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A mathematical account of the NEGF formalism. (English) Zbl 1383.81368

Summary: 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.

MSC:

81V70 Many-body theory; quantum Hall effect
35J08 Green’s functions for elliptic equations
81S22 Open systems, reduced dynamics, master equations, decoherence
82C22 Interacting particle systems in time-dependent statistical mechanics
82C70 Transport processes in time-dependent statistical mechanics
81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
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[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) · Zbl 1034.82008
[2] Aschbacher, W.; Jakšić, V.; Pautrat, Y.; Pillet, C-A; Attal, S. (ed.); Joye, A. (ed.); Pillet, C-A (ed.), Topics in non-equilibrium quantum statistical mechanics, No. 1882 (2006), Berlin
[3] Aschbacher, W., Jakšić, V., Pautrat, Y., Pillet, C.-A.: Transport properties of quasi-free Fermions. J. Math. Phys 48, 032101-1-28 (2007) · Zbl 1137.82331
[4] Araki, H., Moriya, H.: Joint extension of states of subsystems for a CAR system. Commun. Math. Phys. 237, 105-122 (2003) · Zbl 1041.46038 · doi:10.1007/s00220-003-0832-6
[5] Aschbacher, W., Pillet, C.-A.: Non-equilibrium steady states of the XY chain. J. Stat. Phys. 112, 1153-1175 (2003) · Zbl 1032.82020 · doi:10.1023/A:1024619726273
[6] Ben Sâad, R., Pillet, C.-A.: A geometric approach to the Landauer-Büttiker formula. J. Math. Phys. 55, 075202 (2014) · Zbl 1297.82033 · doi:10.1063/1.4879238
[7] Bratelli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics, 2nd edn. Springer, New York (1997) · doi:10.1007/978-3-662-03444-6
[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) · doi:10.1088/0022-3719/4/8/018
[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) · doi:10.1088/0022-3719/4/16/025
[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) · doi:10.1088/0022-3719/5/1/006
[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) · Zbl 1152.81386 · doi:10.1063/1.2992839
[12] Cini, M.: Time-dependent approach to electron transport through junctions: general theory and simple applications. Phys. Rev. B 22, 5887-5899 (1980) · doi:10.1103/PhysRevB.22.5887
[13] Cornean, H.D., Jensen, A., Moldoveanu, V.: A rigorous proof of the Landauer-Büttiker formula. J. Math. Phys. 46, 042106 (2005) · Zbl 1067.82055 · doi:10.1063/1.1862324
[14] Cornean, H.D., Moldoveanu, V.: On the cotunneling regime of interacting quantum dots. J. Phys. A: Math. Theor. 44, 305002 (2011) · Zbl 1221.81080 · doi:10.1088/1751-8113/44/30/305002
[15] Cornean, H.D., Moldoveanu, V., Pillet, C.-A.: Nonequilibrium steady states for interacting open systems: exact results. Phys. Rev. B 84, 075464 (2011) · doi:10.1103/PhysRevB.84.075464
[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) · Zbl 1302.82095 · doi:10.1007/s00220-014-1925-0
[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) · doi:10.1088/0022-3719/4/16/026
[18] Craig, R.A.: Perturbation expansion for real-time Green’s functions. J. Math. Phys. 9, 605-611 (1968) · doi:10.1063/1.1664616
[19] Danielewicz, P.: Quantum theory of nonequilibrium processes I. Ann. Phys. 152, 239-304 (1984) · doi:10.1016/0003-4916(84)90092-7
[20] Dereziński, J., Gérard, C.: Mathematics of Quantization and Quantum Fields. Cambridge University Press, Cambridge (2013) · Zbl 1271.81004 · doi:10.1017/CBO9780511894541
[21] Fröhlich, J., Merkli, M., Ueltschi, D.: Dissipative transport: thermal contacts and tunneling junctions. Ann. Henri Poincaré 4, 897-945 (2004) · Zbl 1106.82021 · doi:10.1007/s00023-003-0150-8
[22] Fujita, S.: Partial self-energy parts of Kadanoff-Baym. Physica 30, 848-856 (1964) · doi:10.1016/0031-8914(64)90127-2
[23] Fetter, A.L., Walecka, J.D.: Quantum Theory of Many-Particle Systems. Dover Publications, New York (2003) · Zbl 1191.70001
[24] Gell-Mann, M., Low, F.: Bound states in quantum field theory. Phys. Rev. 84, 350-354 (1951) · Zbl 0044.23301 · doi:10.1103/PhysRev.84.350
[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)
[26] Imry, Y.: Introduction to Mesoscopic Physics. Oxford University Press, Oxford (1997)
[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) · Zbl 1104.82039 · doi:10.1007/s00220-006-0004-6
[28] Jakšić, V., Ogata, Y., Pillet, C.-A.: Linear response theory for thermally driven quantum open systems. J. Stat. Phys. 123, 547-569 (2006) · Zbl 1101.82029 · doi:10.1007/s10955-006-9075-1
[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) · Zbl 1375.82064 · doi:10.1007/s00023-007-0327-7
[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) · Zbl 0990.82017 · doi:10.1007/s002200200602
[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) · doi:10.1103/PhysRevB.50.5528
[32] Kadanoff, L.P., Baym, G.: Quantum Statistical Mechanics. Benjamin, New York (1962) · Zbl 0115.22901
[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)
[34] Kubo, R., Toda, M., Hashitsume, N.: Statistical Physics II. Nonequilibrium Statistical Mechanics. Springer, Berlin (1985) · Zbl 0996.60501
[35] Langreth, DC; Devreese, JT (ed.); Doren, VE (ed.), Linear and nonlinear response theory with applications, No. 17 (1976), New York
[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) · Zbl 1193.82023 · doi:10.1007/s00023-007-0346-4
[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) · doi:10.1103/PhysRevB.80.115107
[38] Meir, Y., Wingreen, N.S.: Landauer formula for the current through an interacting electron region. Phys. Rev. Lett. 68, 2512-2515 (1992) · doi:10.1103/PhysRevLett.68.2512
[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) · Zbl 1137.82318 · doi:10.1063/1.2712418
[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)
[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) · doi:10.1103/PhysRevB.82.085426
[42] Ness, H.: Nonequilibrium distribution functions for quantum transport: universality and approximation for the steady state regime. Phys. Rev. B 89, 045409 (2014) · doi:10.1103/PhysRevB.89.045409
[43] Stefanucci, G., Almbladh, C.-O.: Time-dependent partition-free approach in resonant tunneling systems. Phys. Rev. B 69, 195318 (2004) · doi:10.1103/PhysRevB.69.195318
[44] Schwinger, J.: Brownian motion of a quantum oscillator. J. Math. Phys. 2, 407-432 (1961) · Zbl 0098.43503 · doi:10.1063/1.1703727
[45] Stefanucci, G., van Leeuwen, R.: Nonequilibrium Many-Body Theory of Quantum System. A Modern Introduction. Cambridge University Press, Cambridge (2013) · Zbl 1276.82002 · doi:10.1017/CBO9781139023979
[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) · doi:10.1103/PhysRevB.82.155108
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