Annales Henri Poincaré

, Volume 19, Issue 2, pp 411–442 | Cite as

A Mathematical Account of the NEGF Formalism

  • Horia D. Cornean
  • Valeriu Moldoveanu
  • Claude-Alain Pillet


The main goal of this paper is to put on solid mathematical grounds the so-called non-equilibrium Green’s function transport formalism for open systems. In particular, we derive the Jauho–Meir–Wingreen formula for the time-dependent current through an interacting sample coupled to non-interacting leads. Our proof is non-perturbative and uses neither complex-time Keldysh contours nor Langreth rules of ‘analytic continuation.’ We also discuss other technical identities (Langreth, Keldysh) involving various many-body Green’s functions. Finally, we study the Dyson equation for the advanced/retarded interacting Green’s function and we rigorously construct its (irreducible) self-energy, using the theory of Volterra operators.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Araki, H., Ho, T.G.: Asymptotic time evolution of a partitioned infinite two-sided isotropic XY-chain. Proc. Steklov Inst. Math. 228, 191–204 (2000)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Aschbacher, W., Jakšić, V., Pautrat, Y., Pillet, C.-A.: Topics in non-equilibrium quantum statistical mechanics. In: Attal, S., Joye, A., Pillet, C.-A. (eds.) Open Quantum Systems III. Recent Developments, Lecture Notes in Mathematics, vol. 1882. Springer, Berlin (2006)Google Scholar
  3. 3.
    Aschbacher, W., Jakšić, V., Pautrat, Y., Pillet, C.-A.: Transport properties of quasi-free Fermions. J. Math. Phys 48, 032101-1–28 (2007)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Araki, H., Moriya, H.: Joint extension of states of subsystems for a CAR system. Commun. Math. Phys. 237, 105–122 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Aschbacher, W., Pillet, C.-A.: Non-equilibrium steady states of the XY chain. J. Stat. Phys. 112, 1153–1175 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ben Sâad, R., Pillet, C.-A.: A geometric approach to the Landauer–Büttiker formula. J. Math. Phys. 55, 075202 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bratelli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics, 2nd edn. Springer, New York (1997)CrossRefGoogle Scholar
  8. 8.
    Caroli, C., Combescot, R., Nozières, P., Saint-James, D.: Direct calculation of the tunneling current. J. Phys. C: Solid State Phys. 4, 916–929 (1971)ADSCrossRefGoogle Scholar
  9. 9.
    Caroli, C., Combescot, R., Lederer, D., Nozières, P., Saint-James, D.: A direct calculation of the tunneling current II. Free electron description. J. Phys. C: Solid State Phys. 4, 2598–2610 (1971)ADSCrossRefGoogle Scholar
  10. 10.
    Caroli, C., Combescot, R., Nozières, P., Saint-James, D.: A direct calculation of the tunneling current IV. Electron–phonon interaction effects. J. Phys. C: Solid State Phys. 5, 21–42 (1972)ADSCrossRefGoogle Scholar
  11. 11.
    Cornean, H.D., Duclos, P., Nenciu, G., Purice, R.: Adiabatically switched-on electrical bias and the Landauer–Büttiker formula. J. Math. Phys. 49, 102106 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cini, M.: Time-dependent approach to electron transport through junctions: general theory and simple applications. Phys. Rev. B 22, 5887–5899 (1980)ADSCrossRefGoogle Scholar
  13. 13.
    Cornean, H.D., Jensen, A., Moldoveanu, V.: A rigorous proof of the Landauer–Büttiker formula. J. Math. Phys. 46, 042106 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Cornean, H.D., Moldoveanu, V.: On the cotunneling regime of interacting quantum dots. J. Phys. A: Math. Theor. 44, 305002 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Cornean, H.D., Moldoveanu, V., Pillet, C.-A.: Nonequilibrium steady states for interacting open systems: exact results. Phys. Rev. B 84, 075464 (2011)ADSCrossRefGoogle Scholar
  16. 16.
    Cornean, H.D., Moldoveanu, V., Pillet, C.-A.: On the steady state correlation functions of open interacting systems. Commun. Math. Phys. 331, 261–295 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Combescot, R.: A direct calculation of the tunneling current III. Effect of localized impurity states in the barrier. J. Phys. C: Solid State Phys. 4, 2611–2622 (1971)ADSCrossRefGoogle Scholar
  18. 18.
    Craig, R.A.: Perturbation expansion for real-time Green’s functions. J. Math. Phys. 9, 605–611 (1968)ADSCrossRefGoogle Scholar
  19. 19.
    Danielewicz, P.: Quantum theory of nonequilibrium processes I. Ann. Phys. 152, 239–304 (1984)ADSCrossRefGoogle Scholar
  20. 20.
    Dereziński, J., Gérard, C.: Mathematics of Quantization and Quantum Fields. Cambridge University Press, Cambridge (2013)CrossRefzbMATHGoogle Scholar
  21. 21.
    Fröhlich, J., Merkli, M., Ueltschi, D.: Dissipative transport: thermal contacts and tunneling junctions. Ann. Henri Poincaré 4, 897–945 (2004)ADSCrossRefzbMATHGoogle Scholar
  22. 22.
    Fujita, S.: Partial self-energy parts of Kadanoff–Baym. Physica 30, 848–856 (1964)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Fetter, A.L., Walecka, J.D.: Quantum Theory of Many-Particle Systems. Dover Publications, New York (2003)zbMATHGoogle Scholar
  24. 24.
    Gell-Mann, M., Low, F.: Bound states in quantum field theory. Phys. Rev. 84, 350–354 (1951)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Haugh, H., Jauho, A.-P.: Quantum Kinetics in Transport and Optics of Semiconductors. Springer Series in Solid State Sciences, vol. 123, 2nd edn. Springer, Berlin (2007)Google Scholar
  26. 26.
    Imry, Y.: Introduction to Mesoscopic Physics. Oxford University Press, Oxford (1997)Google Scholar
  27. 27.
    Jakšić, V., Ogata, Y., Pillet, C.-A.: The Green–Kubo formula and the Onsager reciprocity relations in quantum statistical mechanics. Commun. Math. Phys. 265, 721–738 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Jakšić, V., Ogata, Y., Pillet, C.-A.: Linear response theory for thermally driven quantum open systems. J. Stat. Phys. 123, 547–569 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Jakšić, V., Ogata, Y., Pillet, C.-A.: The Green–Kubo formula for locally interacting fermionic open systems. Ann. Henri Poincaré 8, 1013–1036 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Jakšić, V., Pillet, C.-A.: Non-equilibrium steady states of finite quantum systems coupled to thermal reservoirs. Commun. Math. Phys. 226, 131–162 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Jauho, A.-P., Wingreen, N.S., Meir, Y.: Time-dependent transport in interacting and noninteracting resonant-tunneling systems. Phys. Rev. B 50, 5528–5544 (1994)ADSCrossRefGoogle Scholar
  32. 32.
    Kadanoff, L.P., Baym, G.: Quantum Statistical Mechanics. Benjamin, New York (1962)zbMATHGoogle Scholar
  33. 33.
    Keldysh, L.V.: Diagram technique for nonequilibrium processes. Zh. Eksp. Teor. Fiz. 47, 1515 (1964). English translation in Sov. Phys. JETP 20, 1018–1026 (1965)Google Scholar
  34. 34.
    Kubo, R., Toda, M., Hashitsume, N.: Statistical Physics II. Nonequilibrium Statistical Mechanics. Springer, Berlin (1985)zbMATHGoogle Scholar
  35. 35.
    Langreth, D.C.: Linear and nonlinear response theory with applications. In: Devreese, J.T., van Doren, V.E. (eds.) Linear and Nonlinear Electron Transport in Solids. NATO Advanced Study Institute, Series B: Physics, vol. 17. Plenum Press, New York (1976)Google Scholar
  36. 36.
    Merkli, M., Mück, M., Sigal, I.M.: Theory of non-equilibrium stationary states as a theory of resonances. Ann. Henri Poincaré 8, 1539–1593 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Myohanen, P., Stan, A., Stefanucci, G., van Leeuwen, R.: Kadanoff–Baym approach to quantum transport through interacting nanoscale systems: from the transient to the steady-state regime. Phys. Rev. B 80, 115107 (2009)ADSCrossRefGoogle Scholar
  38. 38.
    Meir, Y., Wingreen, N.S.: Landauer formula for the current through an interacting electron region. Phys. Rev. Lett. 68, 2512–2515 (1992)ADSCrossRefGoogle Scholar
  39. 39.
    Nenciu, G.: Independent electrons model for open quantum systems: Landauer–Büttiker formula and strict positivity of the entropy production. J. Math. Phys. 48, 033302 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Ness, H., Dash, L.K.: Dynamical equations for time-ordered Green’s functions: from the Keldysh time-loop contour to equilibrium at finite and zero temperature. J. Phys.: Condens. Matter 24, 505601 (2012)Google Scholar
  41. 41.
    Ness, H., Dash, L.K., Godby, R.W.: Generalization and applicability of the Landauer formula for nonequilibrium current in the presence of interactions. Phys. Rev. B 82, 085426 (2010)ADSCrossRefGoogle Scholar
  42. 42.
    Ness, H.: Nonequilibrium distribution functions for quantum transport: universality and approximation for the steady state regime. Phys. Rev. B 89, 045409 (2014)ADSCrossRefGoogle Scholar
  43. 43.
    Stefanucci, G., Almbladh, C.-O.: Time-dependent partition-free approach in resonant tunneling systems. Phys. Rev. B 69, 195318 (2004)ADSCrossRefGoogle Scholar
  44. 44.
    Schwinger, J.: Brownian motion of a quantum oscillator. J. Math. Phys. 2, 407–432 (1961)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Stefanucci, G., van Leeuwen, R.: Nonequilibrium Many-Body Theory of Quantum System. A Modern Introduction. Cambridge University Press, Cambridge (2013)CrossRefzbMATHGoogle Scholar
  46. 46.
    von Friesen, P.M., Verdozzi, V., Almbladh, C.-O.: Kadanoff–Baym dynamics of Hubbard clusters: performance of many-body schemes, correlation-induced damping and multiple steady and quasi-steady states. Phys. Rev. B 82, 155108 (2010)ADSCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  • Horia D. Cornean
    • 1
  • Valeriu Moldoveanu
    • 2
  • Claude-Alain Pillet
    • 3
  1. 1.Department of Mathematical SciencesAalborg UniversityAalborgDenmark
  2. 2.National Institute of Materials PhysicsBucharest-MagureleRomania
  3. 3.Aix Marseille UnivUniversité de Toulon, CNRS, CPTMarseilleFrance

Personalised recommendations