( \hat {\mathcal K} + \hat {\mathcal P})} \big)^{l} \big] 
\approx {\rm Tr}\, \big[ 
e^{-\Delta\tau \hat {\mathcal K}} 
e^{-\Delta\tau \hat {\mathcal P}} 
e^{-\Delta\tau \hat {\mathcal K}} 
e^{-\Delta\tau \hat {\mathcal P}} \cdots \big]$. Here $l=\beta/ \Delta \tau$ is the number of incremental time evolution operators. This decomposition has an error proportional to $(\Delta \tau)^2$ and is exact in the limit $\Delta \tau \to 0 $. The resulting isolation of $e^{-\Delta \tau \hat {\mathcal P}}$ allows for the performance of a discrete Hubbard-Stratonovich (HS) transformation so that it can be rewritten in quadratic (single body) form, but with the cost of introducing a discrete auxiliary field with components on each of the space and imaginary time lattice coordinates. The fermions are then integrated out, and the HS field is sampled by the Monte Carlo technique. In the work we report here we choose $t \Delta \tau=0.125$ so that the error from the Trotter-Suzuki decomposition is less than, or comparable to, that from the Monte Carlo sampling. We therefore report error bars from the latter. More details about the method are discussed in Ref. and references therein.

Although DQMC is exact, its low temperature application is restricted to systems with particle-hole or other 