REVIEW non-Fermi liquid behavior (9), which goes beyond the standard theory of metals [Fermi- liquid theory (10)], is another phenomenon that is Heavy Fermions and broadly relevant to the physics of strongly cor- related systems (11, 12). Quantum Phase Transitions Quantum criticality has been implicated to one degree or another in a host of other heavy- fermion metals (4, 13, 14). These include CeCu2Si2, 1 2 Qimiao Si * and Frank Steglich * the first superconductor to be observed among heavy-fermion metals (15), and CeRhIn5 (Fig. Quantum phase transitions arise in many-body systems because of competing interactions that promote 1C) (16). Extensive theoretical studies have led – rivaling ground states. Recent years have seen the identification of continuous quantum phase transitions, to unconventional quantum criticality (17 20). or quantum critical points, in a host of antiferromagnetic heavy-fermion compounds. Studies of the More recently, a plethora of phases have been interplay between the various effects have revealed new classes of quantum critical points and are uncovered in heavy-fermion metals near a QCP b uncovering a plethora of new quantum phases. At the same time, quantum criticality has provided [such as in Ir-doped YbRh2Si2 (8)andin -YbAlB4 fresh insights into the electronic, magnetic, and superconducting properties of the heavy-fermion metals. (21)]. Together with the theoretical studies of the We review these developments, discuss the open issues, and outline some directions for future research. global phase diagram of the heavy-fermion metals (22, 2), these developments open up an entirely new frontier on the interplay between quantum critical- uantum mechanics not only governs the uent particles. In other words, such a parameter ity and unusual phases. subatomic world but also dictates the or- controls quantum-mechanical tunneling dictated Downloaded from Qganization of the microscopic particles in by Heisenberg’s uncertainty principle, changing Quantum Phase Transitions bulk matter at low temperatures. The behavior is the degree of quantum fluctuations. This is the Quantum phase transitions result from the var- strikingly different depending on the spin (the analog of varying the thermal fluctuations in the iation of quantum fluctuations. Tuning a control internal angular momentum) of the constituent case of temperature-driven classical phase tran- parameter at absolute zero temperature tilts the particles. Particles whose spin is an integer mul- sitions, such as the melting of ice or the loss of balance among the competing ground states as- tiple of ħ (Planck’s constant h divided by 2p)are ferromagnetic order in iron. sociated with conflicting interactions of quantum http://science.sciencemag.org/ bosons. When cooled down to sufficiently low The temperature-pressure phase diagram ob- matter. temperatures, they will be described by the same served in the heavy-fermion intermetallic com- Heavy-fermion metals comprise a lattice of wave function, forming a “condensate.” Particles pound CePd2Si2 is illustrated in Fig. 1A (3). At localized magnetic moments and a band of con- whose spin is a half-integer of ħ, on the other ambient pressure, CePd2Si2 orders into an anti- duction electrons (10). The exchange interaction hand, are fermions satisfying the Pauli exclusion ferromagnet, below the Néel temperature TN of between the local moments is primarily that me- principle; no two particles can have the same about 10 K. Applying pressure reduces TN mono- diated by the conduction electrons: the familiar state. At absolute zero, they occupy the states tonically, eventually suppressing the antiferromag- Ruderman-Kittel-Kasuya-Yoshida (RKKY) in- with the lowest energies, up to an energy referred netic order altogether and turning the system into a teraction. This interaction drives the local mo- to as the Fermi energy. In the momentum space, paramagnetic metal. The putative critical pressure ments into an ordered pattern, much like H2O this defines a Fermi surface, enclosing a Fermi p is around 2.8 GPa, at which point an antifer- molecules are condensed into an ordered arrange- c on May 14, 2018 volume in which all the states are occupied. romagnetic QCP is implicated. The QCP, how- ment in ice. The Kondo-exchange interaction be- When the particle-particle interactions are in- ever, is not explicitly observed. Instead, a “dome” tween the local moments and conduction electrons cluded, the behavior of such quantum systems emerges at very low temperatures in the vicinity introduces spin flips, which is a tunneling process becomes even richer. These strongly correlated of pc, under which the system is a superconductor. enabled by quantum mechanics. Corresponding- systems have taken the center stage in the field of This phase diagram exemplifies a general point. ly, increasing the Kondo interaction amounts to quantum matter over the past two decades (1). It suggests that antiferromagnetic quantum criticality enhancing quantum fluctuations, which eventu- High-temperature superconductors, fractional quan- can provide a mechanism for superconductivity— ally destroys the magnetic order and yields a tum Hall systems, colossal magnetoresistive ma- an observation that may be of relevance to a range paramagnetic phase (23, 24). terials, and magnetic heavy-fermion metals are a of other strongly correlated systems, such as high– The theory of classical phase transitions, for- few prominent examples. The central question critical temperature (Tc) cuprates, organic super- mulatedbyLandau(25), is based on the princi- for all these systems is how the electrons are or- conductors, and the recently discovered high-Tc ple of spontaneous symmetry breaking. Consider ganized and, in particular, whether there are prin- iron pnictides. The formation of new phases near CePd2Si2 at ambient pressure. In the paramag- ciples that are universal among the various classes a QCP may be considered the consequence of an netic phase, at T > TN, the spins are free to rotate. of these strongly correlated materials. One such accumulation of entropy, which is a generic fea- Upon entering the magnetically ordered phase, principle, which has come to the forefront in re- ture of any QCP (4) and has recently been observed this continuous spin-rotational symmetry is spon- cent years, is quantum criticality (2). experimentally (4–6). taneously broken; the spins must choose preferred A quantum critical point (QCP) arises when A good example for such an antiferromag- orientations. In the Landau formulation, this sym- matter undergoes a continuous transition from netic QCP is the one observed in the compound metry distinction is characterized by a quantity one phase to another at zero temperature. A non- YbRh2Si2 (4). Here, the nonthermal control pa- called order parameter; in our case, this is the thermal control parameter, such as pressure, tunes rameter is a (small) magnetic field. The studies of staggered magnetization of the antiferromagnet. the amount of zero-point motion of the constit- heavy-fermion antiferromagnets have shown that The order parameter is nonzero in the magneti- accompanying the QCP at zero temperature is a cally ordered phase but vanishes in the paramag- 1 finite parameter range at nonzero temperatures, in netic phase. The critical point arises when the Department of Physics and Astronomy, Rice University, Houston, — TX 77005, USA. 2Max Planck Institute for Chemical Physics of which the metallic state is anomalous (Fig. 1B) phase transition is continuous when the order Solids, 01187 Dresden, Germany. (7, 8). Over this quantum critical regime, the parameter goes to zero smoothly. It is described in *To whom correspondence should be addressed. E-mail: electrical resistivity is linear in temperature—a terms of the spatial fluctuations of the order [email protected] (Q.S.); [email protected] (F.S.) telltale sign for an unusual metallic state. This parameter. These fluctuations occur over a char- www.sciencemag.org SCIENCE VOL 329 3 SEPTEMBER 2010 1161 REVIEW Fig. 1. Quantum phase transitions ic exponent defined in terms of the relationship x º xz A in heavy-fermion metals. (A) Suppres- t , describes the number of effective spa- CePd Si sion of antiferromagnetic order through tial dimensions to which the time dimension 10 2 2 corresponds. pressure in CePd2Si2. TN is the Néel transition temperature, and the cor- However, it has been appreciated that this responding antiferromagnetic order Landau paradigm can break down for QCPs. Con- T N is illustrated in the inset. At the sider the effect of the Kondo-exchange coupling. boundary of the antiferromagnetism, In addition to destabilizing the magnetic order, a phase of unconventional super- the Kondo interaction also introduces quantum conductivity arises. Tc corresponds to coherence between the local moments and con- 5 the superconducting transition tem- duction electrons. Indeed, inside the paramag- perature (3). (B) Field-induced quan- Temperature (K) netic phase a process called Kondo screening tum phase transition in YbRh2Si2.The takes place, which leads to a qualitatively new blue regions label the Fermi-liquid ground state in which the local moments and behavior observed with measurements conduction electrons are entangled. Just as a con- of electrical resistivity and other trans- tinuous onset of magnetic order at zero temper- 2 T C port and thermodynamic properties; ature introduces quantum fluctuations of the they correspond to T < T at B < B 0 N N magnetic order parameter, a critical onset of Kondo and T < T at B > B ,whereB is the 012 3 FL N N entanglement also yields its own quantum critical Pressure (GPa) critical field at T =0.Theorangere- gion describes non-Fermi liquid be- degrees of freedom. When that happens, a new B 0.3 type of QCP ensues. havior that is anchored by the QCP at Downloaded from B B 7 T YbRh Si = N ( ). The ( *) line delineates 2 2 Kondo Effect and Heavy Fermions crossover behavior associated with the B c destructionoftheKondoeffect(8). (C) Historically, the Kondo screening effect was in- 0.2 The pressure-field phase diagram at troduced for dilute magnetic impurities in metal- the lowest measured temperature lic hosts (10).