CÁNCER DE MAMA PRIMARIO

Quantum criticality occurs when a second-order phase transition takes place at zero temperature, driven by some non-thermal control parameter. The transition is not then due to thermal fluctuations as in the classical transition, but may be said to be due to ‘quantum fluctuations’. So called due to the analogy between quantum superposition and fluctuations in time, these quantum fluctuations actually mea- sure how far the true quantum state of the system is from the classical groundstate which we expand about in calculations. At a finite-temperature critical point these quantum effects are overwhelmed by thermal effects, but as we approach zero tem- perature the quantum mechanics become increasingly important. Order parameter fluctuations have a characteristic frequencyωc, when ¯hωc ≫kBT the system behaves quantum mechanically. At the classical critical point the correlation time diverges, so the characteristic frequency goes to zero and the only temperature at which quan- tum mechanics is important is zero temperature. However near a quantum critical point, the only non-zero energy scale is provided by temperature so that ωc ∼ T

(all other energy scales renormalise to zero). Near the quantum critical point then both quantum and classical effects have equal footing, statics and dynamics become intertwined and the properties of the system are radically altered. This region of novel behaviour extends over a surprisingly wide range of parameters and temper- atures, as illustrated in figures 1.4 and 2.4. The identification of non-Fermi liquid behaviour in such regions has become the signature of quantum criticality.

The search for quantum critical points in itinerant systems has revealed that new phases are stabilized in their vicinity. This could be due to an existing instability being favoured as the energy scales of the system tend to zero at the quantum critical point, or quantum fluctuations mediating an entirely new phase. Superconductivity is commonly discovered around the quantum critical points of heavy fermion mate- rials [5] and a new phase appears around the putative quantum critical endpoint of Sr3Ru2O7 [52]. This thesis is concerned with the phase which appears in Sr3Ru2O7. We do not invoke the quantum properties of the critical point, instead the reduction of energy scales allows the formation of a new phase of the normal Fermi liquid.

The question of how quantum criticality relates to the formation of phases is complicated. Bare itinerant ferromagnetic quantum critical points are never ob- served, there always seems to be a new phase intervening to mask the quantum critical point, or the continuous transition becomes first-order before reaching zero temperature. Recent developments to the theory of itinerant quantum criticality are

2. Itinerant magnetism - important developments and recent interest 20

Fig. 2.4: Dome of superconductivity over quantum critical point: Dome of super- conductivity over quantum critical point in CePd2Si2 (left) and CeCu2(Si1−xGex) (right).

For the N´eel temperature Tn the open circles (squares) of the right panel correspond to x= 0.1 (x= 0.25). For the superconducting transition temperature Tc the thin solid line

(full circles) of the right panel corresponds to x= 0 (x= 0.1). Figures from [5].

putting this on firmer theoretical ground. The standard theory of itinerant quantum critical points, known as Hertz-Millis theory [37, 38], has recently been shown to be incomplete [39, 40]. The theory breaks down close to the quantum critical point that it was intended to describe. Extra terms have been shown to appear in the Hertz-Millis action which drive the continuous transition first-order and stabilize the presence of modulated phases [32, 39, 40].

Hertz-Millis theory is the standard description of itinerant quantum critical points. The Hertz-Millis action is typically studied in a Renormalisation Group anal- ysis [53], as is classical criticality, or in a self-consistent renormalisation scheme [54]. It leads to the prediction of scaling laws relating the various parameters of the sys- tem in the critical region. These laws give critical exponents different from their classical counterparts, and depend on the dimension and nature of the ordering, not on the microscopic details of the system (see for example [55]).

Hertz-Millis theory is an extension of Ginzburg-Landau theory to include quan- tum dynamics. Underlying this is the assumption that the action can be expanded in powers of the order parameter and its gradients. This assumption turns out to be incorrect. Analysis of higher-order correction terms to the electron self-energy reveal that non-analytic terms in q and T enter the action. These can alter the low-temperature phase diagram. By renormalising the quadratic and quartic coeffi- cients of the free energy these can induce the transition to become first-order or for a modulated phase to appear as an intermediate phase in the transition.

2. Itinerant magnetism - important developments and recent interest 21

purely mean-field approach. The effects which we predict, although similar to those which may occur due to quantum fluctuations, are entirely due to the effect of the lattice. We will discuss the possible connection between these effects and those due to band effects which are considered in this thesis in section 8.2.1.