Typical high transition temperature superconductors consist of copper-oxide superconducting planes separated by insulating layers. The superconducting layers are coupled by Josephson and electromagnetic interactions. Although, the microscopic theory of high T_c superconductivity has not been constructed yet, the phenomenological description is well-established. The vortex dynamics of a 2D (uncharged) superfluid film can be described by the following models which are assumed to belong to the the same universality class: the two-dimensional sine--Gordon field theory (2D--SG), the XY spin model (2D--XY) and the 2D-Coulomb gas. For charged 2D films, i.e. for superconducting thin films, the vortex behavior is described by the massive 2D--SG, the frustrated 2D--XY and the 2D--Yukawa gas. The gas of topological excitations and the equivalent spin model of a system of coupled layers is known for the Josephson and for the magnetically coupled case, however, no field theoretical model has been constructed. We propose a quantum field theoretical approach to the vortex dynamics of magnetically coupled layered superconductors by constructing a two-dimensional multi-layer sine-Gordon type model which we map onto a gas of topological excitations. The known interaction potentials of magnetically coupled vortices are consistently obtained from our field-theoretical analysis. Originally, this layered SG model has been introduced as the bosonised version of the multi-flavor Schwinger model, and, consequently, it has been used to study quark confinement. We analyze the phase structure of the multi-layer sine--Gordon model by functional renormalization group methods.
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