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Wagner, Ingo (2013): Algebraic approach towards conductivity in ergodic media. Dissertation, LMU München: Fakultät für Mathematik, Informatik und Statistik



This thesis is about an operator algebraic approach towards the derivation of the electrical conductivity in disordered solid states based on the theory of quantum many-particle systems. Such an approach is of interest since it allows for the description of interacting electron gases, which is a feature not present in previous work. In the context of the description of ergodic media, new concepts are introduced, such as covariant states and covariant morphisms. Moreover, the concept of covariant states is combined with the well-known concept of KMS states. In its covariant form, KMS states describe electron gases in ergodic media at thermal equilibrium. Such states are the starting point of the electron gases considered here. An external electric field is applied to the system, influences the electron gas and causes internal electric currents. Thus, the equilibrium position of the system is disturbed, leading to a time evolution of the system, which is described by covariant automorphisms. Summing up, the system is given in a time dependent, covariant state that acts on the algebra of bounded and local operators on the fermionic Fock space defined over some given one-particle Hilbert space. For a discrete model of an extended electron gas in one space dimension with a pair interaction of finite range, explicit constructions of the above states are presented. In addition, for the special case of non-interacting electron gases, the construction of the time dependent covariant state is carried out in arbitrary space dimension. Since measurements in a quantum system are implemented by the action of its state on bounded, local and self-adjoint operators, the concept of a current density operator is introduced. The current density is then defined as the result of the measurement of the current density operator. By an application of Birkhoff’s ergodic theorem, the transformation law of the current density operator together with the covariant transformation law of the state of the electron gas implies the almost sure existence of the spatial mean of the current density. Moreover, the spatial mean current density is almost surely independent of the concrete realisation given. The electric current density describes the linear dependence of the spatial mean current density on the external electric field, for small strengths. Via linear response theory for the noninteracting model of an electron gas, an explicit expression for the current density is derived in terms of a so called Kubo formula. For the derivation the system needs to satisfy a localisation condition, which is specifically designed for non-interacting electron gases. In view of a linear response theory of interacting electron gases, candidates for a generalisation of this localisation criterion that also apply to interacting systems are introduced.