarizabilties are obtained as loops consisting of two normal or two anomalous Green functions:

The sum over the Matsubara frequencies, ${\omega}_{n}$, is performed at zero temperature.

${\Pi}_{n}\left(q\right)$ depends on the population of free carriers, while

${\Pi}_{a}\left(q\right)$, with opposite sign, depends on the population of electron-hole pairs.

${\Pi}_{a}\left(q\right)$ is only non-zero in the superfluid state. There is a competition between of

${\Pi}_{n}\left(q\right)$ and

${\Pi}_{a}\left(q\right)$ which is connected to the competition between the populations of the condensed pairs and the free carriers. Screening is associated with the population of charged free carriers. A system with a large superfluid condensate fraction of strong-coupled pairs thus has very weak screening [

6].

For a given chemical potential

$\mu $, the carrier density

n of one monolayer is the sum of the subband carrier densities

${n}_{b}$ and

${n}_{t}$,

where

are the Bogoliubov amplitudes.

The

${\Delta}_{\beta \beta}$ and

$\mu $ shown in

Figure 2 are determined by self consistently solving Equations (

8) and (9) coupled with Equations (14) and (15).

The condensate fractions

${C}_{\beta \beta}$ used in

Figure 3 to identify the weak- and strong-coupled regimes of the superfluid, are given by [

41,

42],

The Berezinskii-Kosterlitz-Thouless transition temperature

${T}_{BKT}$ [

27] for a 2D system with parabolic bands, is determined from [

43,

44],

${\rho}_{s}\left(T\right)$ is the superfluid stiffness.

We note that unlike DBG, there are no chiral symmetry degrees of freedom in TMDs. Reference [

39] has suggested that these additional degrees of freedom in DBG would lead to topological excitations of quarter vortices. If so, in DBG but not in TMDs, there would be a prefactor of

$1/16$ in Equation (

18). This would dramatically reduce

${T}_{BKT}$.