Quantum phase transitions

We give a general introduction to quantum phase transitions in strongly correlated electron systems. These transitions, which occur at zero temperature when a non-thermal parameter g such as the pressure, chemical concentration or magnetic field is tuned to a critical value, are characterized by a dynamie exponent z related to the energy and length scales δ and ∂. We show how one can derive an effective bosonic model associated with fluctuations in the ordering fields. Simple arguments based on an expansion to first order in the effective interaction u allow us to define an upper critical dimension D c = 4 (where D = d + z and d is the spatial dimension) below which the mean-field description is no longer valid. We present an alternative tricritical crossover approach valid at D < D c in the large-N limit. We emphasize the role of perturbative renormalization group approaches and self-consistent renormalized spin fluctuation theories in understanding the quantum-classical crossover in the vicinity of the quantum critical point with generalization to the Kondo effect in heavy-fermion systems. Finally, we quote some recent inelastic neutron scattering experiments performed on heavy fermions which lead to an unusual scaling law in ω/T for the dynamical spin susceptibility, revealing critical local modes beyond the itinerant magnetism picture. We mention new attempts to describe this local quantum critical point.