Application of Grassmannian techniques and Kubo formula to the transport of fermions in a two-dimensional frustrated spin medium

We review several Grassmann techniques applied to
the transport of itinerant and spinless fermions moving as defects in dilute two-dimensional magnetic media, where they occupy site vacancies. 
Grassmann non-commuting variables are intrinsically related to classical spins, fermions, and field theories. Exact results are provided when a correspondence is made between partition functions of solvable problems and Grassmann integrals which can be expressed as determinants or Pfaffians. These techniques can be applied to both classical models of spins and fermionic problems. In this manuscript, we consider as an example the problem of fermions interacting indirectly via local classical spins in a magnetic medium, and we address the computation of thermal quantities from the Grassmann algebraic tools. The presence of strong non-local interactions between the fluctuations of the magnetic medium and the itinerant fermions can be analyzed via coupled self-consistent mean-field equations, after expressing the spins and fermion Hamiltonian in term of a Grassmannian effective action. We also apply these methods to the electronic transport at finite temperature, and show that Grassmann and Matsubara formalisms are an alternative solution to compute the Kubo conductivities. We also compare the results with the standard technique relying on the spectral decomposition of the Hamiltonian.