  In this paper, we propose a quantum field theoretical renormalization group
approach to the vortex dynamics of magnetically coupled layered
superconductors, to supplement our earlier investigations on the
Josephson-coupled case. We construct a two-dimensional multi-layer sine-Gordon
type model which we map onto a gas of topological excitations. With a special
choice of the mass matrix for our field theoretical model, vortex dominated
properties of magnetically coupled layered superconductors can be described.
The well known interaction potentials of fractional flux vortices are
consistently obtained from our field-theoretical analysis, and the physical
parameters (vortex fugacity and temperature parameter) are also identified. We
analyse the phase structure of the multi-layer sine--Gordon model by a
differential renormalization group method for the magnetically coupled case
from first principles. The dependence of the transition temperature on the
number of layers is found to be in agreement with known results based on other
methods.
