Application of Grassmannian techniques and Kubo formula to the transport of fermions in a two-dimensional frustrated spin medium
Abstract
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. %Contrary to bosonic methods, precise results for non-quadratic actions can be obtained, due to the cancellation of many diagrams involving identical variables or having opposite signs. 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.
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BibTeX
@article{fortin2026application,
title = {{Application of Grassmannian techniques and Kubo formula to the transport of fermions in a two-dimensional frustrated spin medium}},
author = {Jean-Yves Fortin},
journal = {Journal of Physics: Condensed Matter},
year = {2026},
month = jun,
doi = {10.1088/1361-648x/ae6d66},
url = {https://hal.science/hal-05396981v2},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Application of Grassmannian techniques and Kubo formula to the transport of fermions in a two-dimensional frustrated spin medium
AU - Jean-Yves Fortin
PY - 2026
DA - 2026/06/01
JO - Journal of Physics: Condensed Matter
DO - 10.1088/1361-648x/ae6d66
UR - https://hal.science/hal-05396981v2
AB - 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. %Contrary to bosonic methods, precise results for non-quadratic actions can be obtained, due to the cancellation of many diagrams involving identical variables or having opposite signs. 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.
SN - 0953-8984
SN - 1361-648X
ER -