Out of Equilibrium Quantum Statistical Physics-S3

Out of Equilibrium Quantum Statistical Physics

Contact: Pr. Thierry Martin


As a starting point, one proceeds with an introduction of non- relativistic quantum field theory with the goal to describe a system of interacting fermions (Coulomb interaction, electron- phonon interaction, or impurity averaged collision with a potential for disorder). We start with the zero-temperature formalism, which exploits the adiabatic switching hypothesis of the interaction potential in order to compute average values of operators in the Heisenberg picture. On establishes a dictionary between the Heisenberg picture and the Interaction picture were the Wick theorem can be used. Single particle Green’s functions are introduced for fermions and bosons, and one shows how perturbation theory can be used to express these quantities in terms of Feynman diagrams. The analytic properties of such Green function allow to introduce the concept of quasiparticles. The self energy of the Dyson series is introduced at this end of this first part.
Next, the non-equilibrium field theoretical framework of quantum statistical mechanics of Keldysh and Schwinger is introduced. Green’s functions thus become two by two matrices as each time argument can be placed on the top or bottom part of the Keldysh contour. This allows to treat open systems and in particular to compute quantum transport quantities such as current and the noise (the current current correlation function in time, subsequently Fourier transformed).
We apply this formalism to several situations : 1) the calculation of the current through a tunnel barrier (a quantum point contact) to all orders in the tunneling amplitude. 2) the calculation of transport through a generalized nano-object such as a single molecule ; 3) the calculation of transport in superconducting systems using the Nambu formalism. In these systems if the starting Hamiltonian is quadratic in the fermions degrees of freedom, the Dyson series can be summed to all orders.
We will also provide an introduction to one-dimensional fermionic systems where the Tomonaga Luttinger formalism allow to treat interaction in a non-perturbative manner : this is called the bosonization technique, as fermion operators are expressed in terms of an exponential of bosnic fields. The price to pay in the calculation of thermodynamical quantities is that the tunnel Hamiltonian is non-quadratic in the bosonic fields and it has to be dealt with using perturbation theory. An application to the edge state picture of the fractional quantum Hall effect (where quasiparticle excitations propagate on the edges of a two-dimensional electron gas will allow to compute the current and noise, and thus to compute its ratio with provides a direct determination of the non-integer charge of these excitations for the Laughlin sequence of the fractional quantum Hall effect.



  • Abrikosov, Gorkov and Dyalishinsky, "quantum field theory methods fo statistical physics",
  • Rammer "quantum field theory of non-equilibrium states",
  • Altland and Simons "Condensed matter field theory", 
  • Kamenev "Field theory of non- equilibrium systems",
  • T. Martin, "Noise in mesoscopic physics" in «Nanophysics: Coherence and Transport » École d'été de Physique des Houches Session LXXXI H. Bouchiat, S. Gueron, G. Montambaux, J. Dalibard eds