for electrons and holes, respectively.

The Hamiltonian describing electron states in the conduction band, valence band, and a mid-gap defect state is, $$\begin{aligned}
H_{o} & = & \sum_{\vec{k},s}E_{c,s}(\vec{k}) c^{\dagger}_{\vec{k},s} c_{\vec{k},s} + \sum_{\vec{k},s}E_{v,s}(\vec{k}) b^{\dagger}_{\vec{k},s} b_{\vec{k},s} \nonumber \\
& & + \sum_{\sigma}E_{d} d^{\dagger}_{\sigma}d_{\sigma}\end{aligned}$$ Here, $c_{\vec{k},s}$, $b_{\vec{k},s}$, and $d_{\sigma}$ are the destruction operators for the conduction band, valence band, and defect states, respectively. The bandgap is $E_{g_{s,s'}} = E_{c,s}(\vec{k}=0) - E_{v,s'}(\vec{k}=0)$. Since only the smallest bandgap will be relevant in the discussion that follows, we will drop the spin/valley indices from $E_{g_{s,s'}}$ for simplicity.

Electron-Hole Interaction and Exciton States {#sec:Heh}
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The Coulomb interaction between the electrons and holes can be included by adding the following term to the Hamiltonian, $$H_{eh} =  \frac{1}{A} \sum_{\vec{k},\vec{k}',\vec{q},s,s'} V(\vec{q})  F_{s,s'}(\vec{k},\vec{k}',\vec{q}) c^{\dagger}_{\vec{k}+\vec{q},s}  b^{\dagger}_{\vec{k}'-\vec{q},s'} b_{\vec{k}',s'} c_{\vec{k},s}$$ $V(\vec{q})$ is the 2D Coulomb potential and equals $e^{2}/2\epsilon_{o} \epsilon(\vec{q})q$. The wavevector-dependent dielectric constant $\epsilon(\vec{q})$ for monolayer MoS$_{2}$ is given by Zhang et al. [@Changjian14] and Berkelbach et al. [@timothy]. $F_{s,s'}(\vec{k},\vec{k}',\vec{q})$ is [@Efimkin13], $$F_{s,s'}(\vec{k},\vec{k}',\vec{q}) = \langle v_{c,\vec{k}+\vec{q},s}| v_{c,\vec{k},s} \rangle \, \langle v_{v,\vec{k}'-\vec{q},s'} | v_{v,\vec{k}',s'} \rangle \label{eq:F1}$$ Near the conduction band minima, where $\hbar v k << \Delta_{s}$, $\cos(\theta_{\gamma,\vec{k},s}) \approx 1$ and $\sin(\theta_{\gamma,\vec{k},s}) << 1$. Similarly, near the valence band maxima, $\sin(\theta_{\gamma,\vec{k},s}) \approx 1$ 