# Density Operator Methods

### Liouville Space Quantum Fields

Density operator approaches to open quantum systems are relevant in various fields, such as quantum transport, quantum optics, and quantum information. Their most prominent advantage is that they apply to systems with strong local interactions coupled to non-interacting reservoirs. These interactions, however, do not allow the local system to be integrated out and an effective theory can therefore only formulated in terms of superoperators that generate the dissipative dynamics of the reduced density operator. For relatively small local systems the superoperator structure rapidly becomes unmanageable. To simplify this description field-theoretical constructions analogous to the “second quantization” have been introduced.

In recent renormalization-group studies special field superoperators were introduced whose “causal” properties were later found to generally simplify the analysis of open quantum systems in two important ways: (1) the process of integrating out fermionic reservoirs is simplified, revealing a “causal structure” of the resulting effective field theory (2) the solution the resulting effective problem in the reduced system is strongly simplified. These causal “superfermions” open up many new avenues for applying quantum-field theoretical techniques to open quantum systems. For example, they make a 1- and 2-loop analysis of the Anderson model practically feasible and drastically simplify finding exact solutions in simple limits. Most recently, an new exact duality between open fermionic systems with opposite signs of the interaction (and other energies) has been derived which directly impacts studies of time-dependent heat transport in nanostructures.

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R. Saptsov and M. R. Wegewijs, Phys. Rev. B **86**, 235432 (2012)