Demystify the mixed-parity pairing of attractive fermions with spin-orbit coupling in optical lattice

The admixture of spin-singlet and spin-triplet pairing states in superconductors can be typically induced by breaking spatial inversion symmetry. Employing the {\it numerically exact} auxiliary-field Quantum Monte Carlo method, we study such mixed-parity pairing phenomena of attractive fermions with Rashba spin-orbit coupling (SOC) in two-dimensional optical lattice at finite temperature. We systematically demystify the evolution of the essential pairing structure in both singlet and triplet channels versus the temperature, fermion filling, SOC and interaction strengths, via computing the condensate fraction and pair wave function. Our numerical results reveal that the singlet channel dominates in the fermion pairing and the triplet pairing has relatively small contribution to the superfluidity for physically relevant parameters. In contrast to the singlet channel mainly consisted of the on-site Cooper pairs, the triplet pairing has plentiful patterns in real space with the largest contributions from several nearest neighbors. As the SOC strengh increases, the pairing correlation is firstly enhanced and then suppressed for triplet pairing while it's simply weakened in singlet channel. We have also obtained the Berezinskii-Kosterlitz-Thouless transition temperatures through the finite-size analysis of condensate fraction. Our results can serve as quantitative guide for future optical lattice experiments as well as accurate benchmarks for theories and other numerical methods.