Theory of Non-Equilibirum States Driven by Constant Electromagnetic Fields: Non-Commutative Quantum Mechanics in the Keldysh Formalism

Details Theory of Non-Equilibirum States Driven by Constant Electromagnetic Fields: Non-Commutative Quantum Mechanics in the Keldysh Formalism Theory of Non-Equilibirum States Driven by Constant Electromagnetic Fields: Non-Commutative Quantum Mechanics in the Keldysh Formalism

2006-05-14

arxiv.org/abs/cond-mat/0605363v3

Prog.Theor.Phys. 116 (2006) 61

Physics

Condensed Matter

Statistical Mechanics

26 pages, PTPtex; final version to appear in Prog. Theor. Phys

Scientific paper

10.1143/PTP.116.61

We develop a general theory of non-equilibrium states based on the Keldysh formalism, in particular, for charged-particle systems under static uniform electromagnetic fields. The Dyson equation for the uniform stationary state is rewritten in a compact gauge-invariant form by using the Moyal product in the phase space of energy-momentum variables, which originally do not commute in the case of the conventional operator algebra. Expanding the Dyson equation in electromagnetic fields, a systematic method for the order-by-order calculation of linear and non-linear responses from the zeroth-order Green's functions is obtained. In particular, we find that for impurity problems, up to linear order in the electric field, the present approach provides a diagrammatic method for the St\u{r}eda formula. This approach also generalizes the semi-classical Boltzmann transport theory to fully quantum-mechanical and/or multi-component systems. In multi-component systems and/or for Hall transport phenomena, however, this quantum Boltzmann transport theory, constructed from the anti-symmetric combination of two different representations for the Dyson equation, does not uniquely specify the non-equilibrium state, but the symmetric combination is required. We demonstate the formalism to calculate longitudinal and Hall electric conductivities in an isotropic single-band electron system in the clean limit. It is found that the results are fully consistent with those obtained by Mott and Ziman in terms of the semi-classical Boltzmann transport theory.

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