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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 of strongly cor- related systems (11, 12). Quantum 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 , 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 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 , up to an 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 . 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 — 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 , 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 (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) . 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) 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-

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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 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). By the 1970s, the notion that the (T = 0.5 K) in CeRhIn5. The antifer- Kondo phenomenon operates in a dense periodic T (K) romagnetic order, denoted by MO, at array of magnetic Ce ions in intermetallic com- http://science.sciencemag.org/ 0.1 TN ambient pressure gives way to super- pounds, such as CeAl2 (27), was already in place. conductivity, specified by SC, at higher A characteristic scale, at which the Kondo screen- B pressures. At = 0, the antiferromag- ing initially sets in, is the Kondo temperature TK. 0.0 netic order goes away when the pres- The list of heavy-fermion materials is long, 012 P sure exceeds 1.Whenthemagnetic and they are typically compounds containing rare B (T) field exceeds just enough to suppress earths or actinides (including Yb, U, and Np, in superconductivity, the system is anti- C 12 addition to Ce) with partially filled 4f- or 5f- ferromagnetically ordered at lower orbitals. Their defining characteristic is that the P P Magnetically CeRhIn pressures ( < 2) but yields a non- 10 5 Magnetically effective mass of the charge carriers at the lowest ordered (MO) magnetic phase at higher pressures

disordered accessible temperatures is hundreds of times the (P > P2). The hatched line refers to on May 14, 2018 8 bare electron mass. the transition at P2, between these 37 Microscopically, heavy-fermion systems can P2 two phases ( ). 6 MO + SC be modeled as a lattice of localized f-electron mo-

B (T) ments that are coupled to a band of conduction 4 electrons. In the early 1980s, the description of the Kondo effect in the ground state of this Kondo SC 2 lattice was formulated (10). The local moments P1 lose their identity by forming a many-body spin 0 singlet with all the conduction electrons, leading to 1.4 1.6 1.8 2.0 2.2 2.4 P (GPa) an entangled state (Fig. 2A). The Kondo entan- glement in the ground state makes the local mo- ments, which are charge-neutral to begin with, acquire the quantum numbers of the conduction acteristic length scale, which increases on ap- that accompanies the divergent correlation-length electrons, namely spin-ħ/2 and charge-e. Corre- proaching the critical point. At the critical point, scale. When the transition takes place at a finite spondingly, “Kondo resonances” appear as charge the correlation length is infinite. Correspondingly, temperature TN, ħ/kBTN serves as the upper carriers, and they remember their localized-moment physical properties are invariant under a mathe- bound of the correlation time, and the ultimate origin by possessing a heavy mass. Because the matical operation that dilates the lengths; in other critical behavior is still determined by the fluc- Kondo resonances are part of the electronic- words, they are scale-invariant. tuations in space only. When TN is driven to zero excitation spectrum, they must be accounted for in A straightforward generalization of the Lan- temperature, however, a divergent correlation the Fermi surface, leading to the notion of a large dau paradigm to QCPs gives rise to essentially time xt accompanies the divergent correlation Fermi surface (Fig. 2B)—the picture of a heavy the same theoretical description (26). Quantum length x, and both must be taken into account Fermi liquid. mechanics introduces a “time” axis: Quantum even for equilibrium properties. Hence, the quan- The Kondo resonances can alternatively be states evolve in time. (For quantum systems in equi- tum critical fluctuations of the order parameter thought of as the remnants of the original f-electrons. librium, the relevant quantum evolution is along an take place both in space and in time. The effective They are delocalized because the 4f- or 5f-wave imaginary time of length ħ/kBT,wherekB is the dimensionality of the fluctuations is d + z,where function has a finite overlap with the ligand orbitals Boltzmann constant.) This introduces a time scale d is the spatial dimensionality, and z, the dynam- that form the conduction electrons. In other words,

1162 3 SEPTEMBER 2010 VOL 329 SCIENCE www.sciencemag.org REVIEW the f-electrons and conduction electrons are comes zero, the ground state is no longer a atures. In contrast to the case of the Kondo- hybridized. Kondo singlet, and there are no fully developed singlet ground state, these quasiparticles are Kondo resonances. Correspondingly, the Fermi adiabatically connected to the ordinary conduc- QCPs in Heavy Fermions surface is small, incorporating only the conduc- tion electrons and are located at the small Fermi Two types of QCPs. The Kondo singlet in the tion electrons. surface (Fig. 2D). ground state of a heavy-fermion paramagnet rep- In the Kondo-screened paramagnetic phase The large number of available compounds is resents an organized macroscopic pattern of the (Fig. 2A), the large Fermi surface is where the a key advantage in the study of quantum critical quantum many-body system (Fig. 2A). It endows heavy quasiparticles are located in the momentum heavy-fermion systems. At the same time, it raises the paramagnetic phase at zero temperature with space (Fig. 2B). As usual, such sharply defined an important question: Can we classify the quan- a quantum order. This characterization of the phase Fermi surfaces occur below an effective Fermi tum critical behavior observed in these heavy- goes beyond the Landau framework. The Kondo- temperature, TFL. Below this temperature, stan- fermion compounds? Below, we summarize the singlet state does not invoke any spontaneous dard Fermi-liquid properties—such as the inverse evidence for such a classification in the systems breaking of symmetry because the spins can orient quasiparticle lifetime and the electrical resistivity that have been most extensively studied in the in arbitrary directions; no Landau order parameter being quadratically dependent on temperature— present context. can be associated to the Kondo effect. Two types take place. QCP of the SDW type. The phase diagram for of QCPs arise, depending on the behavior of the In the Kondo-destroyed antiferromagnetic CePd2Si2 (Fig. 1A) (3) is reminiscent of theoret- Kondo singlet as we approach the QCP from the phase (Fig. 2C), there is no Kondo singlet in the ical discussions of unconventional superconduc- paramagnetic side. ground state, and correspondingly, static Kondo tivity near an SDW instability. Unfortunately, When the Kondo singlet is still intact across screening is absent. Kondo screening still oper- because of the high pressure necessary to access the antiferromagnetic transition at zero temper- ates dynamically, leading to an enhancement of the QCP in this compound, it has not yet been ature, the only critical degrees of freedom are the the mass of the quasiparticles. The quasiparticles possible to study either the order or the fluctua- Downloaded from fluctuations of the magnetic order parameter. In still have a Fermi-liquid form at low temper- tion spectrum near the QCP. CeCu2Si2 is an ideal this case, the antiferromagnetically ordered phase system for such an investigation in the immediate proximity to the QCP can be because, here, heavy-fermion su- described in terms of a spin-density-wave (SDW) perconductivity forms in the vi- A B order of the heavy quasiparticles of the para- cinity of an antiferromagnetic magnetic phase. The QCP is referred to as of the QCP at ambient/low pressure.

SDW type, which is in the same class as that Neutron diffractometry revealed http://science.sciencemag.org/ already considered by Hertz (26, 28–30). On the the antiferromagnetically ordered other hand, when the Kondo singlet exists only in state to be an incommensurate the paramagnetic phase, the onset of magnetic SDW with small ordered mo- order is accompanied by a breakdown of the ment (~0.1 mB/Ce) because of Kondo effect. The quantum criticality incorpo- the nesting of the renormalized rates not only the slow fluctuations of the anti- Fermi surface (31). Inelastic ferromagnetic order parameter but also the emergent neutron-scattering studies on degrees of freedom associated with the breakup paramagnetic CeCu2Si2 have of the Kondo singlet. The corresponding transi- C D identified fluctuations close to

tion is referred to as locally critical (17, 18); the the incommensurate ordering on May 14, 2018 antiferromagnetic transition is accompanied by a wave vector of the nearby SDW localization of the f-electrons. and have shown that such fluc- This distinction of the two types of QCPs can tuations play a dominant role in also be made in terms of energetics. The key driving superconducting pairing quantity to consider is the energy scale E*, which (32), confirming earlier theoret- dictates the breakup of the entangled Kondo sin- ical predictions. glet state as the system moves from the heavy- Another compound is Fermi-liquid side toward the quantum critical CeNi2Ge2, for which the mag- regime. A reduction of the E* scale upon ap- Fig. 2. Kondo entanglement and its breakdown in heavy-fermion netic instability may be achieved proaching the magnetic side is to be expected metals. (A) Kondo-singlet ground state in a paramagnetic phase, by slight volume expansion. The because the development of antiferromagnetic giving rise to a heavy Fermi liquid. The shapes with orange arrows critical Grüneisen ratio in this correlations among the local moments reduces indicate the mobile conduction electrons, and the thick black arrows system diverges as T −1,which the strength of the Kondo singlet (17–20). When indicate localized magnetic moments. The purple profile describes lends support for a nearby SDW E* remains finite at the antiferromagnetic QCP, the Kondo singlet in the ground state. (B) The Kondo singlet in the QCP (4). the Kondo singlet is still formed, and the quan- ground state gives rise to Kondo resonances, which must be incor- There are also a few exam- porated into the Fermi volume. Correspondingly, the Fermi surface is tum criticality falls in the universality class of the ples of magnetic quantum phase large, with a volume that is proportional to 1 + x,where1andx, SDW type. When the E* scale continuously goes transitions induced by alloy- respectively, refer to the number of local moments and conduction to zero at the antiferromagnetic QCP, a critical electrons per unit cell. An SDW refers to an antiferromagnetic order ing that appear to fall in the Kondo breakdown accompanies the magnetic that develops from a Fermi-surface instability of these quasiparticles. category of the SDW QCP. In transition. The TK scale, in which the Kondo (C)Kondobreakdowninanantiferromagnetic phase. The local moments Ce1−xLaxRu2Si2, for instance, re- screening initially sets in, is always nonzero near arrange into an antiferromagnetic order among themselves, and they do cent inelastic neutron-scattering the QCP, even when E* approaches zero. not form static Kondo singlets with the conduction electrons. (D)Kondo experiments have provided such The consequence of the Kondo breakdown resonances do not form in the absence of static Kondo screening. evidence near its critical concen- for the change of the Fermi surface is illustrated Correspondingly, the Fermi surface is small, enclosing a volume in the tration xc ≈ 0.075 (33). in Fig. 2. When E* is finite, the Kondo-singlet paramagnetic Brillouin zone that is proportional to x. Dynamical Kondo QCP involving a Kondo ground state supports Kondo resonances, and the screening, however, still operates, giving rise to an enhancement of the breakdown. As shown in Fig. Fermi surface is large. When the E* scale be- quasiparticle mass near the small Fermi surface. 3A, inelastic neutron-scattering

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experiments on the quantum critical material A CeCu5.9Au0.1 revealed an energy over tempera- ture (E/T) scaling (34) of the dynamical suscep- CeCu5.9Au0.1

tibility, with a fractional exponent (35). The same ) 100 is found to govern the magnetic –0.25 susceptibility at wave vectors far away from the 0.6 antiferromagnetic wave vector. These features are ) meV 2 B B B incompatible with the predictions of the SDW 2

– µ theory (26, 28 30) and have provided the initial ( 0.4 motivation for the development of local quantum -1 0.75 10 criticality (17). Because such a QCP involves a ) T (q) (meV/ µ B

breakdown of the Kondo effect, it must be man- χ 0.2

ifested in the charge carriers and their Fermi 1/ surfaces as well. S(k 0.0 Direct measurements of Fermi surfaces are 0 1234 typically done by using angle-resolved photoemis- T 0.75(K0.75) sion spectroscopy (ARPES). In spite of impres- -2 10 -2 -1 0 1 2 sive recent developments, ARPES still does not 10 10 10 10 10

have the resolution to study heavy-fermion metals E/kBT in the required sub-Kelvin low-temperature range. Fig. 3. Quantum critical properties of CeCu −xAux The other well-established means to probe Fermi 6 P1 P2 E T Downloaded from surfaces is the de Haas–van Alphen (dHvA) tech- and CeRhIn5.(A) Quantum-dynamical / scaling B S 8 nique, which, however, requires a large magnetic of the inelastic neutron-scattering cross-section field of several teslas. A rare opportunity arises in in CeCu5.9Au0.1. The measurements were performed CeRhIn at the antiferromagnetic wave vectors (where S is 5 CeRhIn5, in which a magnetic field of about 10 T is β maximized), and the scaling collapse is constructed 2 in fact needed to suppress superconductivity and a in the form of T S as a function of E/T.Thetemper- Oe) 6 α expose a quantum phase transition (Fig. 1C). From 7 1 ature and energy exponent is fractional: a = 0.75. α dHvA measurements performed in the field range α α 2 http://science.sciencemag.org/ The different symbols represent data taken in different ( x 10 1

3 of 10 to 17 T, a pronounced jump in the Fermi spectrometers at the different peak wave vectors. surface was seen in CeRhIn at the critical pressure α 5 (Inset) The inverse of the bulk magnetic suscepti- 4 2,3 of 2.3 GPa (Fig. 3B) (36). This, together with the bility, 1/c(q =0)≡ H/M, and that of the static observation of a seemingly diverging cyclotron susceptibility at other wave vectors derived from the a mass of the heavy charge carriers, is commonly dynamical spin susceptibility through the Kramers- A considered as evidence for a Kondo-breakdown Kronig relation (35). (B) Several dHvA frequencies 2 QCP (37). We caution that for CeRhIn , this issue b 5 as a function of pressure in CeRhIn5. The applied dHvA Frequency P remains to be fully settled; an alternative explana- magnetic field ranges between 10 T and 17 T. 1 c tion that is based on a change of the valence state and P2 have the same meaning as in Fig. 1C. The

of the Ce ions has also been made (38). symbols denote different branches of the Fermi 0 on May 14, 2018 The heavy-fermion metal YbRh Si has pro- surface (36). 0123 2 2 Pressure (GPa) vided an opportunity to probe the electronic prop- erties near an antiferromagnetic QCP involving a breakdown of the Kondo effect. As mentioned energy scale that vanishes at the antiferroman- compound. Furthermore, it provides evidence earlier, the very weak antiferromagnetic order of getic QCP (Fig. 4A) (41). The T* scale is distinct that the Kondo-breakdown effect indeed under- YbRh2Si2 is suppressed by a small magnetic field, from the Fermi-liquid scale TFL,belowwhicha lies such quantum critical scaling (43). giving way to non-Fermi liquid behavior (7). Fur- T2 resistivity is observed (Fig. 4A). These prop- thermore, the magnetic field induces a substantial erties are naturally interpreted as signatures of a Global Phase Diagram change of the isothermal Hall coefficient. The lat- breakdown of the Kondo effect at the QCP, with The fact that in YbRh2Si2, the multiple lines de- ter has been shown to probe, at low temperatures, the Fermi surface being large at B > BN (Fig. 2, A fining the Kondo-breakdown scale T*, the the properties of the Fermi surface (39). A new and B) and being small at B < BN (Fig. 2, C and Fermi-liquid scale TFL, and the Néel-temperature temperature scale, T*(B), was identified in the T−B D); T* refers then to the temperature scale ac- scale TN all converge at the same magnetic field phase diagram of YbRh2Si2 (Fig. 1B); across this companying the Kondo-breakdown energy scale in the zero-temperature limit raises the question scale, the isothermal Hall coefficient exhibits a E* introduced earlier. Notably, the E* scale is of what happens when some additional control crossover as a function of the applied magnetic field distinct from the aforementioned TK scale, which parameter is varied. This global phase diagram (B). This crossover sharpens upon cooling. Extrap- serves as the upper cut-off of the quantum-critical has recently been explored by introducing chem- olation to T = 0 suggests an abrupt change of the scaling regime and should therefore remain finite ical pressure to YbRh2Si2 (8). The antiferromag- Fermi surface at the critical magnetic field BN,the near the QCP. For instance, at the critical con- netic order is stabilized or weakened by means of field where TN approaches zero (39). Further evi- centration of CeCu6−xAux TK has been observed volume compression or expansion, respectively dence for the inferred change of Fermi surface has in photoemission spectroscopy to be nonzero (42), (Fig. 4, C to E), in accordance with the well- come from thermotransport measurements (40). even though E* is expected to vanish. established fact that pressure reinforces magnet- Across the T* line, the low-temperature thermo- A recent thorough study of the Hall crossover ism in Yb-based intermetallics. Unexpectedly power shows a sign change, suggesting an evo- on YbRh2Si2 single crystals of substantially im- however, the T*(B) line is only weakly dependent lution between hole-like and electron-like Fermi proved quality showed unequivocally that the on chemical pressure. Under volume compres- surfaces, as illustrated in Fig. 2, B and D. width of the crossover at T*(B) is strictly pro- sion (3% Co-doping), the antiferromagnetic QCP Further thermodynamic and transport inves- portional to temperature (Fig. 4B) (43). This occurs at a field substantially higher than B*, at tigations confirmed T*(B)tobeanintrinsic indicates that the E/T scaling also operates in this which T* → 0 (Fig. 4E). In this situation, T*is

1164 3 SEPTEMBER 2010 VOL 329 SCIENCE www.sciencemag.org REVIEW finite at the antiferromagnetic QCP. One there- ing the parameter G boosts the inherent quantum for a finite range of small Ir concentrations. The fore expects that the SDW description will apply, fluctuations of the local-moment system and cor- extension of this global phase diagram is cur- and this is indeed observed (8). Under a small respondingly weakens the magnetism. In the rently being pursued theoretically (2). When the volume expansion (2.5% Ir-doping), BN and B* two-parameter global phase diagram of (22), quantum fluctuations among the local moments continue to coincide within the experimental ac- each kind of transition appears as a line of critical are even stronger, a possibility exists for a para- curacy (Fig. 4D). With a large volume expansion points: One line is associated with local quantum magnetic phase with a suppressed Kondo entan- (17% Ir-doping), on the other hand, BN has criticality, with the breakdown of the Kondo glement and a concomitant small Fermi surface; vanished, but B* remains finite (Fig. 4C). This effect occurring at the antiferromagnetic-ordering this can be compared with the region highlighted opens up a range of magnetic field in which not transition; the other one is associated with SDW by the question marks in Fig. 4F. This phase only any magnetic ordering is absent but also quantum criticality, in which case the Kondo couldbeaspinliquidorcouldbeanorderedstate the Kondo-breakdown scale vanishes, suggest- breakdown can only take place inside the anti- (such as a spin-Peierls phase) that preserves the ing a small Fermi surface. Hydrostatic-pressure ferromagnetically ordered region. This is con- spin-rotational invariance. Understanding the experiments (44) on undoped YbRh2Si2 give sistent with Fig. 4F, in which BN and B*coincide nature of the phase represents an intriguing results comparable with those problem worthy of further of the Co-doped materials with A B 0.4 study, both theoretically and a similar average unit-cell vol- 0.8 B c YbRh Si experimentally. ume, indicating that the cross- 2 2 Other heavy-fermion sys- T * ing of T (B)andT*(B)as 0.3 tems may also be discussed in N 0.6 observed (8) for 7% Co-doped this two-parameter global phase

YbRh2Si2 originates from the 0.2 diagram. The zero-temperature 0.4 alloying-induced volume com- T (K) transition in CeCu6−xAux as a Downloaded from pression rather than disorder. FWHM (T) function of doping or pressure 0.1 The results can be summa- 0.2 can be described in terms of TFL rized in the global phase dia- TN local quantum criticality. As a gram shown in Fig. 4F. The 0.0 0.0 function of magnetic field, for transition from the small-Fermi- 0.0 0.1 0.2 0.0 0.1 0.2 0.3 0.4 0.5 both CeCu − Au (47)andCeIn B (T) T (K) 6 x x 3 surface antiferromagnet to the (48) the Kondo breakdown heavy-Fermi-liquid state has C D seems to take place inside the http://science.sciencemag.org/ three types. It may go through 1.0 17% Ir 1.0 2.5% Ir antiferromagnetic part of the a large-Fermi-surface antifer- phase diagram. It will be instruc- romagnet, such as in the Co- T * tive to see whether other heavy- doped cases. The transition can fermion materials can be used to also occur directly, such as in T * map the global phase diagram T (K) T (K) 0.5 0.5 the pure and 2.5% Ir-doped com- and, in particular, display a para- pounds. Or, it may go through a magnetic non-Fermi liquid phase small-Fermi-surface paramag- near a Kondo-breakdown QCP. netic phase, such as in the case TN Conclusions and Outlook of the 6% Ir-doped YbRh Si 0.0 0.0

2 2 on May 14, 2018 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 (8). In this phase, the electrical B (T) B (T) Studies in the last decade have resistivity shows a quasi-linear firmly established the exis- E F temperature dependence (8). 0.15 tence of QCPs in heavy-fermion 1.0 Yb(Rh Ir ) Si Theoretically, two kinds of 3% Co ?? 1-y y 2 2 metals. These transitions arise antiferromagnet–to–heavy- T * 0.10 from the suppression of long- FL Fermi-liquid transitions were 0.05 range antiferromagnetic or- already considered in the pre- T=0 dering by means of tuning vious section. One way to 0.00 pressure, chemical composi- T (K) 0.5 Yb(Rh1-xCox)2Si2 connect them is to invoke a T = 0.02 tion, or magnetic field. An im- 0 global phase diagram (22), AF portant property of QCPs is T xy B spanned by two parameters N 0.04 N the accumulation of entropy. B* associated with two types of 0.06 Correspondingly, the Grüneisen quantum fluctuations. One pa- 0.0 ratio or the magnetocaloric ef- 0.0 0.1 0.2 0.3 0.00 0.05 0.10 0.15 0.20 rameter, JK, describes the Kondo B (T) B (T) fect diverges, which serves as an coupling between the conduc- important thermodynamic char- tion electrons and the local Fig. 4. Quantum criticality and global phase diagram in pure and doped YbRh2Si2.(A) acterization of the QCPs. Multiple energy scales in pure YbRh Si . T* is extracted from isothermal crossovers in moments; increasing J enhan- 2 2 Two types of QCPs have K the Hall effect and thermodynamic properties, which is interpreted in terms of a Kondo ces the ability of the conduction been developed for antiferro- breakdown. TFL is the scale below which Fermi-liquid properties occur. Both crossover electrons to screen the local lines merge with the line that specifies the magnetic phase boundary T in the zero- magnetic heavy-fermion sys- moments and reduces the mag- N tems. When a breakdown of temperature limit, at BN (41). (B) Full width at half maximum (FWHM) of the crossover netic order. The other param- in the Hall coefficient of a high-quality single crystal (RRR = 120). It extrapolates to the Kondo entanglement occurs eter, G, is associated with the zero in the T = 0 limit, implying a jump of the Hall coefficient and other properties. It is inside the antiferromagnetically interactions among the local proportional to temperature, suggesting a quantum-dynamical E/T scaling (43). (C to E) ordered phase, the QCP has the moments and refers to, for in- T*(B)andT (B) lines for Ir- and Co-doped YbRh Si , determined via AC susceptibility standard SDW form that con- N 2 2 ’ stance, the degree of geometric measurements (8). Data for the 7% Co-doped YbRh2Si2 show an intersection of the two forms to Landau s paradigm of frustration (45)orsimplythe lines (8). (F)TheT = 0 phase diagram, doping-concentration versus magnetic field, for order-parameter fluctuations. dimensionality (17, 46); rais- Yb(Rh1−xMx)2Si2,M=Co,Ir(8). When such a Kondo breakdown

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tum fluctuations in heavy-fermion systems can be superconductivity, which has featured so promi- 37. T. Park et al., Nature 440, 65 (2006). http://science.sciencemag.org/ 38. S. Watanabe, A. Tsuruta, K. Miyake, J. Flouquet, J. Phys. tunedinmorewaysthanone.Differentphasesand nently in the systems considered here, is probably Soc. Jpn. 78, 104706 (2009). QCPs may arise when a magnetic disordering is pertinent to heavy-fermion metals in general as well 39. S. Paschen et al., Nature 432, 881 (2004). induced by the Kondo coupling between the local as other classes of correlated-electron materials, in- 40. S. Hartmann et al., Phys. Rev. Lett. 104, 096401 moments and conduction electrons or when it is cluding the iron pnictides and organic charge- (2009). 41. P. Gegenwart et al., Science 315, 969 (2007). caused by reduced dimensionality and/or magnetic transfer salts. Quantum phase transitions are also 42. M. Klein et al., Phys. Rev. Lett. 101, 266404 (2008). frustration. being discussed in broader settings, such as ultra- 43. S. Friedemann et al., Proc. Natl. Acad. Sci. U.S.A. Theoretically, an important notion that has cold atomic gases and quark matter. It is con- (2010). emerged from studies in heavy-fermion systems ceivable that issues related to our discussion here 44. Y. Tokiwa, P. Gegenwart, C. Geibel, F. Steglich, J. Phys. Soc. Jpn. 78, 123708 (2009). is that quantum criticality can go beyond the will come into play in those systems as well. 45. L. Balents, Nature 464, 199 (2010).

Landau paradigm of fluctuations in an order pa- 46. H. Shishido et al., Science 327, 980 (2010). on May 14, 2018 rameter associated with a spontaneous symmetry 47. O. Stockert, M. Enderle, H. v. Löhneysen, Phys. Rev. Lett. References and Notes breaking. This notion has affected the develop- 99, 237203 (2007). 1. I. Osborne, R. Coontz, Science 319, 1201 (2008). 48. S. E. Sebastian et al., Proc. Natl. Acad. Sci. U.S.A. 106, ments on quantum criticality in other systems, 2. Special Issue, Phys. Status Solidi B 247, 457 (2010). 7741 (2009). including insulating magnets. More generally, 3. F. M. Grosche et al., J. Phys. Condens. Matter 13, 2845 49. I. R. Klebanov, J. M. Maldacena, Phys. Today 62,28 quantum criticality in heavy-fermion metals epit- (2001). (2009). 4. P. Gegenwart, Q. Si, F. Steglich, Nat. Phys. 4, 186 50. A. R. Schmidt et al., Nature 465, 570 (2010). omizes the richness and complexity of continu- (2008). ous quantum phase transitions as compared with 51. P. Aynajian et al., Proc. Natl. Acad. Sci. U.S.A. 107, 5. Y. Tokiwa, T. Radu, C. Geibel, F. Steglich, P. Gegenwart, 10383 (2010). their classical counterparts. New theoretical meth- Phys. Rev. Lett. 102, 066401 (2009). 52. N. Doiron-Leyraud et al., Nature 447, 565 (2007). ods are needed to study strongly coupled quantum 6. A. W. Rost, R. S. Perry, J.-F. Mercure, A. P. Mackenzie, 53. We thank E. Abrahams, M. Brando, P. Coleman, S. A. Grigera, Science 325, 1360 (2009). S. Friedemann, P. Gegenwart, C. Geibel, F. M. Grosche, critical systems. One promising new route is pro- 7. J. Custers et al., Nature 424, 524 (2003). vided by an approach that is based on quantum S. Kirchner, C. Krellner, M. Nicklas, T. Park, J. Pixley, 8. S. Friedemann et al., Nat. Phys. 5, 465 (2009). O. Stockert, J. D. Thompson, S. Wirth, and S. Yamamoto gravity (49). Using a charged black hole in a 9. M. B. Maple et al., J. Low Temp. Phys. 95, 225 (1994). for useful discussions. This work has been supported by weakly curved space-time to model a finite density 10. A. C. Hewson, The Kondo Problem to Heavy Fermions NSF and the Robert A. Welch Foundation grant C-1411 of electrons, this approach has provided a tantaliz- (Cambridge Univ. Press, Cambridge, 1993). (Q.S.) and by the DFG Research Unit 960 “Quantum 11. R. A. Cooper et al., Science 323, 603 (2009). Phase Transitions” (F.S.). ing symmetry reason for some fermionic spectral 12. S. A. Grigera et al., Science 294, 329 (2001). quantities to display an anomalous frequency 13. G. R. Stewart, Rev. Mod. Phys. 73, 797 (2001). 10.1126/science.1191195

1166 3 SEPTEMBER 2010 VOL 329 SCIENCE www.sciencemag.org Heavy Fermions and Quantum Phase Transitions Qimiao Si and Frank Steglich

Science 329 (5996), 1161-1166. DOI: 10.1126/science.1191195

From Simplicity to Complexity The relatively simple properties of isolated electrons become rich and complex when the particle-particle interactions are strong enough to form a correlated system. Emergence of complex behavior from relatively simple subunits is an intensely studied topic in condensed-matter physics and applies to many systems in superconductivity and magnetism. Si and Steglich (p. 1161) review the physics of heavy fermion intermetallic compounds. These make ideal materials for study because they can exhibit metallic, magnetic, and superconducting behavior showing novel quantum Downloaded from phases and unconventional quantum criticality. http://science.sciencemag.org/

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