Spin-imbalanced superconducting state {#subsec:basecase}
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Having confirmed the accuracy of DMC simulations in the spin-balanced case, we now turn to the simplest class of spin-imbalanced systems with a 2:1 ratio of states on the Fermi surfaces, and so communal pairing theory predicts a $({N_\uparrow},{N_\downarrow}) = (2,1)$ communal pairing instability while FFLO theory predicts pairs with nonzero net momentum. We conduct our study on the triangular lattice with 61 spin-up and 19 spin-down fermions, and in the square lattice with 25 spin-up and 9 spin-down fermions. On both lattices we use $a=6.0{r_\mathrm{s}}$ and ${r_\mathrm{e}}=0$.

![Momentum density $n$ of the majority up (red) and minority down (blue) spin-species on the triangular lattice. The vertical dashed lines indicate the respective Fermi momenta and the the horizontal black dashed line denotes $n=1$. The finite slope at the Fermi momenta are due to the finite resolution of the momentum space lattice.[]{data-label="momden"}](momden3){width="0.9\linewidth"}

### Momentum density

We first examine the momentum density, with the results on the triangular lattice in Fig. \[momden\]. Both spin-species have momentum density close to unity beneath their respective Fermi momenta and close to zero above. A breached superfluid [@breachsupercon1; @breachsupercon2; @breachsupercon3] would have the majority species exhibit depletion at the minority species 