Simple model for coupled magnetic and quadrupolar instabilities in uranium heavy-fermion materials.

We present a mean-field calculation of the phase diagram of a simple model of localized moments, in the hexagonal uranium heavy-fermion compounds. The model considers a non-Kramers quadrupolar doublet ground state magnetically coupled with a singlet excited-state, favoring in-plane van-Vleck magnetism, as has been conjectured for UPt3. The Hamiltonian which defines the model is Heisenberg like in both, magnetic and quadrupolar moments. No Kondo effect physics is included in the calculations. Among our main results are: (i) for zero intersite quadrupolar coupling, the magnetic order is achieved by a first order transition above a critical intersite magnetic coupling value which becomes second order at higher coupling strengths. (ii) for finite intersite quadrupolar coupling, at temperatures below a second order quadrupolar ordering transition, the minimal magnetic coupling value is increased, but (a) the magnetic ordering temperature is enhanced above this value, and (b) the ordering of first and second order transitions in the phase diagram is reversed. By considering the general structure of the Ginsburg-Landau free energy, we argue that the Kondo effect will not modify the shape of the phase diagram, but will modify the quantitative values at which transitions occur.