Group Theoretical Methods Applied to Condensed Matter Problems

Group theoretical methods have important applications in many areas of condensed matter physics. As a first example, we present an investigation of “quadrupolar” phases in UPd3, a system which has a three-component order parameter resulting in an interesting phase diagram with an “octacritical” point. In our second example we discuss an unconventional (tetragonal) superconductor and show that the symmetry of its order parameter is revealed by spontaneous Josephson currents in a bicrystal ring experiment. The final example addresses polymer dynamics through an exact enumeration procedure for interactions and relative motion of two polymer segments on a lattice. 1 Quadrupolar phases in UPd3 In the absence of external fields, UPd3, a hexagonal crystal with space group G = P63/mmc, undergoes a continuous phase transition at a temperature T1 ≈ 7K (cf. Ref. [1] and references therein). According to Landau’s theory [2], continuous phase transitions are characterized by an order parameter η, which belongs to an irreducible representation of the group G. The order parameter is zero in the phase of higher symmetry (disordered phase) and changes continuously at the critical point to a non-zero value in the phase of lower symmetry (ordered phase). From an analysis of neutron scattering data, Walker et al. [1] determined the symmetry of the order parameter and concluded that its three components describe the ordering of the electric quadrupole moments of the uranium ions and a simultaneous ion displacement along three equivalent directions in the basal (x, y) plane of the crystal. The “Landau free energy” is a variational free energy expressed as a power series in the components ηi, which is invariant under the symmetry operations of the group G and respects thermodynamic stability requirements [2]. In the presence of symmetry-breaking fields, e.g. stress or magnetic fields, coupling terms (invariant under operations of G) between the order parameter and the applied fields are included, resulting in a Landau free energy [4]: