# PHYSICS OF QUANTUM CRITICALITY

Quantum criticality is at the forefront of modern condensed matter physics. The recent discovery of an entirely unexpected energy scale [S. Paschen et al., Nature 432, 881 (2004), P. Gegenwart et al., Science 315, 969 (2007); see Fig. 1] in the phase diagram of a prototypical quantum critical compound mystified the community. With our project QuantumPuzzle we address the key issues of this discovery and set out to solve the “quantum puzzle” with techniques that go drastically beyond the state-of-the-art in the field.

# Classical phase transitions

The three different phases of water, the solid, liquid, and gas phase are familiar to most of us and early childhood experience taught us that the ambient temperature ultimately determines the phase water will be in. It is obvious that phase changes must be collective phenomena involving many particles. After all, a single water molecule is not a gas nor a liquid and certainly not a solid. Modern condensed matter relates the solid character of ice (i.e. the rigidity) to the periodic alignment of the water molecules in ice. This periodicity is in fact a characteristic feature of all solids that gives rise to specific patterns in X-ray scattering experiments. The rigidity emerges when liquid undergoes a phase transition to the solid state. While the strength of this rigidity is a material specific quantity (the shear modulus), its mere existence is tied to general symmetry arguments.

# The universality of phase transitions

As it turns out, there are two broad classes of phase transitions: continuous or discontinuous ones. Discontinuous transitions are sometimes called first order transitions. Quantities which are related to the emerging order, so-called generalized susceptibilities, diverge at continuous transitions because they directly probe the amount of fluctuation in the associated quantity. The existence of fluctuations on all length scales right at the transition is the characteristic feature of a continuous phase transition and is commonly referred to as criticality. As the temperature is tuned towards the so-called critical temperature, the length scale on which fluctuations occur tends towards infinity and gives rise to universal behavior.

Figure 2 shows an example of universality for the liquid to gas transition and is one of the earliest scaling plots. Different liquids are described by the same hence universal scaling curve near their respective continuous transition (T_{c} and p_{c} denotes the critical temperature and pressure, respectively). The universal scaling curve is a reflection of the fact that the approach to the critical point is characterized by the vanishing of a single length scale. In fact, this physical behavior is so universal across different systems that the scaling curves will only depend on very general properties like, e.g. the symmetry properties of the emerging order or the space dimensionality. As a result, even the continuous transition from a normal metal to a ferromagnet is described by a scaling curve identical to the one in Figure 2. As Figure 2 already suggests, the transition can be reached by either fixing the ambient pressure to the critical pressure and then tuning the temperature through its critical value or by setting the temperature to the critical value and addressing the phase transition as a function of pressure.

# Quantum phase transitions

If the transition happens to occur at zero temperature, it is customary to call it a quantum phase transition. This alludes to the fact that only in this case quantum mechanical fluctuations can successfully conspire with thermal fluctuations and modify the scenario outlined so far. Since the transition occurs at zero temperature, a different – non-thermal – control parameter is needed to tune through the transition. Depending on the particular system under consideration this tuning parameter may be, for example, pressure, magnetic field, or a sequence of different materials that differ in a well-controlled manner. The materials that are central to our proposal are compounds complex enough to – at least in certain cases – enable tuning across a quantum phase transition by changing the mass relationships among elements that constitute the compounds. This is called stoichiometric tuning. Again, the transition may be continuous or discontinuous in close analogy to the classical case outlined above.

The overall picture that so far has emerged for quantum phase transitions by and large parallels that of classical phase transitions. The most prominent modification comes from a change in the effective spatial dimension that should be used to describe the universal scaling behavior. The approach to a continuous quantum phase transition should therefore be characterized by the vanishing of one length or energy scale, which we will refer to as the correlation length.

The recent discovery of a heavy fermion compound whose approach to quantum criticality is characterized by at least two independent energy scales is the motivation for our project QuantumPuzzle. With the general state of affairs in mind it becomes clear that the mere existence of such additional scales, independent from the scale set by the correlation length, poses a severe challenge to our understanding. It is to be expected that the resolution to this problem will not only have ramifications to the general theory of quantum phase transitions but also to our understanding of emergent quantum phases in complex materials.

# Heavy fermion systems

Ordinary metals, e.g. aluminum, are well described in terms of a Landau Fermi liquid. Electrons in such a Fermi liquid are completely independent of each other except for an adjustment in certain parameters like the electron mass. This in itself is truly remarkable, given that electrons carry charge and therefore should repel each other. This feature is a consequence of the quantum mechanical delocalization of the electron across the metal. More precisely, the fundamental particle in a metal like aluminum is an excitation that carries the charge of an electron but is a quasi-particle to which many electrons contribute as the enhanced effective mass indicates.

In heavy fermion systems two different kinds of electrons determine the physical properties. Heavy fermion systems contain delocalized electrons just like aluminum, but, in addition, contain localized electrons that all by themselves would not carry an electric current. An intricate many body effect, called the Kondo effect, nonetheless allows the localized electrons to contribute to the electronic properties by forming a quasi-particle with the delocalized electrons. The Kondo effect thus entangles the localized electrons with the delocalized ones. The resulting quasi-particles in a heavy fermion compound form a Fermi liquid that typically has an effective mass that exceeds the bare electron mass by a factor of up to 1000.

# The spin density wave scenario

In the conventional approach to quantum criticality, the Fermi liquid may undergo a quantum phase transition to a magnetically ordered state. This is called the spin density wave scenario (SDW). Figure 1 shows the phase diagram of YbRh_{2}Si_{2}. The quantum critical point is accessed through magnetic field tuning. For magnetic fields larger than the critical field, YbRh_{2}Si_{2} is in a heavy fermion state. For field strength below the critical field the compound orders magnetically. Both the black and yellow lines (to left and right from the quantum critical point, respectively) are related to the correlation length via the traditional theory of quantum criticality.

# Local quantum criticality

The puzzling feature unaccounted for by the traditional theory is the crossover region (red line towards top) that vanishes at the QCP. Several alternatives and modifications of the SDW scenario have been put forth. One of the promising alternatives is called local quantum criticality. It predicts a completely new form of quantum criticality where the heavy quasiparticles break up concomitant with the onset of magnetic ordering. Within this picture, the additional scale at the quantum transition is naturally explained as a consequence of the critically destroyed Kondo effect, as suggested in Figure 1.