Imaginary-time formulation of strongly correlated nonequilibrium
Heary, Ryan Joseph
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Strongly correlated nanostructures and lattices of electrons are studied when these systems reside in a steady-state nonequilibrium. Much of the work done to date has made use of the nonequilibrium real-time Keldysh Green function technique. These methods include: the Keldysh Green function perturbation theory, time-dependent numerical renormalization group, density matrix renormalization group, and diagrammatic quantum Monte Carlo. In the special case of steady-state nonequilibrium we construct an imaginary-time theory. The motivation to do this is simple: there exist an abundant number of well-established strongly correlated computational solvers for imaginary-time theory and perturbation theory on the imaginary-time contour is much more straightforward than that of the real-time contour. The first model system we focus on is a strongly interacting quantum dot situated between source and drain electron reservoirs. The steady-state nonequilibrium boundary condition is established by applying a voltage bias Φ across the reservoirs, in turn modifying the chemical potentials of the leads. For a symmetric voltage drop we have μ source = Φ/2 and μ drain = -Φ/2. The dynamics of the electrons are governed by the Hamiltonian [Special characters omitted.] <math> <f> <a><ac><sc>H</sc></ac><ac>&d4;</ac></a></f> </math> which is inherently independent of the imbalance in the source and drain chemical potentials. The statistics though are determined by the operator [Special characters omitted.] <math> <f> <a><ac><sc>H</sc></ac><ac>&d4;</ac></a>-<a><ac><sc>Y</sc></ac><ac>&d4;</ac></a> ,</f> </math> where [Special characters omitted.] <math> <f> <a><ac><sc>Y</sc></ac><ac>&d4;</ac></a></f> </math> imposes the nonequilibrium boundary condition. We show that it is possible to construct a single effective Hamiltonian [Special characters omitted.] <math> <f> <a><ac><sc>K</sc></ac><ac>&d4;</ac></a></f> </math> able to describe both the dynamics and statistics of the system. Upon formulating the theory we explicitly show that it is consistent with the real-time Keldysh theory both formally and through an example using perturbation theory. In these systems there exists a strong interplay between the interactions and nonequilibrium leading to novel nonperturbative phenomena. Therefore, we combine our theory with the Hirsch-Fye quantum Monte Carlo algorithm to study these effects. We then propose a nonequilibrium Hubbard model in the special limit of infinite dimensions (dynamical mean-field theory) where the problem is reduced to solving a self consistent nonequilibrium impurity model. The final chapter concentrates on electron spin and charge filtering through a quantum dot embedded Aharonov-Bohm interferometer.