Topological Landau Theory

We present an extension of Landau theory of phase transitions by incorporating the topology and Berry phase of the order parameter. When the order parameter consists of multiple components arising from the multiplicity of the same irreducible representation, the quadratic term in the Landau free energy acquires a matrix structure. As microscopic parameters are varied, the order parameter can change while remaining in the same representation and have a non-trivial Berry phase. To illustrate the idea, we study the superconducting phase transition of an electronic system with $D_{4h}$ symmetry and an attractive interaction involving two partial waves, both transforming in the trivial representation. The Landau free energy is derived via a weak coupling calculation and the Berry phase structure of the resulting thermodynamic ground state is discussed. Finally, we show that the Berry phase can be measured in a Josephson junction.