Heavy Fermions: Electrons at the Edge of Magnetism

Piers Coleman Rutgers University, Piscataway, NJ, USA

Mathur et al., 1998). The ‘quantum critical point’ (QCP) that 1Introduction:‘AsymptoticFreedom’inaCryostat 95 separates the heavy-electron ground state from the AFM rep- 2LocalMomentsandtheKondoLattice 105 resents a kind of singularity in the material phase diagram 3KondoInsulators 123 that profoundly modifies the metallic properties, giving them aapredispositiontowardsuperconductivityandothernovel 4Heavy-fermionSuperconductivity 126 states of matter. 5QuantumCriticality 135 One of the goals of modern condensed matter research 6ConclusionsandOpenQuestions 142 is to couple magnetic and electronic properties to develop Notes 142 new classes of material behavior, such as high-temperature Acknowledgments 142 superconductivity or colossal magnetoresistance materials, References 143 spintronics, and the newly discovered multiferroic materials. Further Reading 148 Heavy-electron materials lie at the very brink of magnetic instability, in a regime where quantum fluctuations of the magnetic and electronic degrees are strongly coupled. As such, they are an important test bed for the development of 1INTRODUCTION:‘ASYMPTOTIC our understanding about the interaction between magnetic FREEDOM’ IN A CRYOSTAT and electronic quantum fluctuations. Heavy-fermion materials contain rare-earth or actinide The term heavy fermion was coined by Steglich et al.(1976) ions, forming a matrix of localized magnetic moments. The in the late 1970s to describe the electronic excitations in active physics of these materials results from the immersion anewclassofintermetalliccompoundwithanelectronic of these magnetic moments in a quantum sea of mobile con- density of states as much as 1000 times larger than copper. duction electrons. In most rare-earth metals and insulators, Since the original discovery of heavy-fermion behavior in local moments tend to order antiferromagnetically, but, in CeAl3 by Andres, Graebner and Ott (1975), a diversity of heavy-electron metals, the quantum-mechanical jiggling of heavy-fermion compounds, including superconductors, anti- the local moments induced by delocalized electrons is fierce ferromagnets (AFMs), and insulators have been discovered. enough to melt the magnetic order. In the last 10 years, these materials have become the focus of The mechanism by which this takes place involves a intense interest with the discovery that intermetallic AFMs remarkable piece of quantum physics called the Kondo can be tuned through a quantum phase transition into a effect (Kondo, 1962, 1964; Jones, 2007). The Kondo effect heavy-fermion state by pressure, magnetic fields, or chemical describes the process by which a free magnetic ion, with a doping (von Lohneysen¨ et al., 1994; von Lohneysen,¨ 1996; Curie magnetic susceptibility at high temperatures, becomes screened by the spins of the conduction sea, to ultimately Handbook of Magnetism and Advanced Magnetic Materials.Edited by Helmut Kronmuller¨ and Stuart Parkin. Volume 1: Fundamentals form a spinless scatering center at low temperatures and and Theory.  2007 John Wiley & Sons, Ltd. ISBN: 978-0-470- low magnetic fields (Figure 1a). In the Kondo effect, this 02217-7. screening process is continuous, and takes place once the 96 Strongly correlated electronic systems

250

Electrical resistivity

200

UBe13 CeAl3

150

CeCu2 Si2 cm) Ω m ( U2 Zn17 T Free local moment r 1 χ ∼ 100 T

T/TK ~ 1

1.1

50 cm)

Ω CeAl3 m

( 0.9 r 1 Fermi χ ∼ 0.7 liquid TK 0 0.005 0.01 T2 (K2) H 0 H/TK ~ 1 0 100 200 300 (a) (b) T (K)

Figure 1. (a) In the Kondo effect, local moments are free at high temperatures and high fields, but become ‘screened’ at temperatures and magnetic fields that are small compared with the ‘Kondo temperature’ TK,formingresonantscatteringcentersfortheelectronfluid.The magnetic susceptibility χ changes from a Curie-law χ 1 at high temperature, but saturatesataconstantparamagnetic value χ 1 at low ∼ T ∼ TK temperatures and fields. (b) The resistivity drops dramatically at low temperatures in heavy fermion materials, indicating the development of phase coherence between the scatering of the lattice of screened magnetic ions. (Reproduced from J.L. Smith and P.S. Riseborough, J. Mag. Mat. 47–48, 1985, copyright  1985, with permission from Elsevier.) magnetic field, or the temperature drops below a character- net effect of this process is an increase in the volume of istic energy scale called the Kondo temperature TK.Such the electronic Fermi surface, accompanied by a profound ‘quenched’ magnetic moments act as strong elastic scatter- transformation in the electronic masses and interactions. ing potentials for electrons, which gives rise to an increase Aclassicexampleofsuchbehaviorisprovidedbythe in resistivity produced by isolated magnetic ions. When the intermetallic crystal CeCu6.Superficially,thismaterialis 3 same process takes place inside a heavy-electron material, it copper, alloyed with 14% Cerium. The Cerium Ce + ions 3 1 leads to a spin quenching at every site in the lattice, but now, in this material are Ce + ions in a 4f configuration with the strong scattering at each site develops coherence, lead- alocalizedmagneticmomentwithJ 5/2. Yet, at low = ing to a sudden drop in the resistivity at low temperatures temperatures, they lose their spin, behaving as if they were 4 (Figure 1b). Ce + ions with delocalized f electrons. The heavy electrons Heavy-electron materials involve the dense lattice analog that develop in this material are a thousand times ‘heavier’ of the single-ion Kondo effect and are often called Kondo than those in metallic copper, and move with a group velocity lattice compounds (Doniach, 1977). In the lattice, the Kondo that is slower than sound. Unlike copper, which has Fermi effect may be alternatively visualized as the dissolution of temperature of the order 10 000 K, that of CeCu6 is of the localized and neutral magnetic f spins into the quantum order 10 K, and above this temperature, the heavy electrons conduction sea, where they become mobile excitations. Once disintegrate to reveal the underlying magnetic moments of mobile, these free spins acquire charge and form electrons the Cerium ions, which manifest themselves as a Curie-law with a radically enhanced effective mass (Figure 2). The susceptibility χ 1 .Therearemanyhundredsofdifferent ∼ T Heavy fermions: electrons at the edge of magnetism 97

E E mean-field theory (DMFT) approaches to heavy-fermion r(E) physics (Georges, Kotliar, Krauth and Rozenberg, 1996; Cox x Kondo r(E) effect and Grewe, 1988; Jarrell, 1995; Vidhyadhiraja, Smith, Logan ds and Krishnamurthy, 2003) nor the extensive literature on the ∑ p = 1 (a) s order-parameter phenomenology of heavy-fermion supercon- ductors (HFSCs) reviewed in Sigrist and Ueda (1991a). E E r(E) Lattice Kondo x r(E) 1.1 Brief history effect n Heavy-electron materials represent a frontier in a journey of 2 FS = h + h (b) (2p)3 e Spins discovery in electronic and magnetic materials that spans more than 70 years. During this time, the concepts and Figure 2. (a) Single-impurity Kondo effect builds a single understanding have undergone frequent and often dramatic fermionic level into the conduction sea, which gives rise to a reso- nance in the conduction electron density of states. (b) Lattice Kondo revision. effect builds a fermionic resonance into the conduction sea in each In the early 1930s, de Haas, de Boer and van der unit cell. The elastic scattering of this lattice of resonances leads to Berg (1933) in Leiden, discovered a ‘resistance minimum’ the formation of a heavy-electron band, of width TK. that develops in the resistivity of copper, gold, silver, and many other metals at low temperatures (Figure 3). It took a further 30 years before the purity of metals and varieties of heavy-electron material, many developing new alloys improved to a point where the resistance minimum and exotic phases at low temperatures. could be linked to the presence of magnetic impurities This chapter is intended as a perspective on the the current (Clogston et al., 1962; Sarachik, Corenzwit and Longinotti, theoretical and experimental understanding of heavy-electron 1964). Clogston, Mathias, and collaborators at Bell Labs materials. There are important links between the material (Clogston et al., 1962) found they could tune the conditions in this chapter and the proceeding chapter on the Kondo under which iron impurities in Niobium were magnetic, by effect by Jones (2007), the chapter on quantum criticality alloying with molybdenum. Beyond a certain concentration by Sachdev (2007), and the perspective on spin fluctuation of molybdenum, the iron impurities become magnetic and a theories of high-temperature superconductivity by Norman resistance minimum was observed to develop. (2007). For completeness, I have included references to an In 1961, Anderson formulated the first microscopic model extensive list of review articles spanning 30 years of dis- for the formation of magnetic moments in metals. Earlier covery, including books on the Kondo effect and heavy work by Blandin and Friedel (1958) had observed that fermions (Hewson, 1993; Cox and Zawadowski, 1999), gen- localized d states form resonances in the electron sea. eral reviews on heavy-fermion physics (Stewart, 1984; Lee Anderson extended this idea and added a new ingredient: et al., 1986; Ott, 1987; Fulde, Keller and Zwicknagl, 1988; the Coulomb interaction between the d-electrons, which he Grewe and Steglich, 1991), early views of Kondo and mixed modeled by term valence physics (Gruner and Zawadowski, 1974; Varma, 1976), the solution of the Kondo impurity model by renor- HI Un n (1) malization group and the strong coupling expansion (Wil- = ↑ ↓ son, 1976; Nozieres` and Blandin, 1980), the Bethe Ansatz Anderson showed that local moments formed once the method (Andrei, Furuya and Lowenstein, 1983; Tsvelik and Coulomb interaction U became large. One of the unexpected Wiegman, 1983), heavy-fermion superconductivity (Sigrist consequences of this theory is that local moments develop and Ueda, 1991a; Cox and Maple, 1995), Kondo insula- an antiferromagnetic coupling with the spin density of tors (Aeppli and Fisk, 1992; Tsunetsugu, Sigrist and Ueda, the surrounding electron fluid, described by the interaction 1997; Riseborough, 2000), X-ray spectroscopy (Allen et al., (Anderson, 1961; Kondo, 1962, 1964; Schrieffer and Wolff, 1986), optical response in heavy fermions (Degiorgi, 1999), 1966; Coqblin and Schrieffer, 1969) and the latest reviews on non-Fermi liquid behavior and quantum criticality (Stewart, 2001; Coleman, Pepin,´ Si and H J σ(0) S (2) I $ Ramazashvili, 2001; Varma, Nussinov and van Saarlos, 2002; = $ · von Lohneysen,¨ Rosch, Vojta and Wolfe, 2007; Miranda where S is the spin of the local moment and σ(0) is $ $ and Dobrosavljevic, 2005; Flouquet, 2005). There are the spin density of the electron fluid. In Japan, Kondo inevitable apologies, for this chapter is highly selective and, (1962) set out to examine the consequences of this result. 1 partly owing to lack of space, it neither covers dynamical He found that when he calculated the scattering rate τ of 98 Strongly correlated electronic systems

1.04 Mo0.2 Nb0.8 Mo0.4 Nb0.6 1 1

1.02

) K (0)

r

4.2 r

/ 1 Mo0.6 Nb0.4 r ( ∞ )]/

r 0.5 0.5 0.98 ( Τ ) − r 0.96 [ Resistivity ( Resistivity Mo0.9 Nb0.1

0.94 0 010 20 3040 50 60 0.001 0.01 0.1 1 10

(a) Temperature (K) (b) T/TK

Figure 3. (a) Resistance minimum in MoxNb1 x .(ReproducedfromM.Sarachik,E.Corenzwit,andL.D.Longinotti,Phys. Rev. 135,1964, A1041, copyright  by the American Physical− Society, with permission of the APS.) (b) Temperature dependence of excess resistivity produced by scattering off a magnetic ion, showing, universal dependence on a single scale, the Kondo temperature. Original data from White and Geballe (1979). electrons of a magnetic moment to one order higher than exclusive. Tiny concentrations of magnetic impurities pro- Born approximation, duce a lethal suppression of superconductivity in conven- tional metals. Early work on the interplay of the Kondo effect 1 D 2 and superconductivity by Maple et al.(1972)didsuggestthat Jρ 2(Jρ)2 ln (3) τ ∝ + T the Kondo screening suppresses the pair-breaking effects of ! " magnetic moments, but the implication of these results was where ρ is the density of state of electrons in the conduction only slowly digested. Unfortunately, the belief in the mutual sea and D is the width of the electron band. As the exclusion of local moments and superconductivity was so temperature is lowered, the logarithmic term grows, and the deeply ingrained that the first observation of superconductiv- scattering rate and resistivity ultimately rises, connecting the ity in the ‘local moment’ metal UBe13 (Bucher et al., 1975) resistance minimum with the antiferromagnetic interaction was dismissed by its discoverers as an artifact produced by between spins and their surroundings. stray filaments of uranium. Heavy-electron metals were dis- A deeper understanding of the logarithmic term in this covered by Andres, Graebner and Ott (1975), who observed scattering rate required the renormalization group concept that the intermetallic CeAl3 forms a metal in which the Pauli (Anderson and Yuval, 1969, 1970, 1971; Fowler and Zawad- susceptibility and linear specific heat capacity are about 1000 owskii, 1971; Wilson, 1976; Nozieres,` 1976; Nozieres` and times larger than in conventional metals. Few believed their Blandin, 1980). The key idea here is that the physics of a speculation that this might be a lattice version of the Kondo spin inside a metal depends on the energy scale at which it effect, giving rise to a narrow band of ‘heavy’ f electrons in is probed. The ‘Kondo’ effect is a manifestation of the phe- the lattice. The discovery of superconductivity in CeCu2Si2 nomenon of ‘asymptotic freedom’ that also governs quark in a similar f-electron fluid, a year later by Steglich et al. physics. Like the quark, at high energies, the local moments (1976), was met with widespread disbelief. All the measure- inside metals are asymptotically free, but at temperatures ments of the crystal structure of this material pointed to the 3 1 and energies below a characteristic scale the Kondo tem- fact that the Ce ions were in a Ce + or 4f configuration. Yet, perature, this meant one local moment per unit cell – which required an explanation of how these local moments do not destroy 1/(2Jρ) TK De− (4) superconductivity, but, rather, are part of its formation. ∼ Doniach (1977), made the visionary proposal that a heavy- where ρ is the density of electronic states; they interact so electron metal is a dense Kondo lattice (Kasuya, 1956), in strongly with the surrounding electrons that they become which every single local moment in the lattice undergoes screened into a singlet state, or ‘confined’ at low energies, the Kondo effect (Figure 2). In this theory, each spin is ultimately forming a Landau–Fermi liquid (Nozieres,` 1976; magnetically screened by the conduction sea. One of the great Nozieres` and Blandin, 1980). concerns of the time, raised by Nozieres` (1985), was whether Throughout the 1960s and 1970s, conventional wisdom there could ever be sufficient conduction electrons in a dense had it that magnetism and superconductivity are mutually Kondo lattice to screen each local moment. Heavy fermions: electrons at the edge of magnetism 99

Theoretical work on this problem was initially stalled for want of any controlled way to compute properties of the Kondo lattice. In the early 1980s, Anderson (1981) proposed a way out of this log-jam. Taking a cue from the success of the 1/S expansion in spin-wave theory, and the 1/N expansion in statistical mechanics and particle physics, he noted that the large magnetic spin degeneracy N 2j 1 = + of f moments could be used to generate an expansion in the small parameter 1/N about the limit where N .Ander- →∞ son’s idea prompted a renaissance of theoretical development (Ramakrishnan, 1981; Gunnarsson and Schonhammer,¨ 1983; (a) (b) (c) (d) Read and Newns, 1983a,b; Coleman, 1983, 1987a; Auerbach Figure 4. Figure from Monod, Bourbonnais and Emery (1986), one and Levin, 1986), making it possible to compute the X-ray of three path-breaking papers in 1986 to link d-wave pairing to absorption spectra of these materials and, for the first time, antiferromagnetism. (a) The bare interaction, (b), (c), and (d), the examine how heavy f bands form within the Kondo lattice. paramagnon-mediated interaction between antiparallel or parallel By the mid-1980s, the first de Haas van Alphen experiments spins. (Reproduced from M.T.B. Monod, C. Bourbonnais, and (Reinders et al., 1986; Taillefer and Lonzarich, 1988) had V. Emery, Phys. Rev. B. 34,1986,7716,copyright 1986 by the American Physical Society, with permission of the APS.) detected cyclotron orbits of heavy electrons in CeCu6 and UPt3.Withthesedevelopments,theheavy-fermionconcept was cemented. superconductivity. Both the resonating valence bond (RVB) On a separate experimental front, in Ott, Rudigier, Fisk and the spin-fluctuation theory of d-wave pairing in the and Smith (1983), and Ott et al. (1984) returned to the mate- cuprates are, in my opinion, close cousins, if not direct descendents of these early 1986 papers on heavy-electron rial UBe13,and,bymeasuringalargediscontinuityinthe bulk specific heat at the resistive superconducting transition, superconductivity. confirmed it as a bulk heavy-electron superconductor. This After a brief hiatus, interest in heavy-electron physics provided a vital independent confirmation of Steglich’s dis- reignited in the mid-1990s with the discovery of QCPs in covery of heavy electron superconductivity, assuaging the these materials. High-temperature superconductivity intro- old doubts and igniting a huge new interest in heavy-electron duced many important new ideas into our conception of physics. The number of heavy-electron metals and supercon- electron fluids, including ductors grew rapidly in the mid-1980s (Sigrist and Ueda, 1991b). It became clear from specific heat, NMR, and ultra- Non-Fermi liquid behavior: the emergence of metallic • sound experiments on HFSCs that the gap is anisotropic, with states that cannot be described as fluids of renormalized lines of nodes strongly suggesting an electronic, rather than quasiparticles. aphononmechanismofpairing.Thesediscoveriesprompted Quantum phase transitions and the notion that zero tem- • theorists to return to earlier spin-fluctuation-mediated models perature QCPs might profoundly modify finite tempera- of anisotropic pairing. In the early summer of 1986, three ture properties of metal. new theoretical papers were received by Physical Review, the first by Monod, Bourbonnais and Emery (1986) working Both of these effects are seen in a wide variety of heavy- in Orsay, France, followed closely (6 weeks later) by papers electron materials, providing an vital alternative venue for from Scalapino, Loh and Hirsch (1986) at UC Santa Barbara, research on these still unsolved aspects of interlinked, California, and Miyake, Rink and Varma (1986) at Bell Labs, magnetic, and electronic behavior. New Jersey. These papers contrasted heavy-electron super- In 1994 Hilbert von Lohneysen¨ and collaborators discov- conductivity with superfluid He-3. Whereas He-3 is domi- ered that by alloying small amounts of gold into CeCu6,one nated by ferromagnetic interactions, which generate triplet can tune CeCu6 xAux through an antiferromagnetic QCP, − pairing, these works showed that, in heavy-electron sys- and then reverse the process by the application of pressure tems, soft antiferromagnetic spin fluctuations resulting from (von Lohneysen,¨ 1996; von Lohneysen¨ et al., 1994). These the vicinity to an antiferromagnetic instability would drive experiments showed that a heavy-electron metal develops anisotropic d-wave pairing (Figure 4). The almost coinci- ‘non-Fermi liquid’ properties at a QCP, including a linear dent discovery of high-temperature superconductivity the temperature dependence of the resistivity and a logarith- very same year, 1986, meant that these early works on mic dependence of the specific heat coefficient on tempera- heavy-electron superconductivity were destined to exert huge ture. Shortly thereafter, Mathur et al.(1998),atCambridge influence on the evolution of ideas about high-temperature showed that when pressure is used to drive the AFM CeIn3 100 Strongly correlated electronic systems through a quantum phase transition, heavy-electron supercon- important clues (Curro et al., 2005) to the ongoing quest to ductivity develops in the vicinity of the quantum phase tran- discover room-temperature superconductivity. sition. Many new examples of heavy-electron system have come to light in the last few years which follow the same 1.2 Key elements of heavy-fermion metals pattern. In one fascinating development, (Monthoux and Lon- zarich, 1999) suggested that if quasi-two-dimensional ver- Before examining the theory of heavy-electron materials, we sions of the existing materials could be developed, then the make a brief tour of their key properties. Table 1 shows a superconducting pairing would be less frustrated, leading to selective list of heavy fermion compounds ahighertransitiontemperature.Thisledexperimentalgroups to explore the effect of introducing layers into the material 1.2.1 Spin entropy: a driving force for new physics CeIn3,leadingtothediscoveryoftheso-called1 1 5 − − compounds, in which an XIn2 layer has been introduced The properties of heavy-fermion compounds derive from into the original cubic compound. (Petrovic et al., 2001; the partially filled f orbitals of rare-earth or actinide ions Sidorov et al., 2002). Two notable members of this group are (Stewart, 1984; Lee et al., 1986; Ott, 1987; Fulde, Keller CeCoIn5 and, most recently, PuCoGa5 (Sarrao et al., 2002). and Zwicknagl, 1988; Grewe and Steglich, 1991). The large The transition temperature rose from 0.5to2.5K inmoving nuclear charge in these ions causes their f orbitals to collapse from CeIn3 to CeCoIn5.Mostremarkably,thetransitiontem- inside the inert gas core of the ion, turning them into localized perature rises to above 18 K in the PuCoGa5 material. This magnetic moments. amazing rise in Tc,anditscloseconnectionwithquantum Moreover, the large spin-orbit coupling in f orbitals com- criticality, are very active areas of research, and may hold bines the spin and angular momentum of the f states into a

Table 1. Selected heavy-fermion compounds.

1 2 Type Material T ∗ (K) Tc,xc,Bc Properties ρ mJmol− K− References γ n

2 Metal CeCu6 10 – Simple HF T 1600 Stewart, Fisk and Wire (1984a) metal and Onuki and Komatsubara (1987)

2 Super- CeCu2Si2 20 Tc 0.17 K First HFSC T 800–1250 Steglich et al.(1976)and conductors = Geibel et al.(1991a,b) UBe13 2.5 Tc 0.86 K Incoherent ρ 800 Ott, Rudigier, Fisk and Smith = c ∼ metal HFSC 150 µ& cm (1983, 1984) → CeCoIn5 38 Tc 2.3Quasi2D T 750 Petrovic et al.(2001)and = HFSC Sidorov et al.(2002)

'/T Kondo Ce3Pt4Bi3 Tχ 80 – Fully gapped e –Hundleyet al.(1990)and insulators ∼ KI ∼ Bucher, Schlessinger, Canfield and Fisk (1994) CeNiSn Tχ 20 – Nodal KI Poor metal – Takabatake et al.(1990,1992) ∼ and Izawa et al.(1999)

1 T0 Quantum CeCu6 xAux T0 10 xc 0.1Chemically T ln von Lohneysen¨ et al.(1994)and − ∼ = ∼ T0 T critical tuned QCP # $ von Lohneysen¨ (1996) 1 T0 YbRh2Si2 T0 24 B 0.06 T Field-tuned T ln Trovarelli et al.(2000),Paschen ∼ ⊥ = ∼ T0 T B 0.66 T QCP # $ et al.(2004),Custerset al. + = (2003) and Gegenwart et al. (2005)

2 SC other UPd2Al3 110 TAF 14 K, AFM + HFSC T 210 Geibel et al.(1991a),Satoet al. + = order Tsc 2K (2001) and Tou et al.(1995) = 2 URu2Si2 75 T1 17.5K, Hidden order T 120/65 Palstra et al.(1985)andKim = Tsc 1.3K and HFSC et al.(2003) =

Unless otherwise stated, T ∗ denotes the temperature of the maximum in resistivity. Tc, xc,andBc denote critical temperature, doping, and field. ρ denotes the temperature dependence in the normal state. γ CV /T is the specific heat coefficient in the normal state. n = Heavy fermions: electrons at the edge of magnetism 101

state of definite J ,anditistheselargequantumspindegrees 3.0 of freedom that lie at the heart of heavy-fermion physics. Heavy-fermion materials display properties which change 2.5 qualitatively, depending on the temperature, so much so, that UBe13 Specific heat

the room-temperature and low-temperature behavior almost 2.0 mol)

resembles two different materials. At room temperature, high –2 T C magnetic fields, and high frequencies, they behave as local 1.5 v dT′ = Spin entropy (T) 0 T

moment systems, with a Curie-law susceptibility ′ / T (JK T /

ej 1.0 C 2 M 2 2 χ M (gJ µ ) J(J 1) (5) = 3T = B + 0.5 where M is the magnetic moment of an f state with 0 0123456 total angular momentum J and the gyromagnetic ratio gJ . However, at temperatures beneath a characteristic scale, T (K) we call T ∗ (to distinguish it from the single-ion Kondo Figure 5. Showing the specific heat coefficient of UBe13 after (Ott, temperature TK), the localized spin degrees of freedom melt Rudigier, Fisk and Smith, 1985). The area under the CV /T curve up into the conduction sea, releasing their spins as mobile, to a temperature T provides a measure of the amount of unquenched conducting f electrons. spin entropy at that temperature. The condensation entropy of ACuriesusceptibilityisthehallmarkofthedecoupled, HFSCs is derived from the spin-rotational degrees of freedom of the local moments, and the large scale of the condensation entropy rotational dynamics of the f moments, associated with an indicates that spins partake in the formation of the order parameter. unquenched entropy of kB ln N per spin, where N S = = (Reproduced from H.R. Ott, H. Rudigier, Z. Fisk, and J.L. Smith, 2J 1isthespindegeneracyofanisolatedmagnetic in W.J.L. Buyers (ed.): Proceedings of the NATO Advanced Study + moment of angular momentum J .Forexample,inaCerium- Institute on Moment Formation in Solids, Vancouver Island,August heavy electron material, the 4f1 (L 3) configuration of 1983, Valence Fluctuations in Solids (Plenum, 1985), p. 309. with 3 = permission of Springer Science and Business Media.) the Ce + ion is spin-orbit coupled into a state of definite J L S 5/2withN 6. Inside the crystal, the full = − = = rotational symmetry of each magnetic f ion is often reduced connected to those of a noninteracting electron fluid. Heavy- by crystal fields to a quartet (N 4) or a Kramer’s doublet fermion metals are extreme examples of Landau–Fermi = N 2. At the characteristic temperature T ∗, as the Kondo liquids which push the idea of adiabaticity into an regime = effect develops, the spin entropy is rapidly lost from the where the bare electron interactions, on the scale of electron material, and large quantities of heat are lost from the volts, are hundreds, even thousands of times larger than material. Since the area under the specific heat curve the millivolt Fermi energy scale of the heavy-electron determines the entropy, quasiparticles. The Landau–Fermi liquid that develops in these materials shares much in common with the Fermi T CV liquid that develops around an isolated magnetic impurity S(T) dT , (6) = T %0 , (Nozieres,` 1976; Nozieres` and Blandin, 1980), once it is quenched by the conduction sea as part of the Kondo effect. arapidlossofspinentropyatlowtemperaturesforcesasud- There are three key features of this Fermi liquid: den rise in the specific heat capacity. Figure 5 illustrates this phenomenon with the specific heat capacity of UBe13.Notice how the specific heat coefficient C /T rises to a value of Single scale: T ∗ The quasiparticle density of states ρ∗ V • ∼ order 1 J mol 1K2,andstartstosaturateatabout1K,indicat- 1/T ∗ and scattering amplitudes Akσ,k σ T ∗ scale − , , ∼ ing the formation of a Fermi liquid with a linear specific heat approximately with a single scale T ∗. coefficient. Remarkably, just as the linear specific heat starts Almost incompressible: Heavy-electron fluids are ‘almost • to develop, UBe13 becomes superconducting, as indicated by incompressible’, in the sense that the charge suscepti- the large specific heat anomaly. bility χ c dNe/dµ ρ∗ is unrenormalized and typi- = - cally more than an order of magnitude smaller than the quasiparticle density of states ρ .Thisisbecausethe 1.2.2 ‘Local’ Fermi liquids with a single scale ∗ lattice of spins severely modifies the quasiparticle den- The standard theoretical framework for describing metals is sity of states, but leaves the charge density of the fluid Landau–Fermi liquid theory (Landau, 1957), according to ne(µ),anditsdependenceonthechemicalpotentialµ which the excitation spectrum of a metal can be adiabatically unchanged. 102 Strongly correlated electronic systems

Local: Quasiparticles scatter when in the vicinity of a ρ • χ e2 ∗ e2ρ 1 As (11) local moment, giving rise to a small momentum depen- c s ∗ 0 = 1 F0 = − dence to the Landau scattering amplitudes (Yamada, + * + 1975; Yoshida and Yamada, 1975; Engelbrecht and where the quantities Bedell, 1995). F s,a As,a 0 (12) Landau–Fermi liquid theory relates the properties of a 0 = 1 F s,a + 0 Fermi liquid to the density of states of the quasiparticles and asmallnumberofinteractionparameters(BaymandPethick, are the s-wave Landau scattering amplitudes in the charge 1992). If Ekσ is the energy of an isolated quasiparticle, then (s) and spin (a) channels, respectively (Baym and Pethick, the quasiparticle density of states ρ δ(Ekσ µ) 1992). ∗ = kσ − determines the linear specific heat coefficient The assumption of local scattering and incompressibility & in heavy electron fluids simplifies the situation, for, in this 2 2 CV π kB case, only the l 0 components of the interaction remain γ LimT 0 ρ∗ (7) = = → T = 3 and the quasiparticle scattering amplitudes become ' ( In conventional metals, the linear specific heat coefficient is 1 s a A A σσ,A (13) 1 2 kσ,k,σ , 0 0 of the order 1–10 mJ mol− K− .Inasystemwithquadratic = ρ∗ + 2 2 ! k dispersion, Ek ,thequasiparticledensityofstatesand , - = 2m∗ Moreover, in local scattering, the Pauli principle dictates that effective mass m are directly proportional ∗ quasiparticles scattering at the same point can only scatter k when in opposite spin states, so that ρ F m ∗ 2 2 ∗ (8) = π ! (0) s a ' ( A A0 A0 0(14) ↑↑ = + = where kF is the Fermi momentum. In heavy-fermion com- s a pounds, the scale of ρ∗ varies widely, and specific heat and hence A0 A0.Theadditionalassumptionofincom- 1 2 =− 2 s a coefficients in the range 100–1600 mJ mol K have been pressibility forces χ c/(e ρ∗) 1, so that now A A − − - 0 =− 0 ≈ observed. From this simplified perspective, the quasiparticle 1andallthatremainsisasingleparameterρ∗. effective masses in heavy-electron materials are two or three This line of reasoning, first developed for the single orders of magnitude ‘heavier’ than in conventional metals. impurity Kondo model by Nozieres` and Blandin (1980) and, In Landau–Fermi liquid theory, a change δn in the Nozieres` (1976) and later extended to a bulk Fermi liquid by k,σ , quasiparticle occupancies causes a shift in the quasiparticle Engelbrecht and Bedell (1995), enables us to understand two energies given by important scaling trends amongst heavy-electron systems. The first consequence, deduced from equation (11), is that δE f δn (9) the dimensionless Sommerfeld ratio, or ‘Wilson ratio’ W kσ kσ,kσ , k,σ , = π2k2 χ = k σ B s , , 2 γ 2. Wilson (1976) found that this ratio is almost ) µB ≈ ' ( In a simplified model with a spherical Fermi surface, the exactly equal to 2 in the numerical renormalization group Landau interaction parameters only depend on the relative treatment of the impurity Kondo model. The connection angle θ between the quasiparticle momenta, and are between this ratio and the local Fermi liquid theory was k,k, expanded in terms of Legendre Polynomials as first identified by Nozieres` (1976), and Nozieres` and Blandin (1980). In real heavy-electron systems, the effect of spin-orbit 1 s a coupling slightly modifies the precise numerical form for this f (2l 1)Pl(θ )[F σσ,F ](10) kσ,kσ , = ρ + k,k, l + l ratio, nevertheless, the observation that W 1overawide ∗ l ∼ ) range of materials in which the density of states vary by more s,a than a factor of 100 is an indication of the incompressible The dimensionless ‘Landau parameters’ Fl parameterize the detailed quasiparticle interactions. The s-wave (l 0) and local character of heavy Fermi liquids (Figure 6). = Landau parameters that determine the magnetic and charge Asecondconsequenceoflocalityappearsinthetrans- susceptibility of a Landau–Fermi liquid are given by Landau port properties. In a Landau–Fermi liquid, inelastic electron– (1957), and Baym and Pethick (1992) electron scattering produces a quadratic temperature depen- dence in the resistivity

2 ρ∗ 2 a χ s µB a µBρ∗ 1 A0 2 = 1 F = − ρ(T) ρ0 AT (15) 0 = + + * + Heavy fermions: electrons at the edge of magnetism 103

B.A. Jones et al. (1985)

4f 5f Superconducting

Magnetic CeAl3 CeCu Si Not superconducting 2 2 CeCu5

) UBe 1 1000 or magnetic 13

−

)

2 UCd11 U2Zn17 NpBe UPt3 13

YbCuAl NpIr2 NpOs2 CeAl2 UAl2

(mJ (mol) f atom K

g 100

U2PtC2

UIr2 CeRu3Si2 U6Fe YbAl2 a–Np a–U a–Ce 10 10–4 10–3 10–2 10–1 c (O) (emu (mol f atom)−1)

Figure 6. Plot of linear specific heat coefficient versus Pauli susceptibility to show approximate constancy of the Wilson ratio. (Reproduced from P.A. Lee, T.M. Rice, J.W. Serene, L.J. Sham, and J.W. Wilkins, Comments Condens. Matt. Phys. 9212, (1986) 99, with permission from Taylaor & Francis Ltd, www.informaworld.com.)

In conventional metals, resistivity is dominated by electron– The approximate validity of the scaling relations phonon scattering, and the ‘A’coefficientisgenerallytoo χ A small for the electron–electron contribution to the resis- , cons 2 cons (18) tivity to be observed. In strongly interacting metals, the γ ≈ γ ≈ A coefficient becomes large, and, in a beautiful piece of for a wide range of heavy-electron compounds constitutes phenomenology, Kadowaki and Woods (1986), observed excellent support for the Fermi-liquid picture of heavy that the ratio of A to the square of the specific heat electrons. 2 coefficient γ A classic signature of heavy-fermion behavior is the dramatic change in transport properties that accompanies A 5 2 1 the development of a coherent heavy-fermion band structure αKW (1 10− )µ&cm(mol K mJ− ) (16) = γ 2 ≈ × (Figure 6). At high temperatures, heavy-fermion compounds exhibit a large saturated resistivity, induced by incoherent is approximately constant, over a range of A spanning four spin-flip scattering of the conduction electrons of the local orders of magnitude. This can also be simply understood fmoments.Thisscatteringgrows as the temperature is from the local Fermi-liquid theory, where the local scattering lowered, but, at the same time, it becomes increasingly amplitudes give rise to an electron mean-free path given by elastic at low temperatures. This leads to the development of phase coherence. the f-electron spins. In the case of heavy- 1 T 2 fermion metals, the development of coherence is marked by constant 2 (17) kFl∗ ∼ + (T ∗) arapidreductionintheresistivity,butinaremarkableclass of heavy fermion or ‘Kondo insulators’, the development The ‘A’coefficientintheelectronresistivitythatresults of coherence leads to a filled band with a tiny insulating 1 2 from the second term satisfies A 2 γ .Amore gap of the order TK.Inthiscase,coherenceismarked ∝ (T ∗) ∝˜ detailed calculation is able to account for the magnitude of by a sudden exponential rise in the resistivity and Hall the Kadowaki–Woods constant, and its weak residual depen- constant. dence on the spin degeneracy N 2J 1ofthemagnetic The classic example of coherence is provided by metallic = + ions (see Figure 7). CeCu6,whichdevelops‘coherence’andamaximumin 104 Strongly correlated electronic systems

2 10 300 UBe CexLa1 xCu6 13 x 0.094 − N = 2 = CeCu6 A/g2 = 1 × 10−5 0.29 CeAl3 µΩcm(K mol mJ−1)2 200 YbRh2Si2 0.5 CeCu Si 1 2 2 cm/Ce) 10 0.73 N = 8 µ Ω (

N = 6 m 0.9 r 100 N = 4 YbNi2B2C UPt3 N = 2 YbRh2Si2 0.99 N = 4 5f (H = 6T) 1.0 0 10 CeB N = 6 6 0.01 0.1 1 10 100 CeRu2Si2 (K) SmOs4Sb12 USn3 )

2 UAl SmFe4P12

− 2 Figure 8. Development of coherence in Ce1 x Lax Cu6.(Repro- YbCu YbCuAl 5 − −1 UPt UPt duced from Y. Onuki and T. Komatsubara, J. Mag. Mat. 63–64, cm K 10 2 Yb2Co3Ga9 Eu(Pt Ni ) Si 1987, 281, copyright  1987, with permission of Elsevier.) µ Ω 0.8 0.2 2 2 A ( UIn CePd 3 YbCu4.5Ag0.5 3 the spins of the screened local moments are also included in UGa3 Eu(Pt0.75Ni0.25)2Si2 the sum CeNi 10−2 2VFS YbCu4Ag YbInAu2 [ne nspins](19) (2π)3 = + CeNi9Si4 YbNi Ge 2 2 Remarkably, even though f electrons are localized as mag- −3 10 CeSn3 netic moments at high temperatures, in the heavy Fermi N = 8 YbInCu4 liquid, they contribute to the Fermi surface volume. A/g2 = 0.36 × 10−6 YbAl3 The most direct evidence for the large heavy f-Fermi sur- YbAl µΩcm(mol K mJ−1)2 2 faces derives from de Haas van Alphen and Shubnikov de Haas experiments that measure the oscillatory diamagnetism 10−4 23456 23456 23 or and resistivity produced by coherent quasiparticle orbits 10 100 1000 (Figure 9). These experiments provide a direct measure of 1 2 g (mJ mol− K ) the heavy-electron mass, the Fermi surface geometry, and volume. Since the pioneering measurements on CeCu and Figure 7. Approximate constancy of the Kadowaki–Woods ratio, 6 for a wide range of heavy electrons. (After Tsujji, Kontani and UPt3 by Reinders and Springford, Taillefer, and Lonzarich Yoshimora, 2005.) When spin-orbit effects are taken into account, in the mid-1980s (Reinders et al., 1986; Taillefer and Lon- the Kadowaki–Woods ratio depends on the effective degeneracy zarich, 1988; Taillefer et al., 1987), an extensive number of N 2J 1ofthemagneticion,whichwhentakenintoaccount = + such measurements have been carried out (Onuki and Komat- leads to a far more precise collapse of the data onto a single curve. subara, 1987; Julian, Teunissen and Wiegers, 1992; Kimura (Reproduced from H. Tsujji, H. Kontani, and K. Yoshimora, Phys. et al., 1998; McCollam et al., 2005). Two key features are Rev. Lett 94, 2005, copyright  2005 by the American Physical Sdociety, with permisison of the APS.057201.) observed: its resistivity around T 10 K. Coherent heavy-electron AFermisurfacevolumewhichcountsthefelectronsas = • propagation is readily destroyed by substitutional impurities. itinerant quasiparticles. 3 In CeCu6,Ce+ ions can be continuously substituted with Effective masses often in excess of 100 free electron 3 • nonmagnetic La + ions, producing a continuous crossover masses. Higher mass quasiparticle orbits, though inferred from coherent Kondo lattice to single impurity behavior from thermodynamics, cannot be observed with current (Figure 8). measurement techniques. One of the important principles of the Landau–Fermi liq- Often, but not always, the Fermi surface geometry is in • uid is the Fermi surface counting rule, or Luttinger’s theorem accord with band theory, despite the huge renormaliza- (Luttinger, 1960). In noninteracting electron band theory, the tions of the electron mass. volume of the Fermi surface counts the number of conduction electrons. For interacting systems, this rule survives (Martin, Additional confirmation of the itinerant nature of the f 1982; Oshikawa, 2000), with the unexpected corollary that quasiparticles comes from the observation of a Drude peak in Heavy fermions: electrons at the edge of magnetism 105

t [0001] s t

a w s w K

H A L 0 5 10 15 [1120] (a) [1010] (b) dHvA frequency (× 107 Oe)

Figure 9. (a) Fermi surface of UPt3 calculated from band theory assuming itinerant5felectrons(OguchiandFreeman,1985;Wanget al., 1987; Norman, Oguchi and Freeman, 1988), showing three orbits (σ, ω and τ)thatareidentifiedbydHvAmeasurements.(AfterKimura et al., 1998.) (b) Fourier transform of dHvA oscillations identifying σ, ω,andτ orbits shown in (a). (Kimura et al., 1998.) the optical conductivity. At low temperatures, in the coherent where Hatomic describes the atomic limit of an isolated regime, an extremely narrow Drude peak can be observed in magnetic ion and Hresonance describes the hybridization of the optical conductivity of heavy-fermion metals. The weight the localized f electrons in the ion with the Bloch waves of under the Drude peak is a measure of the plasma frequency: the conduction sea. For pedagogical reasons, our discussion the diamagnetic response of the heavy-fermion metal. This initially focuses on the case where the f state is a Kramer’s is found to be extremely small, depressed by the large mass doublet. enhancement of the quasiparticles (Millis and Lee, 1987a; There are two key elements to the Anderson model: Degiorgi, 1999). Atomic limit: The atomic physics of an isolated ion with • dω ne2 asinglef state, described by the model σ qp(ω) (20) ω < TK π = m∗ %| | Hatomic Ef nf Unf nf (22) = + ↑ ↓ Both the optical and. dHvA experiments indicate that the presence of f spins depresses both the spin and diamagnetic Here Ef is the energy of the f state and U is the response of the electron gas down to low temperatures. Coulomb energy associated with two electrons in the same orbital. The atomic physics contains the basic mechanism for local moment formation, valid for f electrons, but also seen in a variety of other contexts, 2LOCALMOMENTSANDTHEKONDO such as transition-metal atoms and quantum dots. LATTICE The four quantum states of the atomic model are

2 2 2.1 Local moment formation f E(f ) 2Ef U | 0 = + nonmagnetic f 0 E(f 0) 0 | 0 = 5 (23) 2.1.1 The Anderson model 1 1 1 f f E(f ) Ef magnetic We begin with a discussion of how magnetic moments form | ↑0 | ↓0 = at high temperatures, and how they are screened again at low In a magnetic ground state, the cost of inducing a temperatures to form a Fermi liquid. The basic model for ‘valence fluctuation’ by removing or adding an electron local moment formation is the Anderson model (Anderson, to the f1 state is positive, that is, 1961) removing: E(f 0) E(f 1) Hresonance − U U † † Ef > 0 >Ef (24) H -knkσ V(k) c fσ f ckσ =− ⇒ 2 + 2 = + kσ + σ /)k,σ )k,σ 01 3 42 adding: E(f 2) E(f 1) − Efnf Unf nf (21) + + ↑ ↓ U U Ef U>0 Ef > (25) Hatomic = + ⇒ + 2 − 2 1 2/ 0 106 Strongly correlated electronic systems

or (Figure 10). The ‘atomic picture’, which starts with the interacting, • but isolated atom (V(k) 0), and considers the effect = U U U of immersing it in an electron sea by slowly dialing up >Ef > (26) 2 + 2 − 2 the hybridization. The ‘adiabatic picture’, which starts with the noninter- Under these conditions, a local moment is well defined, • acting resonant ground state (U 0), and then considers provided the temperature is lower than the valence fluc- = the effect of dialing up the interaction term U. tuation scale TVF max(Ef U, Ef).Atlowertem- = + − peratures, the atom behaves exclusively as a quantum top. These approaches paint a contrasting and, at first sight, Virtual bound-state formation. When the magnetic ion is contradictory picture of a local moment in a Fermi sea. From • immersed in a sea of electrons, the f electrons within the adiabatic perspective, the ground state is always a Fermi the core of the atom hybridize with the Bloch states of liquid (see 1.2.2), but from atomic perspective, provided the surrounding electron sea (Blandin and Friedel, 1958) to hybridization is smaller than U,oneexpectsalocalmagnetic form a resonance described by moment, whose low-lying degrees of freedom are purely rotational. How do we resolve this paradox?

Hresonance -knkσ Anderson’s original work provided a mean-field treatment = k,σ of the interaction. He found that at interactions larger than ) † † Uc π' local moments develop with a finite magnetization V(k)c fσ V(k)∗f ckσ (27) ∼ + kσ + σ M n n .Themean-fieldtheoryprovidesanapprox- k,σ =2 ↑0−2 ↓0 ) 3 4 imate guide to the conditions required for moment formation, where the hybridization matrix element V(k) but it does not account for the restoration of the singlet sym- = metry of the ground state at low temperatures. The resolution k Vatomic f is the overlap of the atomic potential 2 | | 0 between a localized f state and a Bloch wave. In the of the adiabatic and the atomic picture derives from quantum absence of any interactions, the hybridization broadens spin fluctuations, which cause the local moment to ‘tunnel’ the localized f state, producing a resonance of width on a slow timescale τ sf between the two degenerate ‘up’ and ‘down’ configurations. 2 2 ' π V(k) δ(-k µ) πV ρ (28) = | | − = 1 1 k e− f ! e− f (29) ) ↓ + ↑ ↑ + ↓ where V 2 is the average of the hybridization around the These fluctuations are the origin of the Kondo effect. From Fermi surface. the energy uncertainty principle, below a temperature TK, at which the thermal excitation energy k T is of the order There are two complementary ways to approach the B of the characteristic tunneling rate ! ,aparamagneticstate physics of the Anderson model: τ sf with a Fermi-liquid resonance forms. The characteristic width of the resonance is then determined by the Kondo E + U/2 ! f energy kBTK .Theexistenceofthisresonancewasfirst ∼ τ sf deduced by Abrikosov (1965), and Suhl (1965), but it is more frequently called the Kondo resonance.Fromperturbative E + U/2 = U f 0 f renormalization group reasoning (Haldane, 1978) and the Bethe Ansatz solution of the Anderson model (Wiegmann, Local 1980; Okiji and Kawakami, 1983), we know that, for large Charge Kondo effect moments U ',theKondoscaledependsexponentiallyonU.Inthe U 3 symmetric Anderson model, where Ef U/2, =−

1 2U' πU f 2 f TK exp (30) = π 2 − 8' Ef + U/2 = −U 6 ' (

The temperature TK marks the crossover from a a high- 1 Figure 10. Phase diagram for Anderson impurity model in the temperature Curie-law χ susceptibility to a low- ∼ T atomic limit. temperature paramagnetic susceptibility χ 1/TK. ∼ Heavy fermions: electrons at the edge of magnetism 107

2.1.2 Adiabaticity and the Kondo resonance Af (w) Acentralquantityinthephysicsoff-electronsystemsisthe f-spectral function,

1 Af (ω) ImGf (ω iδ) (31) = π −

† iωt where Gf (ω) i ∞ dt Tfσ (t)f (0) e is the Fourier ∆ =− 2 σ 0 transform of the time-ordered−∞ f-Green’s function. When TK 7 1 2 an f electron is added, or removed from the f state, the e− + f → f final state has a distribution of energies described by the f 1 → f 0 + e− f-spectral function. From a spectral decomposition of the w f-Green’s function, the positive energy part of the f-spectral function determines the energy distribution for electron 0 addition, while the negative energy part measures the energy distribution of electron removal:

Energy distribution of state formed by adding one f electron

† 2 w = Ef + U  λ f φ0 δ(ω [Eλ E0]), (ω>0) 2 | σ | 0 − −  /λ 01 2 Af (ω)  ) < <2 (32) = < λ fσ φ0 < δ(ω [E0 Eλ]), (ω<0) U  2 | | 0 − −  λ ) < < w = Ef  Energy< distribution of state< formed by removing an f electron  Kondo   1 2/ 0 where E0is the energy of the ground state, and Eλ is the energy of an excited state λ,formedbyaddingor Anderson removing an f electron. For negative energies, this spectrum U Infinite can be measured by measuring the energy distribution of photoelectrons produced by X-ray photoemission, while for Figure 11. Schematic illustration of the evaluation of the f-spectral positive energies, the spectral function can be measured from function Af (ω) as interaction strength U is turned on continuously, inverse X-ray photoemission (Allen et al., 1986; Allen, Oh, maintaining a constant f occupancy by shifting the bare f-level Maple and Torikachvili, 1983). The weight beneath the Fermi position beneath the Fermi energy. The lower part of diagram is the energy peak determines the f charge of the ion density plot of f-spectral function, showing how the noninteracting resonance at U 0splitsintoanupperandloweratomicpeakat = ω Ef and ω Ef U. 0 = = + nf 2 dωAf (ω) (33) 2 0= %−∞ invariants that do not change, despite the interaction. One such quantity is the phase shift δ associated with the In a magnetic ion, such as a Cerium atom in a 4f1 state, this f scattering of conduction electrons of the ion; another is the quantity is just a little below unity. height of the f-spectral function at zero energy, and it turns Figure 11 illustrates the effect of the interaction on the out that these two quantities are related. A rigorous result f-spectral function. In the noninteracting limit (U 0), the = owing to (Langreth, 1966) tells us that the spectral function f-spectral function is a Lorentzian of width '.Ifweturnon at ω 0isdirectlydeterminedbythef-phaseshift,sothat the interaction U,beingcarefultoshiftingthef-levelposition = its noninteracting value beneath the Fermi energy to maintain a constant occupancy, the resonance splits into three peaks, two at energies ω Ef 2 = sin δf and ω Ef U corresponding to the energies for a valence Af (ω 0) (34) = + = = π' fluctuation, plus an additional central ‘Kondo resonance’ associated with the spin fluctuations of the local moment. is preserved by adiabaticity. Langreth’s result can be heuris- At first sight, once the interaction is much larger than tically derived by noting that δf is the phase of the the hybridization width ',onemightexpecttheretobeno f-Green’s function at the Fermi energy, so that Gf (0 1 1 iδ − spectral weight left at low energies. But this violates the idea i-) G− (0) e f .Now,inaFermiliquid,thescatter- − =| f | − of adiabaticity. In fact, there are always certain adiabatic ing at the Fermi energy is purely elastic, and this implies 108 Strongly correlated electronic systems

1 that ImGf− (0 i-) ',thebarehybridizationwidth. − = 1 1 From this, it follows that ImGf− (0) Gf− (0) sin δf ', iδ =| | = so that Gf (0) e f /(' sin δf ),andtheprecedingresult = follows. The phase shift δf is set via the Friedel sum rule, according (a) CeAI to which the sum of the up-and-down scattering phase shifts, gives the total number of f-bound electrons, or

δ δ fσ 2 f n (35) π π f σ = = ) for a twofold degenerate f state. At large distances, the wave function of scattered electrons ψ (r) sin(kFr δf )/r is f ∼ + ‘shifted inwards’ by a distance δl/kF (λF/2) (δl/π). = × This sum rule is sometimes called a node counting rule (b) CeIr2 because, if you think about a large sphere enclosing the impurity, then each time the phase shift passes through π,a node crosses the spherical boundary and one more electron Intensity (au) per channel is bound beneath the Fermi sea. Friedel’s sum rule holds for interacting electrons, provided the ground state is adiabatically accessible from the noninteracting system (Langer and Ambegaokar, 1961; Langreth, 1966). Since 1 nf 1inanf state, the Friedel sum rule tells us that (c) CeRu2 = the phase shift is π/2 for a twofold degenerate f state. In other words, adiabaticity tell us that the electron is resonantly scattered by the quenched local moment. Photoemission studies do reveal the three-peaked structure characteristic of the Anderson model in many Ce systems, such as CeIr2 and CeRu2 (Allen, Oh, Maple and Torikachvili, 1983) (see Figure 12). Materials in which the Kondo resonance is wide enough to be resolved are more ‘mixed valent’ materials in which the f valence departs significantly −10 0 10 from unity. Three-peaked structures have also been observed Energy above EF (eV) in certain U 5f materials such as UPt3 and UAl2 (Allen et al., 1985) materials, but it has not yet been resolved in UBe13. Figure 12. Showing spectral functions for three different Cerium Athree-peakedstructurehasrecentlybeenobservedin4f f-electron materials, measured using X-ray photoemission (below 13 Yb materials, such as YbPd3,wherethe4f configuration the Fermi energy ) and inverse X-ray photoemission (above the contains a single f hole, so that the positions of the three Fermi energy). CeAl is an AFM and does not display a Kondo resonance. (Reproduced from J.W. Allen, S.J. Oh, M.B. Maple and peaks are reversed relative to Ce (Liu et al., 1992). M.S. Torikachvili: Phys. Rev. 28,1983,5347,copyright 1983 by the American Physical Society, with permission of the APS.)

2.2 Hierarchies of energy scales of times smaller. How can we distill the essential effects of the atomic physics at electron volt scales on the low-energy 2.2.1 Renormalization concept physics at millivolt scales? To understand how a Fermi liquid emerges when a local The essential tool for this task is the ‘renormalization moment is immersed in a quantum sea of electrons, theorists group’ (Anderson and Yuval, 1969, 1970, 1971; Anderson, had to connect physics on several widely spaced energy 1970, 1973; Wilson, 1976; Nozieres` and Blandin, 1980; scales. Photoemission shows that the characteristic energy Nozieres,` 1976), based on the idea that the physics at low- to produce a valence fluctuation is of the order of volts, or energy scales only depends on a small subset of ‘relevant’ tens of thousands of Kelvin, yet the characteristic physics variables from the original microscopic Hamiltonian. The we are interested in occurs at scales hundreds or thousands extraction of these relevant variables is accomplished by Heavy fermions: electrons at the edge of magnetism 109

‘renormalizing’ the Hamiltonian by systematically eliminat- projected out by the renormalization process, and the char- ing the high-energy virtual excitations and adjusting the acter of the Hamiltonian changes qualitatively. In the Ander- low-energy Hamiltonian to take care of the interactions that son model, there are three such important energy scales, these virtual excitations induce in the low energy Hilbert (Figure 13) space. This leads to a family of Hamiltonian’s H(1),each 1 with a different high-energy cutoff 1,whichsharethesame 1I Ef U,wherevalencefluctuationse− f ! • 2 = + 2 + low-energy physics. f into the doubly occupied f state are eliminated. For 1 1I ,thephysicsisdescribedbytheinfinite The systematic passage from a Hamiltonian H(1) to - U Anderson model arenormalizedHamiltonianH(1,) with a smaller cutoff 1, 1/b is accomplished by dividing the eigenstates of † = H -knkσ V(k) c X0σ Xσ 0ckσ H into a a low-energy subspace L and a high-energy = + kσ + { } k,σ k,σ subspace H ,withenergies - <1, 1/b and a - ) ) 3 4 { } | | = | |∈ [1 ,1] respectively. The Hamiltonian is then broken up into Ef Xσσ, (38) , + terms that are block-diagonal in these subspaces, σ ) 1 1 0 1 where Xσσ f : σ f : σ , X0σ f f σ and † HL V 1 =| 0 02 | =| 02 | Xσ 0 f : σ f are ‘Hubbard operators’ that con- H (36) =| 02 | = V H nect the states in the projected Hilbert space with no ! < H " < < double occupancy. < where V and V † provide the matrix elements between L 1II Ef Ef ,wherevalencefluctuationsintothe { } • ∼| |=1 − 0 and H .TheeffectsoftheV are then taken into account by empty state f ! f e− are eliminated to form a local { } + carrying out a unitary (canonical) transformation that block- moment. Physics below this scale is described by the diagonalizes the Hamiltonian, Kondo model. 1 TK,theKondotemperaturebelowwhichthelocal • = moment is screened to form a resonantly scattering local HL 0 H(1) UH(1)U† ˜ (37) Fermi liquid. → = = 0 < ˜H < In the symmetric Anderson model, 1I 1II,andthe < = < transition to local moment behavior occurs in a one-step The renormalized Hamiltonian is then given by H(1,) = crossover process. HL HL δH.TheflowofkeyparametersintheHamil- ˜ = + tonian resulting from this process is called a renormalization group flow. 2.2.2 Schrieffer–Wolff transformation At certain important crossover energy scales, large tracts The unitary or canonical transformation that eliminates of the Hilbert space associated with the Hamiltonian are the charge fluctuations at scales 1I and 1II was first

Hamiltonian Flows Excitations

Λ Anderson model H(Λ) f 0 f 1 f 2 ΛI = Ef + U Valence fluctuations Infinite U Anderson model f 0 f 1 ΛII ~ −Ef Moment formation Kondo model 1 1 f f Local moments

ΛIII = −TK Local Fermi liquid

(a) (b) FP Quasiparticles

Figure 13. (a) Crossover energy scales for the Anderson model. At scales below 1I ,valencefluctuationsintothedoublyoccupiedstate are suppressed. All lower energy physics is described by the infinite U Anderson model. Below 1II,allvalencefluctuations are suppressed, and the physics involves purely the spin degrees of freedom of the ion, coupled to the conduction sea via the Kondo interaction. The Kondo scale renormalizes to strong coupling below 1III,andthelocalmomentbecomesscreenedtoformalocalFermiliquid.(b)Illustrating the idea of renormalization group flows toward a Fermi liquid fixed point. 110 Strongly correlated electronic systems carried out by Schrieffer and Wolff (1966), and Coqblin spin-screening physics of the Kondo effect. The bare inter- and Schrieffer (1969), who showed how this model gives action is weak, but the spin fluctuations it induces have rise to a residual antiferromagnetic interaction between the the effect of antiscreening the interaction at low ener- local moment and conduction electrons. The emergence gies, renormalizing it to larger and larger values. To see of this antiferromagnetic interaction is associated with a this, we follow an Anderson’s ‘Poor Man’s’ scaling pro- process called superexchange:thevirtualprocessinwhich cedure (Anderson, 1973, 1970), which takes advantage of an electron or hole briefly migrates off the ion, to be the observation that at small J the renormalization in the immediately replaced by another with a different spin. When Hamiltonian associated with the block-diagonalization pro- these processes are removed by the canonical transformation, cess δH HL HL is given by second-order perturbation = ˜ − they induce an antiferromagnetic interaction between the theory: local moment and the conduction electrons. This can be seen by considering the two possible spin-exchange processes 1 δHab a δH b [Tab(Ea) Tab(Eb)] (44) =2 | | 0=2 + 1 2 1 e− f f e− f 'EI U Ef where ↑ + ↓1 ↔ 0 ↔ ↓ + ↑1 ∼ + h+ f f h+ f 'EII Ef (39) ↑ + ↓ ↔ ↔ ↓ + ↑ ∼− † Va1V1b Tab(ω) (45) Both processes require that the f electron and incoming = ω E 1 H = 1 > particle are in a spin-singlet. From second-order perturbation | )0∈{ } − theory, the energy of the singlet is lowered by an amount is the many-body ‘t-matrix’ associated with virtual transi- 2J ,where tions into the high-energy subspace H .FortheKondo − { } model, 1 1 J V 2 (40) = 'E1 + 'E2 ! " V HJ S(0) Sd L (46) = P $ · $ P and the factor of two derives from the two ways a singlet where H projects the intermediate state into the high- can emit an electron or hole into the continuum [1] and P energy subspace, while L projects the initial state into V V(kF) is the hybridization matrix element near the P ∼ the low-energy subspace. There are two virtual scatter- Fermi surface. For the symmetric Anderson model, where 2 ing processes that contribute to the antiscreening effect, 'E1 'EII U/2, J 4V /U. = = = involving a high-energy electron (I) or a high-energy If we introduce the electron spin-density operator σ(0) 1 † $ = hole (II). c σ αβck β ,where is the number of sites in the k,k, kα $ , N Process I is denoted by the diagram lattice,N then the effective interaction has the form &

HK 2JPS 0 (41) k ′′l =− = ka k′b 1 1 where PS 0 σ(0) Sf is the singlet projection = = 4 − 2 $ · $ operator. If we3 drop the constant4 term, then the effective interaction induced by the virtual charge fluctuations must s s′′ s′ have the form and starts in state b kα,σ ,passesthroughavirtual HK J σ(0) Sf (42) † | 0=| 0 $ state 1 c ασ,, where -k lies at high energies in the = $ · | 0=| k,, 0 ,, range -k [1/b, 1]andendsinstate a k,β,σ, .The where S is the spin of the localized moment. The complete ,, ∈ | 0=| 0 $f resulting renormalization ‘Kondo Model’, H Hc HK describing the conduction = + I electrons and their interaction with the local moment is k,β,σ, T (E) kα,σ 2 | | 0 1 H - c† c J σ(0) S (43) 2 a b a b k kσ kσ f J (σ βλσ λα)(Sσ σ Sσ σ ) = $ $ + $ · $ = E - × , ,, ,, kσ - [1 δ1,1] ! k,, " ) k,, ∈)− − 2.2.3 The Kondo effect 2 1 a b a b J ρδ1 (σ σ )βα(S S ) (47) ≈ E 1 σ ,σ The antiferromagnetic sign of the superexchange interac- ! − " tion J in the Kondo Hamiltonian is the origin of the In Process II,denotedby Heavy fermions: electrons at the edge of magnetism 111

ka k′b and if we introduce the ‘Kondo temperature’

1 TK D exp (53) = −2g ! o "

s k′′l s′ we see that this can be written s′′ 1 2g(1,) (54) = ln(1/TK) the formation of a virtual hole excitation 1 ck λ σ ,, | 0= ,, | 0 so that once 1, TK,thecouplingconstantbecomesofthe introduces an electron line that crosses itself, introducing ∼ order one – at lower energies, one reaches ‘strong coupling’ a negative sign into the scattering amplitude. The spin where the Kondo coupling can no longer be treated as a operators of the conduction sea and AFM reverse their weak perturbation. One of the fascinating things about this relative order in process II, which introduces a relative minus flow to strong coupling is that, in the limit TK D,all sign into the T-matrix for scattering into a high-energy hole- - explicit dependence on the bandwidth D disappears and the state, Kondo temperature TK is the only intrinsic energy scale in the (II) physics. Any physical quantity must depend in a universal k,βσ, T (E) kασ 2 | | 0 way on ratios of energy to TK, thus the universal part of the 1 free energy must have the form =− E (- - - ) - [ 1, 1 δ1] ! k k, k,, " k,, ∈ −)− + − + − T 2 b a a b F(T) TK4 (55) J (σ σ )βα(S S ) = TK × σ ,σ 2 1 a b a b where 4(x) is universal. We can also understand the resis- J ρδ1 (σ σ )βα(S S ) (48) =− E 1 σ ,σ ! " tance created by spin-flip scattering of a magnetic impurity in − ne2 the same way. The resistivity is given by ρi m τ(T,H), where we have assumed that the energies -k and - are = k, where the scattering rate must also have a scaling form negligible compared with 1. Adding equations (47 and 48) gives ni T H τ(T,H) 42 , (56) 2 = ρ TK TK int I II J ρδ1 a b a b ' ( δHk βσ kασ T T [σ ,σ ]βαS S , ,; = ˆ + =− 1 where ρ is the density of states (per spin) of electrons J 2ρδ1 2 σ βα S (49) and ni is the concentration of magnetic impurities and = 1 $ · $σ ,σ the function 42(t, h) is universal. To leading order in the so the high-energy virtual spin fluctuations enhance or Born approximation, the scattering rate is given by τ 2 2πS(S 1) 2 = 2πρJ S(S 1) + (g0) where g0 g(10) is the ‘antiscreen’ the Kondo coupling constant + = ρ = bare coupling at the energy scale that moments form. We

2 δ1 can obtain the behavior at a finite temperature by replacing J(1,) J(1) 2J ρ (50) = + 1 g0 g(1 2πT),whereupon → = If we introduce the coupling constant g ρJ,recognizing 2πS(S 1) 1 δ1 = τ(T) + (57) that d ln 1 ,weseethatitsatisfies ρ 2 =−1 = 4ln (2πT/TK) ∂g gives the leading high-temperature growth of the resistance β(g) 2g2 O(g3) (51) ∂ ln 1 = =− + associated with the Kondo effect. The kind of perturbative analysis we have gone through This is an example of a negative β function: a signature of here takes us down to the Kondo temperature. The physics at an interaction that grows with the renormalization process. lower energies corresponds to the strong coupling limit of the At high energies, the weakly coupled local moment is Kondo model. Qualitatively, once Jρ 1, the local moment said to be asymptotically free.Thesolutiontothescaling 3 is bound into a spin-singlet with a conduction electron. The equation is number of bound electrons is nf 1, so that by the Friedel = go sum rule (equation (35)) in a paramagnet the phase shift g(1,) (52) δ δ π/2, the unitary limit of scattering. For more = 1 2go ln(1/1,) − ↑ = ↓ = 112 Strongly correlated electronic systems details about the local Fermi liquid that forms, we refer the (Ruderman and Kittel, 1954; Kasuya, 1956; Yosida, 1957) reader to the accompanying chapter on the Kondo effect by magnetic interaction, Jones (2007). JRKKY(x x ) − , 2 HRKKY J χ(x x,) S(x) S(x,) (62) 2.2.4 Doniach’s Kondo lattice concept = − − $ · $ / 01 2 The discovery of heavy-electron metals prompted Doniach where (1977) to make the radical proposal that heavy-electron 2 cos 2kFr materials derive from a dense lattice version of the Kondo JRKKY(r) J ρ (63) ∼− k r effect, described by the Kondo Lattice model (Kasuya, F 1956) In alloys containing a dilute concentration of magnetic transition-metal ions, the oscillatory RKKY interaction gives † † i(k k) Rj H -kc ckσ J Sj c σ αβc e ,− · (58) rise to a frustrated, glassy magnetic state known as a spin = kσ + $ · kα $ k,β j glass.Indensesystems,theRKKYinteractiontypically )kσ ) gives rise to an ordered antiferromagnetic state with a Neel´ 2 In effect, Doniach was implicitly proposing that the key temperature TN of the order J ρ.Heavy-electronmetals physics of heavy-electron materials resides in the interaction narrowly escape this fate. of neutral local moments with a charged conduction electron Doniach argued that there are two scales in the Kondo sea. lattice, the single-ion Kondo temperature TK and TRKKY, Most local moment systems develop an antiferromagnetic given by order at low temperatures. A magnetic moment at location 1/(2Jρ) x0 induces a wave of ‘Friedel’ oscillations in the electron TK De− = 2 spin density (Figure 14) TRKKY J ρ (64) =

When Jρ is small, then TRKKY is the largest scale and an σ(x) Jχ(x x0) S(x0) (59) 2$ 0=− − 2$ 0 antiferromagnetic state is formed, but, when the Jρ is large, the Kondo temperature is the largest scale so a dense Kondo where lattice ground state becomes stable. In this paramagnetic f(- ) f(- ) state, each site resonantly scatters electrons with a phase shift k k, i(k k ) x χ(x) 2 − e − , · (60) - - π/2. Bloch’s theorem then insures that the resonant elastic = k, k ∼ )k,k, ' − ( scattering at each site acts coherently, forming a renormalized band of width TK (Figure 15). ∼ is the nonlocal susceptibility of the metal. The sharp dis- As in the impurity model, one can identify the Kondo continuity in the occupancies f(-k) at the Fermi surface is lattice ground state with the large U limit of the Anderson responsible for Friedel oscillations in induced spin density lattice model. By appealing to adiabaticity, one can then that decay with a power law link the excitations to the small U Anderson lattice model. According to this line of argument, the quasiparticle Fermi cos 2k r σ(r) Jρ F surface volume must count the number of conduction and f 3 (61) 2$ 0∼− kFr | | electrons (Martin, 1982), even in the large U limit, where it corresponds to the number of electrons plus the number of where ρ is the conduction electron density of states and r is spins the distance from the impurity. If a second local moment is introduced at location x,itcouplestothisFriedeloscillation 2 FS n n (65) with energy J S(x) σ(x) ,givingrisetothe‘RKKY’ V 3 e spins 2$ ·$ 0 (2π) = +

Figure 14. Spin polarization around magnetic impurity contains Friedel oscillations and induces an RKKY interaction between the spins. Heavy fermions: electrons at the edge of magnetism 113

Since the scaling enhancement effect stretches out across decades of energy, it is largely robust against crystal fields (Mekata et al., 1986). Nozieres’` exhaustion paradox (Nozieres,` 1985). If one • TK ~ Dexp[−1/Jr] considers each local moment to be magnetically screened by a cloud of low-energy electrons within an energy TK of the Fermi energy, one arrives at an ‘exhaus- T tion paradox’. In this interpretation, the number of electrons available to screen each local moment is of the order TK/D 1perunitcell.Oncetheconcen- Fermi T ~ J 2r ? - TK N liquid tration of magnetic impurities exceeds D 0.1% for 4 ∼ (TK 10 K, D 10 K), the supply of screening elec- = = trons would be exhausted, logically excluding any sort of AFM dense Kondo effect. Experimentally, features of single- ion Kondo behavior persist to much higher densities.

Jr Jrc The resolution to the exhaustion paradox lies in the more modern perception that spin screening of local moments extends up in energy, from the Kondo scale TK out to the bandwidth. In this respect, Kondo screening is reminis-

TKTRKKY.Experimenthastoldusinrecenttimesthatthetran- asuperconductor(Burdin,GeorgesandGrempel,2000), sition between these two regimes is a quantum critical point. The effective Fermi temperature of the heavy Fermi liquid is indicated defining the length scale over which the conduction spin as a solid line. Circumstantial experimental evidence suggests that and local moment magnetization are coherent without this scale drops to zero at the antiferromagnetic quantum critical setting any limit on the degree to which the correlation point, but this is still a matter of controversy. clouds can overlap (Figure 16).

Using topology, and certain basic assumptions about the response of a Fermi liquid to a flux, Oshikawa (2000) was 2.3 The large N Kondo lattice able to short circuit this tortuous path of reasoning, proving that the Luttinger relationship holds for the Kondo lattice 2.3.1 Gauge theories, large N, and strong correlation model without reference to its finite U origins. The ‘standard model’ for metals is built upon the expansion There are, however, aspects to the Doniach argument that to high orders in the strength of the interaction. This leave cause for concern: approach, pioneered by Landau, and later formulated in the language of finite temperature perturbation theory by Landau It is purely a comparison of energy scales and does (1957), Pitaevskii (1960), Luttinger and Ward (1960), and • not provide a detailed mechanism connecting the heavy- Nozieres` and Luttinger (1962), provides the foundation for fermion phase to the local moment AFM. our understanding of metallic behavior in most conventional Simple estimates of the value of Jρ required for heavy- metals. • electron behavior give an artificially large value of the The development of a parallel formalism and approach coupling constant Jρ 1. This issue was later resolved for strongly correlated electron systems is still in its infancy, ∼ by the observation that large spin degeneracy 2j 1of and there is no universally accepted approach. At the heart + the spin-orbit coupled moments, which can be as large of the problem are the large interactions, which effectively as N 8inYbmaterials,enhancestherateofscaling remove large tracts of Hilbert space and impose strong = to strong coupling, leading to a Kondo temperature constraints on the low-energy electronic dynamics. One way (Coleman, 1983) to describe these highly constrained Hilbert spaces is through the use of gauge theories. When written as a field theory, 1 1 local constraints manifest themselves as locally conserved TK D(NJρ) N exp (66) = −NJρ quantities. General principles link these conserved quantities ! " 114 Strongly correlated electronic systems

D In the slave boson, the gauge charges Overlap? E † † Screening cloud Qj fjσfjσ bj bj (70) T = σ + K ) −TK are conserved and the physical Hilbert space corresponds to vF / TK Exhaustion Q 1ateachsite.Thegaugesymmetryisnowf N(0)TK << 1 ? j jσ iθ = iθ → e j fjσ, bj e j bj . These two examples illustrate the link → (a) −D between strong correlation and gauge theories. D Strong correlation Constrained Hilbert space E ↔ Gauge theories (71) TK ↔

−TK Akeyfeatureofthesegaugetheoriesistheappearanceof Composite heavy ‘fractionalized fields’, which carry either spin or charge, but fermion not both. How, then, can a Landau–Fermi liquid emerge −D New states injected within a Gauge theory with fractional excitations? (b) into fermi sea Some have suggested that Fermi liquids cannot reconsti- tute themselves in such strongly constrained gauge theories. Figure 16. Contrasting (a) the ‘screening cloud’ picture of the Kondo effect with (b) the composite fermion picture. In (a), Others have advocated against gauge theories, arguing that low-energy electrons form the Kondo singlet, leading to the the only reliable way forward is to return to ‘real-world’ exhaustion problem. In (b), the composite heavy electron is a highly models with a full fermionic Hilbert space and a finite inter- localized bound-state between local moments and high-energy action strength. A third possibility is that the gauge theory electrons, which injects new electronic states into the conduction approach is valid, but that heavy quasiparticles emerge as sea at the chemical potential. Hybridization of these states with bound-states of gauge particles. Quite independently of one’s conduction electrons produces a singlet ground state, forming a Kondo resonance in the single impurity, and a coherent heavy position on the importance of gauge theory approaches, the electron band in the Kondo lattice. Kondo lattice poses a severe computational challenge, in no small part, because of the absence of any small parameter with a set of gauge symmetries. For example, in the Kondo for resumed perturbation theory. Perturbation theory in the lattice, if a spin S 1/2operatorisrepresentedbyfermions, Kondo coupling constant J always fails below the Kondo = temperature. How, then, can one develop a controlled com- † σ putational tool to explore the transition from local moment Sj f $ fjβ (67) $ = jα 2 magnetism to the heavy Fermi liquid? ' (αβ One route forward is to seek a family of models that then the representation must be supplemented by the con- interpolates between the models of physical interest, and a straint nf (j) 1ontheconservedfnumberateachsite. limit where the physics can be solved exactly. One approach, = This constraint means one can change the phase of each f as we shall discuss later, is to consider Kondo lattices in fermion at each site arbitrarily variable dimensions d,andexpandinpowersof1/d about the limit of infinite dimensionality (Georges, Kotliar, Krauth iφ fj e j fj (68) → and Rozenberg, 1996; Jarrell, 1995). In this limit, electron self-energies become momentum independent, the basis of without changing the spin operator S or the Hamiltonian. $j the DMFT. Another approach, with the advantage that it This is the local gauge symmetry. can be married with gauge theory, is the use of large N Similar issues also arise in the infinite U Anderson or expansions. The idea here is to generalize the problem to a Hubbard models where the ‘no double occupancy’ constraint family of models in which the f-spin degeneracy N 2j 1 = + can be established by using a slave boson representation is artificially driven to infinity. In this extreme limit, the (Barnes, 1976; Coleman, 1984) of Hubbard operators: key physics is captured as a mean-field theory, and finite N

† † properties are obtained through an expansion in the small Xσ 0(j) f bj ,X0σ (j) b fjσ (69) = jσ = j parameter 1/N.SuchlargeN expansions have played an important role in the context of the spherical model of † † where f creates a singly occupied f state, f 0 statistical mechanics (Berlin and Kac, 1952) and in field jσ jσ| 0≡ f 1,jσ ,whileb† creates an empty f 0 state, b† 0 f 0,j . theory (Witten, 1978). The next section discusses how the | 0 j | 0=| 0 Heavy fermions: electrons at the edge of magnetism 115 gauge theory of the Kondo lattice model can be treated in a Although this is a theorists’ idealization – a ‘spherical large N expansion. cow approximation’, it nevertheless captures key aspects of the physics. This model ascribes a spin degeneracy of N 2j 1toboththefelectronsand the conduction 2.3.2 Mean-field theory of the Kondo lattice = + electrons. While this is justified for a single impurity, a more Quantum large N expansions are a kind of semiclassical realistic lattice model requires the introduction of Clebsch– limit, where 1/N ! plays the role of a synthetic Planck’s ∼ Gordon coefficients to link the spin-1/2conductionelectrons constant. In a Feynman path integral with the spin-j conduction electrons. i To obtain a mean-field theory, each term in the Hamil- xf (t) xi , 0 [x]exp S[x,x] (72) tonian must scale as N. Since the interaction contains two 2 | 0= D ! ˙ % ! " sums over the spin variables, this criterion is met by rescaling J where S is the classical action and the quantum action ˜ the coupling constant replacing J N .Anotherimportant 1 1 1 → A S is ‘extensive’ in the variable .When , aspect to this model is the constraint on charge fluctuations, = ! ! ! →∞ fluctuations around the classical trajectory vanish and the which in the Kondo limit imposes the constraint nf 1. = transition amplitude is entirely determined by the classical Such a constraint can be imposed in a path integral with a action to go from i to f .AlargeN expansion for the partition Lagrange multiplier term λ(nf 1).However,withnf 1, − = function Z of a quantum system involves a path integral in this is not extensive in N,andcannotbetreatedusinga imaginary time over the fields φ mean-field value for λ.Theresolutionistogeneralizethe constraint to nf Q,whereQ is an integer chosen so that as NS[φ,φ] = Z [φ]e− ˙ (73) N grows, q Q/N remains fixed. Thus, for instance, if we = D = 1 % are interested in N 2, this corresponds to q nf /N . = = = 2 where NS is the action (or free energy) associated with the In the large N limit, it is then sufficient to apply the con- field configuration in space and time. By comparison, we see straint on the average nf Q through a static Lagrange 2 0= that the large N limit of quantum systems corresponds to multiplier coupled to the difference (nf Q). − an alternative classical mechanics, where 1/N ! emulates The next step is to carry out a ‘Hubbard–Stratonovich’ ∼ Planck’s constant and new types of collective behavior not transformation on the interaction pertinent to strongly interacting electron systems start to appear. J † † HI (j) cjβfjβ fjαcjα (78) Our model for a Kondo lattice of spins localized at sites =−N # $# $ j is J † Here, we have absorbed the term nf c cjα derived − N jα † from the spin-diagonal part of (equation (76)) by a shift H -kckσ ckσ HI (j) (74) = + Jnf kσ j µ µ in the chemical potential. This interaction has ) ) N2 → − † J † the form gA A,withg and A f cjα,whichwe where − = N = jα factorize using a Hubbard–Stratonovich transformation, J † HI (j) Sαβ(j)c cjα (75) = N jβ VV gA†A A†V VA (79) is the Coqblin Schrieffer form of the Kondo interaction − → + + g Hamiltonian (Coqblin and Schrieffer, 1969) between an f so that (Lacroix and Cyrot, 1979; Read and Newns, 1983a) spin with N 2j 1spincomponentsandtheconduction = + sea. The spin of the local moment at site j is represented as † † a bilinear of Abrikosov pseudofermions HI (j) HI [V,j] V j c fjσ f cjσ Vj → = jσ + jσ # $ # $ † nf V j Vj Sαβ(j) fjαfjβ δαβ (76) N (80) = − N + J and This is an exact transformation, provided the Vj (τ) are 1 V † † ik Rj treated as fluctuating variables inside a path integral. The j c c e− · $ (77) jσ = √ kσ can be regarded as a spinless exchange boson for the Kondo k N ) effect. In the parallel treatment of the infinite Anderson creates an electron localized at site j,where is the number model (Coleman, 1987a), Vj Vbj is the ‘slave boson’ field N = of sites. associated with valence fluctuations. 116 Strongly correlated electronic systems

In diagrams: The saddle point is determined by the condition that the Free energy F T ln Z is stationary with respect to =− variations in V and λ.Toimposethiscondition,weneed J to diagonalize H and compute the Free energy. First we d (t − t′) MFT N rewrite the mean-field Hamiltonian in momentum space, † † J/N cs fs f s′cs′

† † -k V ckσ HMFT ckσ ,fkσ J † † = Vλ fkσ − (c f ) ( f s′cs′) kσ ! "' ( N s s ) # $ (81) VV Nn λq (86) + J − The path integral for the Kondo lattice is then G H

Tr Texp β H [V,λ]dτ = − 0 # 7 $ β * + † † Z [V,λ] [c, f ]exp c ∂τ ckσ f ∂τ fjσ H [V,λ] (82) = D / D −  kσ 01 + jσ + 2 0 kσ jσ % % % ) )    where where † H [V,λ] -kckσ ckσ = 1 kσ † † ik Rj f f e $· $ (87) ) kσ jσ $ = √ HI [Vj ,j] λj [nf (j) Q] (83) N j + + − ) j ) , - is the Fourier transform of the f-electron field. This Hamil- This is the ‘Read–Newns’ path integral formulation (Read tonian can then be diagonalized in the form and Newns, 1983a; Auerbach and Levin, 1986) of the Kondo lattice model. The path integral contains an outer integral † † Ek 0 akσ HMFT akσ ,bkσ + [V,λ]overthegaugefieldsVj and λj (τ),andaninner = 0 Ek bkσ D kσ ! −"' ( integral [c, f ]overthefermionfieldsmovinginthe ) # $ 7 D V 2 environment of the gauge fields. The inner path integral N s | | λq (88) 7 + N J − is equal to a trace over the time-ordered exponential of ' ( H [V,λ]. † † † Since the action in this path integral grows extensively where akσ and bkσ are linear combinations of ckσ and f † ,whichdescribethequasiparticlesofthetheory.The with N,thelargeN limit is saturated by the saddle point kσ $ configurations of V and λ,eliminatingthetheouterintegral momentum state eigenvalues E Ek are the roots of the = $ in equation (83). We seek a translationally invariant, static, equation ± saddle point, where λj (τ) λ and Vj (τ) V .Sincethe = = Hamiltonian is static, the interior path integral can be written -k V 2 Det E1 (E -k)(E λ) V as the trace over the Hamiltonian evaluated at the saddle − Vλ = − − −| | ! ' (" point, 0(89) = βHMFT Z Tre− (N ) (84) = →∞ so

1 where 2 2 -k λ -k λ 2 Ek + − V (90) † † † ± = 2 ± 2 +| | HMFT H [V,λ] -kc ckσ Vc fjσ Vf cjσ =' ( > = = kσ + jσ + jσ kσ j,σ ) ) # are the energies of the upper and lower bands. The dispersion † VV described by these energies is shown in Figure 17. Notice λfjσfjσ Nn λoq (85) + + G J − H that: $ Heavy fermions: electrons at the edge of magnetism 117

E(k) E Indirect gap ∆g Direct gap 2V l m

Heavy fermion ‘hole’ Fermi surface

(a)k (b) r∗(E)

Figure 17. (a) Dispersion produced by the injection ofacompositefermionintotheconductionsea. (b) Renormalized density of states, showing ‘hybridization gap’ ('g).

hybridization between the f-electron states and the con- instructively, if n n /aD is the electron density, • e c duction electrons builds an upper and lower Fermi band, = separated by an indirect ‘hybridization gap’ of width e− density quasi particle density 'g Eg( ) Eg( ) TK,where = + − − ∼ VFS Q ne N (94) = (2π)3 − aD V 2 /012 / 01 2 Eg( ) λ (91) ± = ± D positive background ∓ 12/0 so the electron density n divides into a contribution and D are the top and bottom of the conduction band. c ± ± carried by the enlarged Fermi sea, whose enlargement is The ‘direct’ gap between the upper and lower bands is compensated by the development of a positively charged 2 V . | | background. Loosely speaking, each neutral spin in the From (89), the relationship between the energy of the • Kondo lattice has ‘ionized’ to produce Q negatively heavy electrons (E)andtheenergyoftheconduc- charged heavy fermions, leaving behind a Kondo singlet tion electrons (-)isgivenby- E V 2/(E λ), = −| | − of charge Qe (Figure 18). so that the density of heavy-electron states ρ∗(E) + ( ) = δ(E E ± ) is related to the conduction electron k, − k To obtain V and λ, we must compute the free energy density± of states ρ(-) by & F V 2 βEk 2 T ln 1 e− ± s | | λq (95) d- V N =− + + N J − ρ∗(E) ρ ρ(-) 1 | | k, ! " ' ( = dE = + (E λ)2 )± ' − ( V 2 −n e ρ 1 | | outside hybridization gap, c (E λ)2 E(k) −(Q + n )e + − c ∼ I 0# $ inside hybridization gap, (92) − − − − Heavy electrons so the ‘hybridization gap’ is flanked by two sharp peaks − + of approximate width T . + + K + + + The Fermi surface volume expands in response to the + + Kondo singlets: • charged background. injection of heavy electrons into the conduction sea, (a) (b) +Qe

Figure 18. Schematic diagram from Coleman, Paul and Rech VFS 1 NaD n Q n (2005a). (a) High-temperature state: small Fermi surface with a 3 kσ c (93) (2π) = s = + JN kσ K background of spins; (b) Low-temperature state, where large Fermi ) surface develops against a background of positive charge. Each spin ‘ionizes’ into Q heavy electrons, leaving behind a a Kondo where aD is the unit cell volume, n a† a kσ kσ kσ singlet with charge Qe.(ReproducedfromP.Coleman,I.Paul, † = + + bkσ bkσ is the quasiparticle number operator and nc is and J. Rech, Phys. Rev. B 72,2005,094430,copyright 2005 by the number of conduction electrons per unit cell. More the American Physical Society, with permission of the APS.) 118 Strongly correlated electronic systems

At T 0, the free energy converges the ground-state energy The large N theory provides an intriguing answer. The = E0,givenby passage from the original Hamiltonian equation (75) to the mean-field Hamiltonian equation (85) is equivalent to the 0 2 E0 V substitution ρ∗(E)E | | λq (96) N s = + J − N %−∞ ' ( J † † † Sαβ(j)c cjα Vf cjα Vc fjα (102) Using equation (92), the total energy is N jβ −→ jα + jα

E 0 0 E o -ρE E Eρ V 2 In other words, the composite combination of spin and d d d 2 N s = D + D | | (E λ) conduction electron are contracted into a single Fermi N %− %− − V 2 field | | λq + J − ' ( † † Ec/(N s ) EK/(N s ) J J N N † † Sαβ(j)cjβ fjαfjβcjβ Vfjα (103) D2ρ ' λe N = N  → ln λq (97) = − 2 + π T − / 01 2 / ' 01K ( 2   † where we have assumed that the upper band is empty and J J † 1 The amplitude V N fjβcjβ N cjβfjβ involves elec- the lower band is partially filled. TK De− Jρ as before. = =− 2 0 = tron states that extend over decades of energy out to the The first term in (97) is the conduction electron contribution band edges. In this way, the f electron emerges as a compos- to the energy Ec/Nns,whilethesecondtermisthelattice ite bound-state of a spin and an electron. More precisely, in ‘Kondo’ energy EK/N s .Ifnowweimposetheconstraint the long-time correlation functions, ∂Eo N ' nf Q 0thenλ so that the ground-state ∂λ =2 0− = = πq energy can be written † Sγα(i)ciγ (t) Sαβ(j)cjβ (t,) 2 2 0 t t !/TK N V EK ' 'e * | − ,|3 + * f (t)f+ † (t ) ln (98) | 2 | iα jα , (104) N s = π πqTK −−−−−−−→ J 2 0 N ' ( This energy functional has a ‘Mexican Hat’ form, with a Such ‘clustering’ of composite operators into a single entity minimum at is well-known statistical mechanics as part of the operator product expansion (Cardy, 1996). In many-body physics, πq ' T we are used to the clustering of fermions pairs into a 2 K (99) = e composite boson, as in the BCS model of superconductiv- confirming that ' TK.Ifwenowreturntothequasiparticle ∼ ity, gψ (x)ψ (x,) '(x x,).Theunfamiliaraspect − ↓ → − density of states ρ∗, we find it has the value of the Kondo↑ effect is the appearance of a composite q fermion. ρ∗(0) ρ (100) The formation of these composite objects profoundly mod- = + TK ifies the conductivity and plasma oscillations of the electron at the Fermi energy so the mass enhancement of the heavy fluid. The Read–Newns path integral has two U(1) gauge electrons is then invariances – an external electromagnetic gauge invariance associated with the conservation of charge and an internal m q qD ∗ 1 (101) gauge invariance associated with the local constraints. The f m = + ρTK ∼ TK electron couples to the internal gauge fields rather than the external electromagnetic fields, so why is it charged? 2.3.3 The charge of the f electron The answer lies in the broken symmetry associated with How does the f electron acquire its charge? We have the development of the amplitude V .ThephaseofV emphasized from the beginning that the charge degrees of transforms under both internal and external gauge groups. freedom of the original f electrons are irrelevant, indeed, When V develops an amplitude, its phase does not actually absent from the physics of the Kondo lattice. So how are order, but it does develop a stiffness which is sufficient to charged f electrons constructed out of the states of the lock the internal and external gauge fields together so that, Kondo lattice, and how do they end up coupling to the at low frequencies, they become synonymous. Written in a electromagnetic field? schematic long-wavelength form, the gauge-sensitive part of Heavy fermions: electrons at the edge of magnetism 119 the Kondo lattice Lagrangian is external and internal potentials. The minimum energy static configuration is when

D † d x cσ (x)( i∂t e4(x) -p eA)cσ (x) L = − + + $ δλ(x) φ(x) e4(x) (110) σ % ! − + ˙ = )† f (x)( i∂t λ(x))fσ (x) + σ − + so when the external potential changes slowly, the internal † V(x)cσ (x)fσ (x) H.c (105) potential tracks it. It is this effect that keeps the Kondo + + ' (" resonance pinned at the Fermi surface. We can always choose the gauge where the phase velocity φ is absorbed into the where p i .SupposeV(x) V(x)eiφ(x).Thereare ˙ =−∇$ =| | local gauge field λ.RecentworkbyColeman,Marstonand two independent gauge transformations that increase the Schofield (2005b) has extended this kind of reasoning to the phase φ of the hybridization. In the external, electromagnetic case where RKKY couplings generate a dispersion jp for gauge transformation, the change in phase is absorbed onto −A the spinons, where is an internal vector potential, which the conduction electron and electromagnetic field, so if A suppresses currents of the gauge charge n .Inthiscase,the V Veiα, f → long-wavelength action has the form iα(x) φ φ α, c(x) c(x)e− , 2 → + → 1 3 e4(x) e4(x) α(x), eA eA α(x) (106) S d xdτ ρs eA φ → +˙ $ → $ − ∇$ = 2 $ + ∇$ − A$ % ! # $ 2 where (4, A) denotes the electromagnetic scalar and vector χ Q(e4 φ δλ) (111) $ − − ˙ − potential at site j and α ∂t α i∂τ α denotes the deriva- " ˙ = ≡− tive with respect to real time t.Bycontrast,intheinternal In this general form, heavy-electron physics can be seen gauge transformation, the phase change of V is absorbed to involve a kind of ‘Meissner effect’ that excludes the onto the f fermion and the internal gauge field (Read and difference field eA from within the metal, locking the iβ $ − A$ Newns, 1983a), so if V Ve , internal field to the external electromagnetic field, so that → the f electrons, which couple to it, now become charged φ φ β, f(x) f(x)eiβ(x), (Figure 19). → + → λ(x) λ(x) β(x) (107) → − ˙ 2.3.4 Optical conductivity of the heavy-electron fluid If we expand the mean-field free energy to quadratic order in small, slowly varying changes in λ(x),thenthechangein One of the interesting consequences of the heavy-electron the action is given by charge is a complete renormalization of the electronic plasma frequency (Millis, Lavagna and Lee, 1987b). The electronic χ δS Q dDxdτδλ(x)2 (108) =− 2 % where χ δ2F/δλ2 is the f-electron susceptibility eval- Q =− uated in the mean-field theory. However, δλ(x) is not gauge A(x) A(x) invariant, so there must be additional terms. To guarantee gauge invariance under both the internal and external trans- formation, we must replace δλ by the covariant combination A(x) A(x) δλ φ e4. The first two terms are required for invariance + ˙ − under the internal gauge group, while the last two terms are required for gauge invariance under the external gauge group. The expansion of the action to quadratic order in the gauge fields must therefore have the form (a) (b) χ S Q dτ (φ δλ(x) e4(x))2 (109) ∼− 2 ˙ + − Figure 19. (a) Spin liquid, or local moment phase, internal field j % ) decoupled from electromagnetic field. (b) Heavy-electron phase, internalA gauge field ‘locked’ together with electromagnetic field. so the phase φ acquires a rigidity in time that generates Heavy electrons are now charged and difference field [eA(x) $ a‘mass’orenergycostassociatedwithdifference of the (x)]isexcludedfromthematerial. − A 120 Strongly correlated electronic systems

plasma frequency is related via a f-sum rule to the integrated )

w ∆w~ V~ TKD optical conductivity ( s 2 ∞ dω π nce σ(ω) f1 (112) m π = = 2 m (t∗)−1 = t−1 %0 ' ( m ∗ n ‘Interband’ where e is the density of electrons [2]. In the absence of pne 2 f1 = local moments, this is the total spectral weight inside the t 2m 2 2 Drude peak of the optical conductivity. m pne ne f2 = When the heavy-electron fluid forms, we need to consider 2m∗ the plasma oscillations of the enlarged Fermi surface. If the original conduction sea was less than half filled, then the renormalized heavy-electron band is more than half filled, forming a partially filled hole band. The density of electrons D w in a filled band is N/a ,sotheeffectivedensityofhole ~ TKD carriers is then Figure 20. Separation of the optical sum rule in a heavy-fermion D D nHF (N Q c)/a (N Q)/a nc (113) system into a high-energy ‘interband’ component of weight f2 = − − N = − − 2 2 ∼ ne /m and a low-energy Drude peak of weight f1 ne /m . ∼ ∗ The mass of the excitations is also renormalized, m m . → ∗ The two effects produce a low-frequency ‘quasiparticle’ Notice that this means that the residual resistivity Drude peak in the conductivity, with a small total weight m m ρ ∗ V 2 o 2 2 (117) ∼ π nHFe = ne τ ∗ = ne τ dωσ(ω) f2 f1 = = 2 m ∼ %0 ∗ is unaffected by the effects of mass renormalization. This m nHF f1 (114) can be understood by observing that the heavy-electron ×m n - ∗ ' c ( Fermi velocity is also renormalized by the effective mass, m Optical conductivity probes the plasma excitations of the vF∗ ,sothatthemean-freepathoftheheavy-electron = m∗ electron fluid at low momenta. The direct gap between the quasiparticles is unaffected by the Kondo effect. upper and lower bands of the Kondo lattice are separated by l v τ v τ (118) adirecthybridizationgapoftheorder2V √DTK.This ∗ F∗ ∗ F ∼ = = scale is substantially larger than the Kondo temperature, and it defines the separation between the thin Drude peak of the The formation of a narrow Drude peak, and the presence heavy electrons and the high-frequency contribution from the of a direct hybridization gap, have been seen in optical conduction sea. measurements on heavy-electron systems (Schlessinger, Fisk, In other words, the total spectral weight is divided up into a Zhang and Maple, 1997; Beyerman, Gruner, Dlicheouch and Maple, 1988; Dordevic et al., 2001). One of the interesting small ‘heavy fermion’ Drude peak, of total weight f2,where features about the hybridization gap of size 2V is that the n e2 1 mean-field theory predicts that the ratio of the direct to the σ(ω) HF (115) = m (τ ) 1 iω indirect hybridization gap is given by 2V 1 m∗ , ∗ ∗ − − TK ∼ √ρTK ∼ me so that the effective mass of the heavy electrons shouldL scale separated off by an energy of the order V √TKD from an ∼ as square of the ratio between the hybridization gap and the ‘interband’ component associated with excitations between characteristic scale T ∗ of the heavy Fermi liquid the lower and upper Kondo bands (Millis and Lee, 1987a; Degiorgi, Anders, Gruner and Society, 2001). This second m 2V 2 ∗ (119) term carries the bulk f1 of the spectral weight (Figure 20). ∼ me ∝ TK Simple calculations, based on the Kubo formula, confirm ' ( this basic expectation, (Millis and Lee, 1987a; Degiorgi, In practical experiments, TK is replaced by the ‘coherence Anders, Gruner and Society, 2001) showing that the relation- temperature’ T ∗,wheretheresistivityreachesamaximum. ship between the original relaxation rate of the conduction This scaling law is broadly followed (see Figure 21) in sea and the heavy-electron relaxation rate τ ∗ is measured optical data (Dordevic et al., 2001), and provides further confirmation of the correctness of the Kondo lattice 1 m 1 (τ ∗)− (τ)− (116) picture. = m∗ Heavy fermions: electrons at the edge of magnetism 121

1000

) YbFe Sb 1 4 12 CePd3 − 3 CeAl cm 3 1 −

Ω 2 CeCu 3 6 100 CeRu4Sb12 (10

1 1 s ∆ = 90 cm−1 UPd2Al3 0 0 100 200 300 400 500 CeCu5 Frequency (cm−1) 2 )

∗ UCu5

T

10 ) / CeRu Sb 1 2 4 12 YbFe Sb

− 4 12 ∆ ( cm 1 − Ω

3 1 U2PtC2 URu2Si2 (10 1 1 s ∆ = 380 cm−1 0 0 1000 2000 3000 −1 Frequency (cm ) UPt3

0.1 0.1 1 10 100 1000 ∗ m /mb

Figure 21. Scaling of the effective mass of heavy electrons with thesquareoftheopticalhybridizationgap.(Reproducedfrom S.V. Dordevic, D.N. Basov, N.R. Dilley, E.D. Bauer, and M.B. Maple, Phys. Rev. Lett. 86,2001,684,copyright by the American Physical Society, with permission from the APS.)

2.4 Dynamical mean-field theory idea of DMFT is to reduce the lattice problem to the physics of a single magnetic ion embedded within a self-consistently The fermionic large N approach to the Kondo lattice provides determined effective medium (Georges, Kotliar, Krauth and an invaluable description of heavy-fermion physics, one that Rozenberg, 1996; Kotliar et al., 2006). The effective medium can be improved upon beyond the mean-field level. For is determined self-consistently from the self-energies of the example, the fluctuations around the mean-field theory can be electrons that scatter off the single impurity. In its more used to compute the interactions, the dynamical correlation advanced form, the single impurity is replaced by a cluster functions, and the optical conductivity (Coleman, 1987b; of magnetic ions. Millis and Lee, 1987a). However, the method does face a Early versions of the DMFT were considered by Kuramoto number of serious outstanding drawbacks: and Watanabe (1987), and Cox and Grewe (1988), and others, who used diagrammatic means to extract the physics of False phase transition: In the large N limit, the crossover • asingleimpurity.Themodernconceptualframeworkfor between the heavy Fermi liquid and the local moment DMFT was developed by Metzner and Vollhardt (1989), physics sharpens into a phase transition where the 1/N and Georges and Kotliar (1992). The basic idea behind expansion becomes singular. There is no known way of DMFT is linked to early work of Luttinger and Ward (1960), eliminating this feature in the 1/N expansion. and Kotliar et al.(2006),whofoundawayofwritingthe Absence of magnetism and superconductivity: The large • free energy as a variational functional of the full electronic N approach, based on the SU(N) group, cannot form Green’s function atwo-particlesingletforN>2. The SU(N) group is fine for particle physics, where baryons are bound- † states of N quarks, but, for condensed matter physics, ij Tψ (τ)ψ (0) (120) G =−2 i j 0 we sacrifice the possibility of forming two-particle or two-spin singlets, such as Cooper pairs and spin- Luttinger and Ward showed that the free energy is a singlets. Antiferromagnetism and superconductivity are variational functional of F [ ] from which Dyson’s equation consequently absent from the mean-field theory. G relating the to the bare Green’s function 0 G G Amongst the various alternative approaches currently 1 1 under consideration, one of particular note is the DMFT. The [ − − ]ij 6ij [ ](121) G0 − G = G 122 Strongly correlated electronic systems

quantity is self-consistently updated by the DMFT. There are, by now, a large number of superb numerical methods to solve g (w) an Anderson model for an arbitrary environment, including the use of exact diagonalization, diagrammatic techniques, Environment and the use of Wilson’s renormalization group (Bulla, 2006). Time Each of these methods is able to take an input ‘environment’ Green’s function providing as output the impurity self-energy g (w) 6[ 0] 6(iωn). G = Briefly, this is how the DMFT computational cycle works. One starts with an estimate for the environment Green’s function 0 and uses this as input to the ‘impurity solver’ to G compute the first estimate 6(iωn) of the local self-energy. Impurity The interaction strength is set within the impurity solver. This local self-energy is used to compute the Green’s functions of Figure 22. In the dynamical mean-field theory, the many-body the electrons in the environment. In an Anderson lattice, the physics of the lattice is approximatedbyasingleimpurityinaself- consistently determined environment. Each time the electron makes Green’s function becomes asortiefromtheimpurity,itspropagationthroughtheenvironment 1 is described by a self-consistently determined local propagator (ω), V 2 − G G(k,ω) ω Ef 6(ω) (124) represented by the thick gray line. = − − ω -k − ! − "

The quantity 6[ ]isafunctional,amachinewhichtakesthe where V is the hybridization and -k the dispersion of the full propagator ofG the electron and outputs the self-energy of conduction electrons. It is through this relationship that the the electron. Formally, this functional is the sum of the one- physics of the lattice is fed into the problem. From G(k,ω), particle irreducible Feynman diagrams for the self-energy: the local propagator is computed while its output depends on the input Greens function, the 1 actual the machinery of the functional is determined solely V 2 − (ω) ω Ef 6(ω) (125) by the interactions. The only problem is that we do not know G = − − ω -k − how to calculate it. )k ! − " DMFT solves this problem by approximating this func- Finally, the new estimate for the bare environment Green’s tional by that of a single impurity or a cluster of magnetic 1 function 0 is then obtained by inverting the equation − impurities (Figure 22). This is an ideal approximation for 1 G G = 0− 6,sothat a local Fermi liquid, where the physics is highly retarded G − in time, but local in space. The local approximation is also 1 0(ω) G− (ω) 6(ω) (126) asymptotically exact in the limit of infinite dimensions (Met- G = + * + zner and Vollhardt, 1989). If one approximates the input This quantity is then reused as the input to an ‘impurity Green function to 6 by its on-site component ij δij , G ≈ G solver’ to compute the next estimate of 6(ω).Thewholepro- then the functional becomes the local self-energy functional cedure is then reiterated to self-consistency. For the Anderson of a single magnetic impurity, lattice, Cyzcholl (Schweitzer and Czycholl, 1991) has shown that remarkably good results are obtained using a perturba- 6ij [ ls] 6ij [ δls] 6impurity[ ]δij (122) 2 G ≈ G ≡ G tive expansion for 6 to the order of U (Figure 23). Although this approach is not sufficient to capture the limiting Kondo DMFT extracts the local self-energy by solving an Ander- behavior much, the qualitative physics of the Kondo lattice, son impurity model embedded in an arbitrary electronic envi- including the development of coherence at low temperatures, ronment. The physics of such a model is described by a path is already captured by this approach. However, to go to the integral with the action strongly correlated regime, where the ratio of the interaction β to the impurity hybridization width U/(π') is much larger † 1 S dτdτ ,fσ (τ) 0− (τ τ ,)fσ (τ ,) than unity, one requires a more sophisticated solver. =− 0 G − % β There are many ongoing developments under way using U dτn (τ)n (τ) (123) this powerful new computational tool, including the incor- + 0 ↑ ↓ % poration of realistic descriptions of complex atoms, and the where G0(τ) describes the bare Green’s function of the extension to ‘cluster DMFT’ involving clusters of magnetic felectron,hybridizedwithitsdynamicenvironment.This moments embedded in a self-consistent environment. Let me Heavy fermions: electrons at the edge of magnetism 123

0.060 x10−5 11 UU 0.048 10

0.036 9

p (1) 0.024 8

(2) (3) 0.012 7 (4) s = 1/2 0 6 g = 3.92 0 0.3 0.6 0.9 1.2 1.5 g ∆ = 750°K x T 5

Figure 23. Resistivity for the Anderson lattice, calculated using 4 the DMFT, computing the self-energy to order U 2.(1),(2), (3), and (4) correspond to a sequence of decreasing electron 3 FeSi density corresponding to nTOT (0.8, 0.6, 0.4, 0.2) respectively. (Reproduced from H. Schweitzer= and G. Czycholl, Phys. Rev. Lett. 2 67,1991,3724copyright by the American Physical Society, with permission of the APS.) 1

0 end this brief summary with a list of a few unsolved issues 0 100 200 300 400 500 600 700 800 900 with this technique Temperature (K )

There is, at present, no way to relate the thermodynamics Figure 24. Schematic band picture of Kondo insulator, illustrating • how a magnetic field drives a metal-insulator transition. Modified of the bulk to the impurity thermodynamics. from Aeppli and Fisk (1992). (Reproduced from V. Jaccarino, At present, there is no natural extension of these methods • G.K. Wertheim, J.H. Wernick, C.R. Walker and S. Arajs, Phys. Rev. to the infinite U Anderson or Kondo models that 160,1967,476copyright 1967 by the American Physical Society, incorporates the Green’s functions of the localized f- with permission of the APS.) electron degrees of freedom as an integral part of the DMFT. Kondo insulators are highly renormalized ‘band insulators’ The method is largely a numerical black box, making (Figure 24). The d-electron Kondo insulator FeSi has been • it difficult to compute microscopic quantities beyond referred to as renormalized silicon.However,likeMott– the electron-spectral functions. At the human level, Hubbard insulators, the gap in their spectrum is driven by it is difficult for students and researchers to separate interaction effects, and they display optical and magnetic themselves from the ardors of coding the impurity properties that cannot be understood with band theory. solvers, and make time to develop new conceptual and There are about a dozen known Kondo insulators, includ- qualitative understanding of the physics. ing the rare-earth systems SmB6 (Menth, Buehler and Geballe, 1969), YB12 (Iga, Kasaya and Kasuya, 1988), CeFe4P12 (Meisner et al., 1985), Ce3Bi4Pt3 (Hundley et al., 3KONDOINSULATORS 1990), CeNiSn (Takabatake et al., 1992, 1990; Izawa et al., 1999) and CeRhSb (Takabatake et al., 1994), and the d- 3.1 Renormalized silicon electron Kondo insulator FeSi. At high temperatures, Kondo insulators are local moment metals, with classic Curie sus- The ability of a dense lattice of local moments to transform ceptibilities, but, at low temperatures, as the Kondo effect ametalintoaninsulator,a‘Kondoinsulator’isoneofthe develops coherence, the conductivity and the magnetic sus- remarkable and striking consequences of the dense Kondo ceptibility drop toward zero. Perfect insulating behavior is, effect (Aeppli and Fisk, 1992; Tsunetsugu, Sigrist and Ueda, however, rarely observed due to difficulty in eliminating 1997; Riseborough, 2000). Kondo insulators are heavy- impurity band formation in ultranarrow gap systems. One of electron systems in which the the liberation of mobile charge the cleanest examples of Kondo-insulating behavior occurs through the Kondo effect gives rise to a filled heavy-electron in the d-electron system FeSi (Jaccarino et al., 1967; DiTusa band in which the chemical potential lies in the middle et al., 1997). This ‘flyweight’ heavy-electron system provides of the hybridization gap. From a quasiparticle perspective, arathercleanrealizationofthephenomenaseeninother 124 Strongly correlated electronic systems

H = 0 H<<∆0/gJmB H = ∆0/gJmB

k BT<<∆0 k BT<<∆0 k BT<<∆0

E

KKK

Figure 25. Temperature-dependent susceptibility in FeSi (after Jaccarino et al., 1967), fitted to the activated Curie form χ(T) '/(kBT) 2 = (C/T )e− ,withC (gµB) j(j 1),andg 3.92, ' 750 K. The Curie tail has been subtracted. (Reproduced from G. Aeppli and Z. Fisk, Comm. Condens.= Matter Phys.+ 16 (1992)= 155, with= permission from Taylor & Franics Ltd, www/.nformaworld.com.)

Kondo insulators, with a spin and charge gap of about 750 K Figure 26(a) shows the sharp rise in the resistivity of (Schlessinger, Fisk, Zhang and Maple, 1997). Unlike its Ce3Bi4Pt3 as the Kondo-insulating gap forms. In Kondo rare-earth counterparts, the small spin-orbit coupling in this insulators, the complete elimination of carriers at low tem- materials eliminates the Van Vleck contribution to the sus- peratures is also manifested in the optical conductivity. ceptibility at T 0, giving rise to a susceptibility which Figure 26(b) shows the temperature dependence of the opti- = almost completely vanishes at low temperatures (Jaccarino cal conductivity in Ce3Bi4Pt3,showingtheemergenceofa et al., 1967) (Figure 25). gap in the low-frequency optical response and the loss of Kondo insulators can be understood as ‘half-filled’ Kondo carriers at low energies. lattices in which each quenched moment liberates a nega- The optical conductivity of the Kondo insulators is of tively charged heavy electron, endowing each magnetic ion particular interest. Like the heavy-electron metals, the devel- an extra unit of positive charge. There are three good pieces opment of coherence is marked by the formation of a direct of support for this theoretical picture: hybridization gap in the optical conductivity. As this forms, a pseudogap develops in the optical conductivity. In a noninter- Each Kondo insulator has its fully itinerant semiconduct- • acting band gap, the lost f-sum weight inside the pseudogap ing analog. For example, CeNiSn is isostructural and would be deposited above the gap. In heavy-fermion metals, isoelectronic with the semiconductor TiNiSi containing a small fraction of this weight is deposited in the Drude peak Ti4 ions, even though the former contains Ce3 ions + + –however,mostoftheweightissentofftoenergiescom- with localized f moments. Similarly, Ce Bi Pt ,witha 3 4 3 parable with the bandwidth of the conduction band. This is tiny gap of the order 10 meV is isolectronic with semi- one of the most direct pieces of evidence that the formation conducting Th Sb Ni ,witha70meVgap,inwhichthe 3 4 3 of Kondo singlets involves electron energies that spread out 5f-electrons of the Th4 ion are entirely delocalized. + to the bandwidth. Another fascinating feature of the heavy- Replacing the magnetic site with isoelectronic nonmag- • electron ‘pseudogap’ is that it forms at a temperature that is netic ions is equivalent to doping, thus in Ce1 xLax Bi4 3 − significantly lower than the pseudogap. This is because the Pt3,eachLa+ ion behaves as an electron donor in a lat- 4 pseudogap has a larger width given by the geometric mean of tice of effective Ce + ions. Ce3 xLax Pt4Bi3 is, in fact, − the coherence temperature and the bandwidth 2V √TKD. very similar to CePd3,whichcontainsapseudogapin ∼ its optical conductivity, with a tiny Drude peak (Bucher The extreme upward transfer of spectral weight in Kondo et al., 1995). insulators has not yet been duplicated in detailed theoretical The magnetoresistance of Kondo insulators is large and models. For example, while calculations of the optical con- • negative, and the ‘insulating gap’ can be closed by the ductivity within the DMFT do show spectral weight transfer, it is not yet possible to reduce the indirect band gap to the action of physically accessible fields. Thus, in Ce3Bi4Pt3, a30Tfieldissufficienttoclosetheindirecthybridization point where it is radically smaller than the interaction scale U gap. (Rozenberg, Kotliar and Kajueter, 1996). It may be that these discrepancies will disappear in future calculations based on These equivalences support the picture of the Kondo effect the more extreme physics of the Kondo model, but these liberating a composite fermion. calculations have yet to be carried out. Heavy fermions: electrons at the edge of magnetism 125

The nature of the metal-insulator transition induced by • 10 (Ce La ) Bi Pt doping. 0 1−x x 3 4 3

0.005

cm) 3.2 Vanishing of RKKY interactions

Ω 1 (m r There are a number of experimental indications that the low- 0.07 energy magnetic interactions vanish at low frequencies in a 0.1 1 Kondo lattice. The low-temperature product of the suscepti- bility and temperature χT reported (Aeppli and Fisk, 1992) 0 100 200 300 (a) Temperature (K) to scale with the inverse Hall constant 1/RH ,representing the exponentially suppressed density of carriers, so that

1 e '/T χ − (127) ∼ RHT ∼ T 4000 The important point here is that the activated part of the susceptibility has a vanishing Curie–Weiss temperature. A similar conclusion is reached from inelastic neutron scatter- 1 − ing measurements of the magnetic susceptibility χ ,(q, ω) ∆C ∼

cm) 2000 in CeNiSn and FeSi, which appears to lose all of its momen- Ω (

1 4000 tum dependence at low temperatures and frequencies. There s is, to date, no theory that can account for these vanishing interactions. T 0 K 0 10000 0 3.3 Nodal Kondo insulators 0 200 400 600 800 1000 (b) Wave numbers (cm−1) The narrowest gap Kondo insulators, CeNiSn and CeRhSb, are effectively semimetals, for although they do display Figure 26. (a) Temperature-dependent resistivity of Ce3Pt4Bi3 showing the sharp rise in resistivity at low temperatures. (Repro- tiny pseudogaps in their spin and charge spectra, the purest duced from M.F. Hundley, P.C. Canfield, J.D. Thompson, Z. Fisk, samples of these materials develop metallic behavior (Izawa and J.M. Lawrence, Phys. Rev. B. 42,1990,6842,copyright et al., 1999). What is particularly peculiar (Figure 27) about 1990 by the American Physical Society, with permission of the these two materials is that the NMR relaxation rate 1/(T1) APS.) Replacement of local moments with spinless La ions acts shows a T 3 temperature dependence from about 10 to 1 K, like a dopant. (b) Real part of optical conductivity σ (ω) for Kondo 1 followed by a linear Korringa behavior at lower temperatures. insulator Ce3Pt4Bi3.(ReproducedfromB.Bucher,Z.Schlessinger, P.C. Canfield, and Z. Fisk 03/04/2007 Phys. Rev. Lett 72,1994, The usual rule of thumb is that the NMR relaxation rate is 522, copyright  1994 by the American Physical Society, with proportional to a product of the temperature and the thermal permission of the APS.) The formation of the pseudogap associ- average of the electronic density of states N ∗(ω) ated with the direct hybridization gap leads to the transfer of f-sum spectral weight to high energies of order the bandwidth. The pseu- 1 2 2 dogap forms at temperatures that are much smaller than its width T N(ω) T [N(ω T)] (128) T1 ∼ ∼ ∼ (see text). Insert shows σ 1(ω) in the optical range.

where N(ω)2 d- ∂f(ω) N(ω)2 is the thermally smea- = − ∂ω There are, however, a number of aspects of Kondo red average of the squared density of states. This suggests 7 # $ insulators that are poorly understood from the the simple that the electronic density of states in these materials has hybridization picture, in particular, a‘V ’shapedform,withafinitevalueatω 0. Ikeda = and Miyake (1996) have proposed that the Kondo-insulating The apparent disappearance of RKKY magnetic interac- state in these materials develops in a crystal-field state with • tions at low temperatures. an axially symmetric hybridization vanishing along a single The nodal character of the hybridization gap that devel- crystal axis. In such a picture, the finite density of states • ops in the narrowest gap Kondo insulators CeNiSn and does not derive from a Fermi surface, but from the angular CeRhSb. average of the coherence peaks in the density of states. The 126 Strongly correlated electronic systems

60 Local minima (c) 1000 b1 0

N 40 )/ w 100 ( N 20 ) 1

− 0 10 −0.025 0 0.025

(sec w

1 (c) T b2

1/ 1 CeRhSb H = 0 H = 3.6kOe 0 40 0.1 CeNiSn N )/

H = 2.2kOe w ( 20 N 0.01 Global minima (d) 0 0.1 1 10 100 −0.025 0 0.025 (a)Temperature (K) (b) (d) w

3 Figure 27. (a) NMR relaxation rate 1/T1 in CeRhSb and CeNiSn, showing a T relaxation rate sandwiched between a low- and a high-temperature T -linear Korringa relaxation rate, suggesting a V -shaped density of states. (Reproduced from K. Nakamura, Y. Kitaoka, K. Asayama, T. Takabatake, H. Tanaka, and H. Fujii, J. Phys. Soc Japan 63,1994,33,withpermissionofthePhysicalSocietyofJapan.)(b) Contour plot of the ground-state energy in mean-field theory for the narrow gap Kondo insulators, as a function of the two first components of the unit vector b (the third one is taken as positive). The darkest regions correspond to lowest values of the free energy. Arrows point ˆ to the three global and three local minima that correspond to nodalKondoinsulators.(ReproducedfromJ.MorenoandP.Coleman,Phys. Rev. Lett. 84,2000,342,copyright 2000 by the American Physical Society, with permission of the APS.) (c) Density of states of Ikeda and Miyake (1996) state that appears as the global minimum of theKineticenergy.(ReproducedfromH.Ikeda,andK.MiyakeJ. Phys. Soc. Jpn. 65,1996,1769,withpermissionofthePhysicalSocietyofJapan.)(d) Density of states of the MC state (Moreno and Coleman, 2000) that appears as a local minimum of the Kinetic energy, with more pronounced ‘V’-shaped density of states. odd thing about this proposal is that CeNiSn and CeRhSb are couplings select the orbitals in multi f electron heavy-electron monoclinic structures, and the low-lying Kramers doublet of metals such as UPt3.MorenoandColeman(2000)proposea 1 3 the f state can be any combination of the 2 , 2 ,or similar idea in which virtual valence fluctuations into the 5 |± 0 |± 0 2 2 states: f state generate a many-body or a Weiss effective field |± 0 that couples to the orbital degrees of freedom, producing an b1 1/2 b2 5/2 b3 3/2 (129) effective crystal field that adjusts itself in order to minimize |± = |± 0+ |± 0+ |∓ 0 the kinetic energy of the f electrons. This hypothesis is where b (b1,b2,b3) could, in principle, point anywhere consistent with the observation that the Ikeda Miyake state ˆ = on the unit sphere, depending on details of the monoclinic corresponds to the Kondo-insulating state with the lowest crystal field. The Ikeda Miyake model corresponds to three kinetic energy, providing a rational for the selection of symmetry-related points in the space of crystal-field ground the nodal configurations. Moreno and Coleman also found states, another nodal state with a more marked V -shaped density of states that may fit the observed properties more precisely. ( √2 , √5 , 3 ) b 4 4 4 (130) This state is also a local minimum of the electron Kinetic ˆ ∓ − = I (0, 0, 1) energy. These ideas are still in their infancy, and more work needs to be done to examine the controversial idea of a Weiss where a node develops along the x, y,orz axis, respectively. crystal field, both in the insulators and in the metals. But the nodal crystal-field states are isolated ‘points’ amidst acontinuumoffullygappedcrystal-fieldstates.Equally strangely, neutron scattering results show no crystal-field satellites in the dynamical spin susceptibility of CeNiSn, 4HEAVY-FERMION suggesting that the crystal electric fields are quenched SUPERCONDUCTIVITY (Alekseev et al., 1994), and that the nodal physics is a many- body effect (Kagan, Kikoin and Prokof’ev, 1993; Moreno and 4.1 A quick tour Coleman, 2000). One idea is that Hund’s interactions provide the driving force for this selection mechanism. Zwicknagl, Since the discovery (Steglich et al., 1976) of superconduc- Yaresko and Fulde (2002) have suggested that Hund’s tivity in CeCu2Si2, the list of known HFSCs has grown to Heavy fermions: electrons at the edge of magnetism 127

include more than a dozen (Sigrist and Ueda, 1991b) mate- superconductors, such as UPt2Al3 (Geibel et al., 1991a,b), rials with a great diversity of properties (Sigrist and Ueda, CePt3Si, and URu2Si2, develop another kind of order before 1991a; Cox and Maple, 1995). In each of these materials, going superconducting at a lower temperature. In the case the jump in the specific heat capacity at the superconducting of URu2Si2,thenatureoftheupperorderingtransition phase transition is comparable with the normal state specific is still unidentified, but, in the other examples, the upper heat transition involves the development of antiferromagnetism. s n ‘Quantum critical’ superconductors, including CeIn3 and (Cv Cv ) − 1 2(131)CeCu2(Si1 x Gex)2,developsuperconductivitywhenpressure CV ∼ − − is tuned close to a QCP. CeIn3 develops superconductivity at and the integrated entropy beneath the Cv/T curve of the the pressure-tuned antiferromagnetic quantum critical point superconductor matches well with the corresponding area for at 2.5 GPa (25 kbar). CeCu2 (Si,Ge)2 has two islands, one the normal phase obtained when superconductivity is sup- associated with antiferromagnetism at low pressure and a pressed by disorder or fields second at still higher pressure, thought to be associated with critical valence fluctuations (Yuan et al., 2006). Tc (Cs Cn) ‘Strange’ superconductors, which include UBe ,the115 dT v − v 0(132) 13 T = material CeCoIn ,andPuCoGa ,condenseintothesupercon- %0 5 5 ducting state out of an incoherent or strange metallic state. Since the normal state entropy is derived from the f moments, UBe has a resistance of the order 150 µ&cm at its transi- it follows that these same degrees of freedom are involved 13 tion temperature. CeCoIn bears superficial resemblance to in the development of the superconucting state. With the 5 ahigh-temperaturesuperconductor,withalineartempera- exception of a few anomalous cases, (UBe ,PuCoGa,and 13 5 ture resistance in its normal state, while its cousin, PuCoGa CeCoIn ), heavy-fermion superconductivity develops out of 5 5 transitions directly from a Curie paramagnet of unquenched the coherent, paramagnetic heavy Fermi liquid, so heavy f spins into an anisotropically paired, singlet superconductor. fermion superconductivity can be said to involve the pairing These particular materials severely challenge our theoretical of heavy f electrons. understanding, for the heavy-electron quasiparticles appear to Independent confirmation of the ‘heavy’ nature of the pair- form as part of the condensation process, and we are forced ing electrons comes from observed size of the London pen- to address how the f-spin degrees of freedom incorporate into etration depth λ and superconducting coherence length ξ in L the superconducting parameter. these systems, both of which reflect the enhanced effective mass. The large mass renormalization enhances the penetra- tion depth, whilst severely contracting the coherence length, 4.2 Phenomenology making these extreme type-II superconductors. The Lon- don penetration depth of HFSCs agree well with the value The main body of theoretical work on heavy-electron sys- expected on the assumption that only spectral weight in the tems is driven by experiment, and focuses directly on the quasiparticle Drude peak condenses to form a superconduc- phenomenology of the superconducting state. For these pur- tor by poses, it is generally sufficient to assume a Fermi liquid 1 ne2 dω ne2 of preformed mobile heavy electrons, an electronic analog 2 σ(ω) (133) of superfluid Helium-3, in which the quasiparticles interact µoλL = m∗ = ω D.P π - m % ∈ through a phenomenological BCS model. For most purposes, London penetration depths in these compounds are a factor of the Landau–Ginzburg theory is sufficient. I regret that, in this 20–30 times longer (Broholm et al., 1990) than in supercon- short review, I do not have time to properly represent and ductors, reflecting the large enhancement in effective mass. discuss the great wealth of fascinating phenomenology, the By contrast, the coherence lengths ξ vF/' hkF/(m ') wealth of multiple phases, and the detailed models that have ∼ ∼ ∗ are severely contracted in a HFSC. The orbitally limited been developed to describe them. I refer the interested reader upper critical field is determined by the condition that an area to reviews on this subject. (Sigrist and Ueda, 1991a). 2 2 h ξ contains a flux quantum ξ Bc .InUBe13,asuper- On theoretical grounds, the strong Coulomb interac- ∼ 2e conductor with 0.9 K transition temperature, the upper critical tions of the f electrons that lead to moment formation in field is about 11 T, a value about 20 times larger than a con- heavy-fermion compounds are expected to heavily suppress ventional superconductor of the same transition temperature. the formation of conventional s-wave pairs in these sys- Table 2 shows a selected list of HFSCs. ‘Canonical’ tems. A large body of evidence favors the idea that the HFSCs, such as CeCu2Si2 and UPt3,developsuperconductiv- gap function and the anomalous Green function between † † ity out of a paramagnetic Landau–Fermi liquid. ‘Preordered’ paired heavy electrons Fαβ(x) c (x)c (0) is spatially =2 α β 0 128 Strongly correlated electronic systems

Table 2. Selected HFSCs.

Type Material Tc (K) Knight shift Remarks Gap symmetry References (singlet)

CeCu Si 0.7 Singlet First HFSC Line nodes Steglich et al.(1976) Canonical 2 2 UPt3 0.48 ? Double transition to Line and point Stewart, Fisk, Willis and T-violating state nodes Smith (1984b)

UPd2Al3 2SingletNeel´ AFM Line nodes Geibel et al.(1991a), Preordered TN 14 K ' cos 2χ Sato et al.(2001)and = ∼ Tou et al.(1995) URu2Si2 1.3 Singlet Hidden order at Line nodes Palstra et al.(1985)and T0 17.5K Kim et al.(2003) = CePt3Si 0.8 Singlet and Parity-violating Line nodes Bauer et al.(2004) Triplet crystal. TN 3.7K = Quantum CeIn3 0.2 (2.5 GPa) Singlet First quantum critical Line nodes Mathur et al.(1998) critical HFSC CeCu2 (Si1 x Gex )2 0.4 (P 0) Singlet Two islands of SC as Line nodes Yuan et al.(2006) − 0.95 (5.4= GPa) function of pressure

Quadrupolar PrOs4Sb12 1.85 Singlet Quadrupolar Point nodes Isawa et al.(2003) fluctuations

CeCoIn5 2.3 Singlet Quasi-2D Line nodes Petrovic et al.(2001) Strange ρn T dx2 y2 ∼ − UBe13 0.86 ? Incoherent metal at Tc Line nodes Andres, Graebner and Ott (1975) PuCoGa5 18.5 Singlet Direct transition Curie Line nodes Sarrao et al.(2002) metal HFSC → anisotropic, forming either p-wave triplet or d-wave singlet This expression can also be used for a triplet superconductor pairs. that does not break the time-reversal symmetry by making 2 † In BCS theory, the superconducting quasiparticle excita- the replacement 'k d (k) d(k). | | ≡ $ · $ tions are determined by a one-particle Hamiltonian of the Heavy-electron superconductors are anisotropic supercon- form ductors, in which the gap function vanishes at points, or, more typically, along lines on the Fermi surface. Unlike † † † s-wave superconductors, magnetic and nonmagnetic impu- H -kfkαfkα [fkα'αβ(k)f kβ = + − rities are equally effective at pair breaking and suppressing k,σ k ) ) Tc in these materials. A node in the gap is the result of sign f kβ 'βα(k)fkα](134) + − changes in the underlying gap function. If the gap function vanishes along surfaces in momentum space, the intersection where of these surfaces with the Fermi surface produces ‘line nodes’ of gapless quasiparticle excitations. As an example, consider '(k)(iσ ) (singlet) UPt3,where,accordingtoonesetofmodels(Blount,Varma ' (k) 2 αβ (135) αβ d(k) (iσ σ) (triplet) and Aeppli, 1990; Joynt, 1988; Puttika and Joynt, 1988; Hess, = $ 2 αβ M · $ Tokuyasu and Sauls, 1990; Machida and Ozaki, 1989), pair- ing involves a complex d-wave gap For singlet pairing, '(k) is an even parity function of k, while for triplet pairing, d(k) is a an odd-parity function of 2 2 2 2 $ 'k kz(kx iky ), 'k k (k k ) (137) k with three components. ∝ ˆ ± | | ∝ z x + y The excitation spectrum of an anisotropically paired sin- Here 'k vanishes along the basal plane kz 0, producing a glet superconductor is given by = line of nodes around the equator of the Fermi surface, and along the z axis, producing a point node at the poles of the 2 2 Ek - 'k (136) Fermi surface. = k +| | L Heavy fermions: electrons at the edge of magnetism 129

One of the defining properties of line nodes on the Fermi 102 surface is a quasiparticle density of states that vanishes linearly with energy UPd2Al3 27AlNQR

N ∗(E) 2 δ(E Ek) E (138) 1 = − ∝ 10 k ) TN = 14.5 K The quasiparticles surrounding the line node have a ‘rela- Tc = 1.98 K tivistic’ energy spectrum. In a plane perpendicular to the 0 2 2 10 node, Ek (vF k1) (αk2) ,whereα d'/dk2 is the ∼ + = gradient of the gap function and k and k the momen- N 1 2 tum measured in the plane perpendicular to the line node. ) 1 For a two-dimensional relativistic particle with dispersion − E 10−1 (Sec

E ck,thedensityofstatesisgivenbyN(E) .For 1 4|πc|2 = = T the anisotropic case, we need to replace c by the geometric 1/ 2 mean of vF and α,soc vF α.Thisresultmustthenbe → ~ 3 doubled to take account of the spin degeneracy and averaged T + −2 over each line node: 10 dk E N(E) 2 + | | E = 2π 4πv α =| | −−45° nodes % F ) −3 dk 10 + (139) × 4π 2v α ' F ( nodes) % + In the presence of pair-breaking impurities and in a vortex 10−4 state, the quasiparticle nodes are smeared, adding a small 10−1 100 101 102 constant component to the density of states at low energies. Temperature (K) This linear density of states is manifested in a variety of 27 power laws in the temperature dependence of experimental Figure 28. Temperature dependence of the Al NQR relaxation 3 properties, most notably rate 1/T1 for UPd2Al3 (after Tou et al., 1995) showing T depen- dence associated with lines of nodes. Inset showing nodal struc- ture ' cos(2θ) proposed from analysis of anisotropy of ther- Quadratic temperature dependence of specific heat CV mal conductivity∝ in Won et al.(2004).(ReproducedfromH.Tou, • ∝ T 2,sincethespecificheatcoefficientisproportionalto Y. Kitaoka, K. Asayama, C. Geibel, C. Schank, and F. Steglich, the thermal average of the density of states 1995, J. Phys. Soc. Japan 64,1995725,withpermissionofthe Physical Society of Japan.) T ∝ C V N(E) T (140) Although power laws can distinguish line and point nodes, T ∝ ∼ / 01 2 they do not provide any detailed information about the triplet or singlet character of the order parameter or the location where N(E) denotes the thermal average of N(E). AubiquitousT 3 temperature dependence in the nuclear of the nodes. The observation of upper critical fields that • magnetic relaxation (NMR) and nuclear quadrupole are ‘Pauli limited’ (set by the spin coupling, rather than the diamagnetism), and the observation of a Knight shift in most relaxation (NQR) rates 1/T1.Thenuclearrelaxationrate is proportional to the thermal average of the squared HFSCs, indicates that they are anisotropically singlet paired. density of states, so, for a superconductor with line Three notable exceptions to this rule are UPt3,UBe13,and nodes, UNi2Al3,whichdonotdisplayeitheraKnightshiftora Pauli-limited upper critical field, and are the best candidates T 2 for odd-parity triplet pairing. In the special case of CePt3Sn, ∝ 1 the crystal structure lacks a center of symmetry and the T N(E)2 T 3 (141) T1 ∝ ∼ resulting parity violation must give a mixture of triplet and / 01 2 singlet pairs. Figure 28 shows the T 3 NMR relaxation rate of the Until comparatively recently, very little was known Aluminum nucleus in UPd2Al3. about the positions of the line nodes in heavy-electron 130 Strongly correlated electronic systems

v superconductors. In one exception, experiments carried out by an energy of the order 'E ! F ,whereR is the ∼ 2R almost 20 years ago on UPt3 observed marked anisotropies in average distance between vortices in the flux lattice. Writing 2 2 h the ultrasound attenuation length and the penetration depth πHR 40,andπHc2ξ 40 where 40 is the flux ∼ ∼ = 2e (Bishop et al., 1984; Broholm et al., 1990) that appear to quantum, Hc2 is the upper critical field, and ξ is the support a line of nodes in the basal plane. The ultrasonic coherence length, it follows that 1 1 H .Puttingξ R ξ Hc2 attenuation αs(T )/αn in single crystals of UPt3 has a T lin- ∼ ∼ vF/',where' is the typical size of theL gap, the typical ear dependence when the polarization lies in the basal plane shift in the energy of nodal quasiparticles is of the order 3 of the gap nodes, but a T dependence when the polarization H EH ' .Nowsincethedensityofstatesisofthe is along the c axis. ∼ Hc2 E An interesting advance in the experimental analysis of order N(E)L | | N(0),whereN(0) is the density of states = ' nodal gap structure has recently occurred, owing to new in the normal phase, it follows that the smearing of the nodal insights into the behavior of the nodal excitation spectrum quasiparticle energies will produce a density of states of the in the flux phase of HFSCs. In the 1990s, Volovik (1993) order observed that the energy of heavy-electron quasiparticles in afluxlatticeis‘Dopplershifted’bythesuperflowaroundthe H N ∗(H ) N(0) (143) vortices, giving rise to a finite density of quasiparticle states ∼ OHc2 around the gap nodes. The Doppler shift in the quasiparticle energy resulting from superflow is given by This effect, the ‘Volovik effect’, produces a linear component H to the specific heat CV /T .Thisenhancementof ! ∝ Hc2 Ek Ek p vs Ek vF φ (142) the density of states is largestL when the group velocity V → +$·$ = +$ · 2 ∇$ $F at the node is perpendicular to the applied field H ,and $ where vs is the superfluid velocity and φ the superfluid when the field is parallel to vF at a particular node, the $ $ phase. This has the effect of shifting quasiparticle states node is unaffected by the vortex lattice (Figure 29). This

0.10

a a 0.09 ) a

( 0.08 H 0 N

0.07

b b 0.06 (a) (c) 180 a

0.335 T = 0.38 K H = 5 T

H a 0.330

−90 −45 0 45 90 (b) (d) Angle (degrees) CeCoIn5

Figure 29. Schematic showing how the nodal quasiparticle density of states depends on field orientation (after Vekhter, Hirschfield, Carbotte and Nicol, 1999). (a) Four nodes are activated when the field points toward an antinode, creating a maximum in density of states. (b) Two nodes activated when the field points toward a node, creating a minimum in the density of states. (c) Theoretical dependence of density of states on angle. (After Vekhter, Hirschfield, Carbotte and Nicol, 1999.) (d) Measured angular dependence of Cv/T (after Aoki et al., 2004) is 45◦ out of phase with prediction. This discrepancy is believed to be due to vortex scattering, and is expected to vanish at lower fields. (Reproduced from I. Vekhter, P. Hirschfield, J.P. Carbotte, and E.J. Nicol, Phys. Rev. B 59,1998,R9023,copyright 1998 by the American Physical Society, with permission of the APS.) Heavy fermions: electrons at the edge of magnetism 131 gives rise to an angular dependence in the specific heat this idea with a concrete demonstration that antiferromag- coefficient and thermodynamics that can be used to measure netic interactions drive an attractive BCS interaction in the the gap anisotropy. In practice, the situation is complicated at d-wave pairing channel. It is a fascinating thought that at the higher fields where the Andreev scattering of quasiparticles same time that this set of authors was forging the foundations by vortices becomes important. The case of CeCoIn5 is of of our current thoughts on the link between antiferromag- particular current interest. Analyses of the field-anisotropy netism and d-wave superconductivity, Bednorz and Mueller of the thermal conductivity in this material was interpreted were in the process of discovering high-temperature super- early on in terms of a gap structure with dx2 y2 ,while conductivity. the anisotropy in the specific heat appears to− suggest a The BBE and SLH papers develop a paramagnon theory dxy symmetry. Recent theoretical work by Vorontsov and for d-wave pairing in a Hubbard model with a contact Vekhter (2006) suggests that the discrepancy between the interaction I, having in mind a system, which in the modern two interpretations can be resolved by taking into account context, would be said to be close to an antiferromagnetic the effects of the vortex quasiparticle scattering that were QCP. The MSV paper starts with a model with a preexisting ignored in the specific heat interpretation. They predict that, antiferromagnetic interaction, which, in the modern context, at lower fields, where vortex scattering effects are weaker, would be associated with the ‘t–J’ model. It is this approach the sign of the anisotropic term in the specific heat reverses, that I sketch here. The MSV model is written accounting for the discrepancy † It is clear that, despite the teething problems in the inter- H -ka akσ Hint (144) = kσ + pretation of field-anisotropies in transport and thermodynam- ) ics, this is an important emerging tool for the analysis of gap where anisotropy, and, to date, it has been used to give tentative 1 assignments to the gap anisotropy of UPd2Al3,CeCoIn5,and Hint J(k k,)σ αβ σ γδ = 2 − $ ·$ PrOs4Sb12. q )k,k, ) a† a† a a (145) k q/2α k q/2γ k, q/2δ k, q/2β × + − + − + + # $ , - 4.3 Microscopic models describes the antiferromagnetic interactions. There are a number of interesting points to be made here: 4.3.1 Antiferromagnetic fluctuations as a pairing force The authors have in mind a strong coupled model, • such as the Hubbard model at large U,wherethe The classic theoretical models for heavy-fermion supercon- interaction cannot be simply derived from paramagnon ductivity treat the heavy-electron fluids as a Fermi liquid theory. In a weak-coupled Hubbard model, a contact with antiferromagnetic interactions amongst their quasipar- interaction I and bare susceptibility χ (q),theinduced ticles (Monod, Bourbonnais and Emery, 1986; Scalapino, 0 magnetic interaction can be calculated in a random phase Loh and Hirsch, 1986; Monthoux and Lonzarich, 1999). approximation (RPA) (Miyake, Rink and Varma, 1986) UPt provided the experimental inspiration for early theories 3 as of heavy-fermion superconductivity, for its superconduct- ing state forms from within a well-developed Fermi liquid. I Neutron scattering on this material shows signs of antifer- J(q) (146) =−2[1 Iχ0(q)] romagnetic spin fluctuations (Aeppli et al., 1987), making it + natural to presuppose that these might be the driving force MSV make the point that the detailed mechanism that for heavy-electron pairing. links the low-energy antiferromagnetic interactions to Since the early 1970s, theoretical models had predicted the microscopic interactions is poorly described by a that strong ferromagnetic spin fluctuations, often called para- weak-coupling theory, and is quite likely to involve other magnons,couldinducep-wavepairing,andthismechanism processes, such as the RKKY interaction, and the Kondo was widely held to be the driving force for pairing in super- effect that lie outside this treatment. fluid He–3. An early proposal that antiferromagnetic interac- Unlike phonons, magnetic interactions in heavy-fermion • tions could provide the driving force for anisotropic singlet systems cannot generally be regarded as retarded inter- pairing was made by Hirsch (1985). Shortly thereafter, three actions, for they extend up to an energy scale ω0 that is seminal papers, by Monod, Bourbonnais and Emery (1986) comparable with the heavy-electron bandwidth T ∗.Ina (BBE), Scalapino, Loh and Hirsch (1986) (SLH) and by classic BCS treatment, the electron energy is restricted Miyake, Miyake, Rink and Varma (1986) (MSV), solidified to lie within a Debye energy of the Fermi energy. But 132 Strongly correlated electronic systems

† here, ω0 T ∗,soallmomentaareinvolvedinmagnetic † 1 † † 1 † ∼ where 7k 2 akα( iσ2)αβ a kβ and 7k,q 2 akα interactions, and the interaction can transformed to real = − − $ = † # $ # space as ( iσσ2)αβ a kβ create Cooper pairs and − $ − $ † 1 (s) H -kakσ akσ J(Ri Rj )σ i σj (147) V 3[J(k k,) J(k k,)]/2 = + 2 − $ ·$ k,k, i,j =− − + + ) ) (t) V [J(k k,) J(k k,)]/2(153) k,k, = − − + iq R where J(R) q e · J(q) is the Fourier transform of = † the interaction and σ i a σ αβaiβ is the spin density at are the BCS pairing potentials in the singlet and triplet chan- & $ = iα $ site i.Writteninrealspace,theMSVmodelisseentobe nel, respectively. (Notice how the even/odd-parity symmetry an early predecessor of the t –J model used extensively of the triplet pairs pulls out the corresponding symmetrization in the context of high-temperature superconductivity. of J(k k,).) − For a given choice of J(q),itnowbecomespossibleto To see that antiferromagnetic interactions favor d-wave decouple the interaction in singlet and triplet channels. For pairing, one can use the ‘let us decouple the interaction’ in example, on a cubic lattice of side length, if the magnetic real space in terms of triplet and singlet pairs. Inserting the interaction has the form identity [3] J(q) 2J(cos(qx a) cos(qy a) cos(qza)) (154) 3 1 = + + σ αβ σ γδ (σ 2)αγ (σ 2)βδ (σσ2)αγ (σ 2σ)βδ $ ·$ =−2 + 2 $ · $ which generates soft antiferromagnetic fluctuations at the (148) staggered Q vector Q (π/a, π/a, π/a),thenthepairing = into equation (147) gives interaction can be decoupled into singlet and triplet compo- nents, 1 † H J 37† 7 7 7 (149) int ij ij ij $ ij $ ij =−4 − · 3J i,j 3 4 s ) Vk,k s(k)s(k,) dx2 y2 (k)dx2 y2 (k,) , =− 2 + − − where d2z2* r2 (k)d2z2 r2 (k,) + − − t J V pi (k)pi (k,) + (155) † † † k,k, = 2 7ij aiα( iσ)αγ ajγ i x,y,z = − =) † # † $† 7ij aiα( iσσ2)αγ ajγ (150) $ = − $ where # $ create singlet and triplet pairs with electrons located at sites 2 i and j respectively. In real space, it is thus quite clear that s(k) (cos(kx a) = 3 an antiferromagnetic interaction Jij > 0inducesattraction Lcos(kya) cos(kza)) (extended s-wave) + + in the singlet channel, and repulsion in the triplet channel. dx2 y2 (k) (cos(kxa) cos(ky a) − = 1 − Returning to momentum space, substitution of equation (148) d 2 2 (k) (cos(k a) 2z r √3 x (d-wave) into (145) gives − =  cos(ky a) 2cos(kza)) + − (156) † †  Hint J(k k,) 37 7 7 7 are the gap functions for singlet pairing and =− − k, q k,, q − $ k, q · $ k,, q k,k,,q 3 4 ) (151) pi (q) √2sin(qia), (i x,y,z), (p-wave) (157) † 1 † † † = = where 7k,q 2 ak q/2 α( iσ2)αγ a k q/2 γ and 7k,q = + − − − $ = 1 † # † $ describe three triplet gap functions. For J>0, this particular 2 ak q/2 α( iσσ2)αγ a k q/2 γ create singlet and triplet + − $ − − BCS model then gives rise to extended s- and d-wave pairs# at momentum q respectively.$ Pair condensation is described by the zero momentum component of this inter- superconductivity with approximately the same transition action, which gives temperatures, given by the gap equation

† - 1 2 2 H V (s) 7† 7 V (t) 7 7 (152) k s(k) int k,k k k, k,k k k, tanh 2 (158) = , + , $ · $ d 2 2 (k) k,k 2Tc -k x y = 3J ), 3 4 )k ' ( M − 5 Heavy fermions: electrons at the edge of magnetism 133

4.3.2 Toward a unified theory of HFSC 1987) of high-temperature superconductivity. Anderson pro- posed (Anderson, 1987; Baskaran, Zou and Anderson, 1987; Although the spin-fluctuation approach described provides a Kotliar, 1988) that the parent state of the high-temperature good starting point for the phenomenology of heavy-fermion superconductors is a two-dimensional spin liquid of RVBs superconductivity (HFSC), it leaves open a wide range of between spins, which becomes superconducting upon doping questions that suggest this problem is only partially solved: with holes. In the early 1990s, Coleman and Andrei (1989) adapted this theory to a Kondo lattice. Although an RVB How can we reconcile heavy-fermion superconductivity • spin liquid is unstable with respect to the antiferromagnetic with the local moment origins of the heavy-electron order in three dimensions, in situations close to a magnetic quasiparticles? instability, where the energy of the antiferromagnetic state is How can the incompressibility of the heavy-electron comparable with the Kondo temperature, EAFM TK,con- • ∼ fluid be incorporated into the theory? In particular, duction electrons partially spin-compensate the spin liquid, extended s-wave solutions are expected to produce a stabilizing it against magnetism (Figure 30a). In the Kondo- large singlet f-pairing amplitude, giving rise to a large stabilized spin liquid, the Kondo effect induces some RVBs Coulomb energy. Interactions are expected to signifi- in the f-spin liquid to escape into the conduction fluid where cantly depress, if not totally eliminate such extended they pair charged electrons to form a heavy-electron super- s-wave solutions. conductor. Is there a controlled limit where a model of heavy- • AkeyobservationoftheRVBtheoryisthat,whencharge electron superconductivity can be solved? fluctuations are removed to form a spin fluid, there is no What about the ‘strange’ HFSCs UBe13,CeCoIn5,and • distinction between particle and hole (Affleck, Zou, Hsu and PuCoGa5,whereTc is comparable with the Kondo tem- Anderson, 1988). The mathematical consequence of this is perature? In this case, the superconducting order parame- that the the spin-1/2 operator ter must involve the f spin as a kind of ‘composite’ order parameter. What is the nature of this order parameter, σ S f † f †,n1(159) and what physics drives T so high that the Fermi liquid $f iα $ α f c = 2 αβ = forms at much the same time as the superconductivity ' ( iφ develops? is not only invariant under a change of phase fσ e fσ , → but it also possesses a continuous particle-hole symmetry One idea that may help understand the heavy-fermion † † pairing mechanism is Anderson’s RVB theory (Anderson, fσ cos θfσ sgnσ sin θf σ (160) → + −

Energy T AFM 0 T= Spin liquid ~JH 1.0 ESL AFM EAFM AFM ~TK ESL – TK (a) Kondo–stabilized spin liquid K

T /

T e− e− 0.5 (Extended Fermi liquid s wave) f–spins

e− d wave e− Spin-liquid superconductor

f–spins 0 03 6

(b) (c) JH/TK

Figure 30. Kondo-stabilized spin liquid, diagram fromColemanandAndrei(1989).(a)SpinliquidstabilizedbyKondoeffect,(b)Kondo effect causes singlet bonds to form between spin liquid and conduction sea. Escape of these bonds into the conduction sea induces superconductivity. (c) Phase diagram computed using SU(2) mean-field theory of Kondo Heisenberg model. (Reproduced from P. Coleman and N. Andrei, 1989, J. Phys. Cond. Matt. C 1 (1989) 4057, with permission of IOP Publishing Ltd.) 134 Strongly correlated electronic systems

These two symmetries combine to create a local SU(2) where gj is an SU(2) matrix. In this kind of model, one gauge symmetry.Oneoftheimplicationsisthattheconstraint can ‘gauge fix’ the model so that the Kondo effect occurs nf 1associatedwiththespinoperatorisactuallypartof in the particle-hole channel (αi 0). When one does so, = = a triplet of Gutzwiller constraints however, the spin-liquid instability takes place preferentially in an anisotropically paired Cooper channel. Moreover, the † † † † fi fi fi fi 0,fi fi 0,fi fi 0(161) constraint on the f electrons not only suppresses singlet ↑ − ↓ = = ↓ ↑ = ↑ ↓ ↑ ↓ s-wave pairing, it also suppresses extended s-wave pairing If we introduce the Nambu spinors (Figure 30). One of the initial difficulties with both the RVB and fi † † the Kondo-stabilized spin liquid approaches is that, in its fi †↑ , fi (fi ,fi ) (162) ˜ ≡ fi ˜ = ↑ ↓ original formulation, it could not be integrated into a large ' ↓( N approach. Recent work indicates that both the fermionic then this means that all three components of the ‘isospin’ of RVB and the Kondo-stabilized spin-liquid picture can be the f electrons vanish, formulated as a controlled SU(2) gauge theory by carrying out a large N expansion using the group SP(N) (Read and † † 01 0 i 10 fi τfi (fi ,fi ) , − , Sachdev, 1991), originally introduced by Read and Sachdev ˜ $ ˜ = ↓ 10 i 0 0 1 ↑ !' ( ' ( ' − (" for problems in frustrated magnetism, in place of the group fi †↑ 0(163)SU(N).Thelocalparticle-holesymmetryassociatedwith × fi = ' ↓( the spin operators in SU(2) is intimately related to the symplectic property of Pauli spin operators where τ is a triplet of Pauli spin operators that act on the $ f-Nambu spinors. In other words, in the incompressible f T σ 2σ σ 2 σ (167) fluid, there can be no s-wave singlet pairing. $ =−$ This symmetry is preserved in spin-1/2 Kondo models. where σ T is the transpose of the spin operator. This relation, When applied to the Heisenberg Kondo model $ which represents the sign reversal of spin operators under † time-reversal, is only satisfied by a subset of the SU(N) H -kc ckσ JH Si Sj = kσ + · spins for N>2. This subset defines the generators of the kσ (i,j) ) † ) symplectic subgroup of SU(N),calledSP(N). JK c σ c Sj (164) + jσ $ σσ, jσ, · Concluding this section, I want to briefly mention the j ) challenge posed by the highest Tc superconductor, PuCoGa5 (Sarrao et al., 2002; Curro et al., 2005). This material, dis- † σ where Si fiα $ fiβ represents an f spin at site i, it = 2 αβ covered some 4 years ago at Los Alamos, undergoes a direct leads to an SU(#2) $gauge theory for the Kondo lattice with transition from a Curie paramagnet into a heavy-electron Hamiltonian superconductor at around Tc 19 K (Figure 31). The Curie = paramagnetism is also seen in the Knight shift, which scales † † † H -kc τ 3ck λj f τfj [f Uij fj H.c] with the bulk susceptibility (Curro et al., 2005). The remark- = ˜k ˜ + $ ˜j $ ˜ + ˜i ˜ + )k ) )(i,j) able feature of this material is that the specific heat anomaly 1 † † 1 † has the large size (110 mJ mol 1 K2 (Sarrao et al., 2002)) Tr[U Uij ] [f Vi ci H.c] Tr[V Vi ] − +J ij + ˜i ˜ + + J i H i K characteristic of heavy-fermion superconductivity, yet there ) (165) are no signs of saturation in the susceptibility as a precursor to superconductivity, suggesting that the heavy quasiparti- where λj is the Lagrange multiplier that imposes the ck cles do not develop from local moments until the transition. Gutzwiller constraint τ 0ateachsite,ck † ↑ and $ = ˜ = c k This aspect of the physics cannot be explained by the spin- cj − ↓ cj †↑ are Nambu conduction electron spinors, in- the fluctuation theory (Bang, Balatsky, Wastin and Thompson, ˜ = cj ↓ 2004), and suggests that the Kondo effect takes place simul- momentum, - and position basis, respectively, while taneously with the pairing mechanism. One interesting possi-

hij 'ij Vi αi bility here is that the development of coherence between the Uij Vi (166) = 'ij hij = αi V i Kondo effect in two different channels created by the differ- ' − ( ' − ( ent symmetries of the valence fluctuations into the f 6 and are matrix order parameters associated with the Heisenberg f 4 states might be the driver of this intriguing new super- and Kondo decoupling, respectively. This model has the local conductor (Jarrell, Pang and Cox, 1997; Coleman, Tsvelik, † gauge invariance fj gj fj , Vj gj Vj Uij gi Uij g , Andrei and Kee, 1999). ˜ → ˜ → → j Heavy fermions: electrons at the edge of magnetism 135

3.6 300 point where their antiferromagnetic ordering temperature is driven continuously to zero to produce a ‘QCP’ (Stewart, 3.0 250 2001, 2006; Coleman, Pepin,´ Si and Ramazashvili, 2001; Varma, Nussinov and van Saarlos, 2002; von Lohneysen,¨ )

1 2.8 − Rosch, Vojta and Wolfe, 2007; Miranda and Dobrosavljevic, 200 2005). The remarkable transformation in metallic properties, 2.0 often referred to as ‘non-Fermi liquid behavior’, which is cm) emu.mol

6 150 − induced over a wide range of temperatures above the QCP, µ Ω (

10 1.6 r together with the marked tendency to develop supercon- × 100 ductivity in the vicinity of such ‘quantum critical points’ 1.0 M/H ( has given rise to a resurgence of interest in heavy-fermion

50 materials. 0.8 The experimental realization of quantum criticality returns us to central questions left unanswered since the first 0 0 0 50 100 150 200 250 300 discovery of heavy-fermion compounds. In particular: T (K) What is the fate of the Landau quasiparticle when • Figure 31. Temperature dependence of the magnetic susceptibility interactions become so large that the ground state is of PuCoGa5.(AfterSarraoet al., 2002.) The susceptibility shows a direct transition from Curie–Weiss paramagnet into HFSC, without no longer adiabatically connected to a noninteracting any intermediate spin quenching. (Reproduced from Sarrao, J.L., system? L.A. Morales, J.D. Thompson, B.L. Scott, G.R. Stewart, F. Wastlin, What is the mechanism by which the AFM transforms • J. Rebizant, P. Boulet, E. Colineau, and G.H. Lander, 2002, with into the heavy-electron state? Is there a breakdown of the permission from Nature Publishing.  2002.) Kondo effect, revealing local moments at the quantum phase transition, or is the transition better regarded as a 5QUANTUMCRITICALITY spin-density wave transition? 5.1 Singularity in the phase diagram Figure 32 illustrates quantum criticality in YbRh2Si2 (Custers Many heavy electron systems can be tuned, with pres- et al., 2003), a material with a 90 mK magnetic transition sure, chemical doping, or applied magnetic field, to a that can be tuned continuously to zero by a modest magnetic

0.3 2.0

YbRh2Si2 B || c 4.2 1.5 4.0 1.5 e 0.2 1 InT 3.8 ∂ / r

0.5 ∆ 3.6

In 2 2

∂ T (K ) (K) 0 T 3.4 0.0 0.2 0.4 cm) 1.0 0.01 0.1 1 10 100 µ Ω 4.0

T ( 3.2 T (K) 0.1 r 3.0 3.5 cm) Ω

m 0 T 2.8 ( 3.0 1 T r T 2 T 2 6 T 2.6 2.5 14 T

0 2.4 0120 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (a)B (T) (b) Temperature (K)

Figure 32. (a) Grayscale plot of the logarithmic derivative of resistivity dln ρ/dlnT .(ReproducedfromCusterset al., 2003, with permission from Nature Publishing.  2003.) (b) Resistivity of YbRh2Si2 in zero magnetic field. Inset shows logarithmic derivative of resistivity. (Reproduced from O. Trovarelli, C. Geibel, S. Mederle, C. Langhammer, F. Grosche, P. Gegenwart, M. Lang, G. Sparn, and F. Steglich, Phys. Rev. Lett. 85,2000,626,copyright 2000 by the American Physical Society, with permission of the APS.) 136 Strongly correlated electronic systems

field. In wedge-shaped regions, either side of the transition, ratio between the change in the size of the resistivity 'ρ to the resistivity displays the T 2 dependence ρ(T ) ρ AT 2 the zero temperature (impurity driven) resistivity ρ = 0 + 0 (black) that is the hallmark of Fermi-liquid behavior. Yet, in atornadoshapedregionthatstretchesfarabovetheQCPto 'ρ/ρ0 1(171) about 20 K, the resistivity follows a linear dependence over 3 more than three decades. The QCP thus represents a kind of CeCoIn5 is particularly interesting, for, in this case, this resis- ‘singularity’ in the material phase diagram. tance ratio exceeds 102 for current flow along the c axis Experimentally, quantum critical heavy-electron materi- (Tanatar, Paglione, Petrovic and Taillefer, 2007). This obser- als fall between two extreme limits that I shall call ‘hard’ vation excludes any explanation which attributes the unusual and ‘soft’ quantum criticality. ‘Soft’ quantum critical sys- resistivity to an interplay between spin-fluctuation scatter- tems are moderately well described in terms quasiparticles ing and impurity scattering (Rosch, 1999). Mysteriously, interacting with the soft quantum spin fluctuations created 3/2 CeCoIn5 also exhibits a T resistivity for resistivity for by a spin-density wave instability. Theory predicts (Moriya current flow in the basal plane below about 2 K (Tanatar, and Kawabata, 1973) that, in a three-dimensional metal, the Paglione, Petrovic and Taillefer, 2007). Nakasuji, Pines and quantum spin-density wave fluctuations give rise to a weak Fisk (2004) have proposed that this kind of behavior may √ T singularity in the low-temperature behavior of the spe- derive from a two fluid character to the underlying conduc- cific heat coefficient tion fluid.

CV In quantum critical YbRh2Si2 x Gex,thespecificheat γ γ √T (168) 1/3 − T = 0 − 1 coefficient develops a 1/T divergence at the very lowest temperature. In the approach to a QCP, Fermi liquid behavior Examples of such behavior include CeNi2Ge2 (Grosche is confined to an ever-narrowing range of temperature. et al., 2000; Kuchler¨ et al., 2003) chemically doped Ce2 x Moreover, both the linear coefficient of the specific heat and − Lax Ru2Si2 and ‘A’-type antiferromagnetic phases of CeCu2 the the quadratic coefficient A of the resistivity appear to Si2 at a pressure-tuned QCP. diverge (Estrela et al., 2002; Trovarelli et al., 2000). Taken At the other extreme, in ‘hard’ quantum critical heavy together, these results suggests that the Fermi temperature materials, many aspects of the physics appear consistent renormalizes to zero and the quasiparticle effective masses with a breakdown of the Kondo effect associated with diverge a relocalization of the f electrons into ordered, ordered local moments beyond the QCP. Some of the most heav- m∗ T ∗ 0 (172) ily studied examples of this behavior occur in the chem- F → m →∞ ically tuned QCP in CeCu6 xAux (von Lohneysen¨ et al., 1994; von Lohneysen,¨ 1996;− Schroeder et al., 1998, 2000). at the QCP of these three-dimensional materials. A central and YbRh2Si2 x Gex (Custers et al., 2003; Gegenwart et al., property of the Landau quasiparticle is the existence of − 2005) and the field-tuned QCP of YbRh2Si2 (Trovarelli et al., afiniteoverlap‘Z’, or ‘wave function renormalization’ 2000) and YbAgGe (Bud’ko, Morosan and Canfield, 2004, between a single quasiparticle state, denoted by qp− and † | 0 2005; Fak et al., 2005; Niklowitz et al., 2006). Hallmarks of abareelectronstatedenotedby e c 0 , | −0= kσ | 0 hard quantum criticality include a logarithmically diverging specific heat coefficient at the QCP, 2 m Z e− qp− (173) =|2 | 0| ∼ m∗ Cv 1 T0 ln (169) If the quasiparticle mass diverges, the overlap between the T ∼ T0 T ' ( quasiparticle and the electron state from which it is derived and a quasilinear resistivity is driven to zero, signaling a complete breakdown in the quasiparticle concept at a ‘hard’ QCP (Varma, Nussinov and 1 η ρ(T) T + (170) van Saarlos, 2002). ∼ Table 3 shows a tabulation of selected quantum criti- where η is in the range 0–0.2. The most impressive results cal materials. One interesting variable that exhibits singular to date have been observed at field-tuned QCPs in YbRh2Si2 behavior at both hard and soft QCPs is the Gruneisen¨ ratio. and CeCoIn5,wherelinearresistivityhasbeenseentoextend This quantity, defined as the ratio over more than two decades of temperature at the field-tuned QCP (Steglich et al., 1976; Paglione et al., 2003, 2006; Ron- α 1 ∂ ln T 1 9 (174) ning et al., 2006). Over the range where linear, where the = C =−V ∂P ∝ T -

Table 3. Selected heavy-fermion compounds with quantum critical points.

Cv a α Compound xc/Hc ρ T 9(T ) Other References T ∼ = CP

1 To CeCu6 xAux xc 0.1 ln T c – χ Q,, (ω,T) von Lohneysen¨ et al.(1994), − = T0 T + 0 = # $ 1 F ω von Lohneysen¨ (1996) and T 0.7 T Schroeder et al.(1998,2000) * + Hard YbRh2Si2 Bc 0.66 T – T –JumpinHallTrovarelli et al.(2000)and + = constant Paschen et al.(2004) YbRh Si Ge x 0.1 1 TT0.7 –Custerset al.(2003)and 2 2 x x c T 1/3 − − = ↔ Gegenwart et al.(2005) 1 ln To T0 T # $ 1 T0 YbAgGe Bc 9T ln T – NFL over Bud’ko, Morosan and Canfield | = T0 T Bc 5T # $ range of (2004), Fak et al.(2005)and ⊥ = fields Niklowitz et al.(2006)

1 T0 1.5 Soft CeCoIn5 Bc 5T T ln T T /T – ρc T , Paglione et al.(2003,2006), = 0 ∝ 1.5 # $ ρab T Ronning et al.(2006)and ∝ Tanatar, Paglione, Petrovic and Taillefer (2007) 1.2 1.5 1 CeNi2Ge2 Pc 0 γ 0 γ 1√TT− T − –Groscheet al.(2000)and = − Kuchler¨ et al.(2003) of the thermal expansion coefficient α 1 dV to the specific = V dT heat C,divergesataQCP.TheGruneisen¨ ratio is a sensi- T τ tive measure of the rapid acquisition of entropy on warming Classical 1/T ξ x τ away from QCP. Theory predicts that - 1ata3Dspinden- Quantum = Tc sity wave critical point, as seen in CeNi2Ge2.Inthe‘hard’ quantum critical material YbRh2Si2 x Gex, - 0.7indicates τ − = aseriousdeparturefroma3Dspin-densitywaveinstability Quantum ∗ (Kuchler¨ et al., 2003). AFM TF critical Q 1/T metal ξτ Q (A) Q Q x (B2) Pc FL (B1) P 5.2 Quantum versus classical criticality (a) Quantum (b)

Figure 33 illustrates some key distinctions between classical Figure 33. Contrasting classical and quantum criticality in heavy- electron systems. At a QCP, an external parameter P ,suchas and quantum criticality (Sachdev, 2007). Passage through a pressure or magnetic field, replaces temperature as the ‘tuning classical second-order phase transition is achieved by tuning parameter’. Temperature assumes the new role of a finite size the temperature. Near the transition, the imminent arrival of cutoff lτ 1/T on the temporal extent of quantum fluctuations. ∝ order is signaled by the growth of droplets of nascent order (a) Quantum critical regime, where lτ <ξtau probes the interior whose typical size ξ diverges at the critical point. Inside of the quantum critical matter. (b) Fermi-liquid regime, where each droplet, fluctuations of the order parameter exhibit a lτ >ξτ ,wherelikesoda,bubblesofquantumcriticalmatter fleetingly form within a Fermi liquid that is paramagnetic (B1), universal power-law dependence on distance or antiferromagnetically ordered (B2). 1 ψ(x)ψ( ) ,(xξ) 0 d 2 η (175) critical point, order-parameter fluctuations are ‘classical’, 2 0∼x − + - for the characteristic energy of the critical modes !ω(q0), 1 Critical matter ‘forgets’ about its microscopic origins: Its evaluated at a wave vector q0 ξ − ,inevitablydropsbelow ∼ thermodynamics, scaling laws, and correlation exponents the thermal energy !ω(q0) kBTc as ξ . - →∞ associated with critical matter are so robust and universal In the 1970s, various authors, notably Young (1975) and that they recur in such diverse contexts as the Curie point Hertz (1976), recognized that, if the transition temperature of of iron or the critical point of water. At a conventional acontinuousphasetransitioncanbedepressedtozero,the 138 Strongly correlated electronic systems critical modes become quantum-mechanical in nature. The region either side of the transition, in a quantum critical partition function for a quantum phase transition is described region (a), fluctuations persist up to temperatures where lτ by a Feynman integral over order-parameter configurations becomes comparable the with the microscopic short-time ψ(x,τ) in both space and imaginary time (Sachdev, 2007; cutoff in the problem (Kopp and Chakravarty, 2005) (which { } Hertz, 1976) for heavy-electron systems is most likely, the single-ion Kondo temperature lτ !/TK). S[ψ] ∼ Zquantum e− (176) = space–time) configurations 5.3 Signs of a new universality where the action

! The discovery of quantum criticality in heavy-electron sys- kB T S[ψ] dτ ∞ dd xL[ψ(x,τ)](177)tems raises an alluring possibility of quantum critical matter, = 0 % %−∞ auniversalstateofmatterthat,likeitsclassicalcounter- part, forgets its microscopic, chemical, and electronic origins. contains an integral of the Lagrangian L over an infinite There are three pieces of evidence that are particularly fas- range in space, but a finite time interval cinating in this respect: ! lτ (178) 1. Scale invariance, as characterized by E/T scaling in the ≡ kBT quantum-critical inelastic spin fluctuations observed in CeCu1 xAux (Schroeder et al., 1998, 2000). (x xc Near a QCP, bubbles of quantum critical matter form within − = = 0.016), ametal,withfinitesizeξ x and duration ξ τ (Figure 33). These two quantities diverge as the quantum critical point 1 is approached, but the rates of divergence are related by a χ ,, (E, T ) F(E/T) (182) Q0 = T a dynamical critical exponent (Hertz, 1976), a where a 0.75 and F [x] (1 ix)− . Similar behavior ξ (ξ )z (179) ≈ ∝ − τ ∼ x has also been seen in powder samples of UCu5 x Pdx (Aronson et al., 1995). − One of the consequences of this scaling behavior is that time 2. A jump in the Hall constant of YbRh2Si2 when field counts as z spatial dimensions, [τ] [Lz]ingeneral. = tuned through its QCP (Paschen et al., 2004). (see At a classical critical point, temperature is a tuning Figure 34a). parameter that takes one through the transition. The role of 3. A sudden change in the area of the extremal Fermi temperature is fundamentally different at a quantum critical surface orbits observed by de Haas van Alphen at a point: it sets the scale lτ 1/T in the time direction, ∼ pressure-tuned QCP in CeRhin5 (Shishido, Settai, Harima introducing a finite size correction to the QCP. When the and Onuki, 2005). (see Figure 34b). temperature is raised, lτ reduces and the quantum fluctuations are probed on shorter and shorter timescales. There are then Features 2 and 3 suggest that the Fermi surface jumps from two regimes to the phase diagram, a‘small’to‘large’Fermisurfaceasthemagneticorderis lost, as if the phase shift associated with the Kondo effect

(a) Quantum critical: lτ ξ τ (180) collapses to zero at the critical point, as if the f component of - the electron fluid Mott-localizes at the transition. To reconcile where the physics probes the ‘interior’ of the quantum critical a sudden change in the Fermi surface with a second-order bubbles, and phase transition, we are actually forced to infer that the quasiparticle weights vanish at the QCP. (b) Fermi liquid/AFM lτ ξ τ (181) These features are quite incompatible with a spin-density 3 wave QCP. In a spin-density wave scenario, the Fermi where the physics probes the quantum fluid ‘outside’ the surface and Hall constant are expected to evolve continuously quantum critical bubbles. The quantum fluid that forms in through a QCP. Moreover, in an SDW description, the this region is a sort of ‘quantum soda’, containing short- dynamical critical exponent is z 2sotimecountsas = lived bubbles of quantum critical matter surrounded by a z 2dimensionsinthescalingtheory,andtheeffective = paramagnetic (B1) or antiferromagnetically ordered (B2) dimensionality Deff d 2 > 4 lies above the upper critical = + Landau–Fermi liquid. Unlike a classical phase transition, dimension, where mean-field theory is applicable and scale- in which the critical fluctuations are confined to a narrow invariant behavior is no longer expected. Heavy fermions: electrons at the edge of magnetism 139

∗ P Pc 8 0.3 YbRh Si 2 2 CeRhln5 β2 T Oe) Hall 7 6 α1 α 0.2 α1 2 α3

4 α2,3 (K)

T a 0.1 ∗ T T A N 2 b dHvA Frequency (x10 B = B1 = 11 B2 c AF LFL 0.0 0 0 1 2 3 0 1 2 3 (a) B (T) (b) Pressure (GPa)

80

60 CeRhIn5

β2 )

0 40 (m c

∗ α3 m

20 α1

α2,3

0 012 3 Pressure (GPa)

Figure 34. (a) Hall crossover line for sudden evolution of Hall constant in YbRh2Si2.(ReproducedfromPaschen,S.,T.Luhmann,¨ S. Wirth, P. Gegenwart, O. Trovarelli, C. Geibel, F. Steglich, P. Coleman, and Q. Si, 2004, Nature 432,881.)(b)SuddenchangeindHvA frequencies and divergence of quasiparticle effective masses at pressure-tuned, finite field QCP in CeRhIn5.(ReproducedfromH.Shishido, R. Settai, H. Harima, and Y. Onuki, Journal of the Physical Society of Japan 74,2005,1103weithpermissionofthePhysicalSocietyof Japan.)

These observations have ignited a ferment of theoretical This behavior suggests that the critical behavior associated interest in the nature of heavy-fermion criticality. We con- with the heavy-fermion QCP contains some kind of local clude with a brief discussion of some of the competing ideas critical excitation (Schroeder et al., 1998; Coleman, 1999). currently under consideration. One possibility is that this local critical excitation is the spin itself, so that (Coleman, 1999; Sachdev and Ye, 1993; Sengupta, 2000) 5.3.1 Local quantum criticality 1 S(τ)S(τ ) One of the intriguing observations (Schroeder et al., 1998) , 2 - (184) 2 0=(τ τ ,) − in CeCu6 xAux is that the uniform magnetic susceptibil- − 1 − a ity, χ − T C, a 0.75 displays the same power-law is a power law, but where - 0signalsnon-Fermiliquid ∼ + = ;= dependence on temperature observed in the inelastic neutron behavior. This is the basis of the ‘local quantum criticality’ scattering at the critical wave vector Q0. A more detailed set theory developed by Smith and Si (2000) and Si, Rabello, of measurements by Schroeder et al.(2000)revealedthatthe Ingersent and Smith (2001, 2003). This theory requires that scale-invariant component of the dynamical spin susceptibil- the local spin susceptibility χ loc q χ(q,ω)ω 0 diverges = = ity appears to be momentum independent, at a heavy-fermion QCP. Using an extension of the methods & of DMFT (Georges, Kotliar, Krauth and Rozenberg, 1996; 1 a 1 χ − (q,E) T [4(E/T )] χ − (q) (183) Kotliar et al., 2006) Si et al.findthatitispossibletoaccount = + 0 140 Strongly correlated electronic systems

for the local scaling form of the dynamical susceptibility, smaller correlation length ξ 1 is associated with the transition obtaining exponents that are consistent with the observed from AFM to spin liquid, and the second correlation length properties of CeCu6 x Aux (Grempel and Si, 2003). ξ 2 is associated with the confinement of spinons to form a However, there are− some significant difficulties with this valence bond solid (Figure 35). theory. First, as a local theory, the quantum critical fixed It is not yet clear how this scenario will play out for heavy point of this model is expected to possess a finite zero- electron systems. Senthil, Sachdev and Vojta (2005) have point entropy per spin, a feature that is, to date, inconsistent proposed that, in a heavy-electron system, the intermediate with thermodynamic measurements (Custers et al., 2003). spin liquid state may involve a Fermi surface of neutral Second, the requirement of a divergence in the local spin (fermionic) spinons coexisting with a small Fermi surface susceptibility imposes the requirement that the surrounding of conduction electrons, which they call an FL∗ state. In spin fluid behaves as layers of decoupled two-dimensional this scenario, the QCP involves an instability of the heavy- 1 spin fluids. By expanding χ 0− (q) (183) about the critical electron fluid to the FL∗ state, which is subsequently unstable wave vector Q,onefindsthatthesingulartemperature to antiferromagnetism. Recent work suggests that the Hall dependence in the local susceptibility is given by constant can indeed jump at such a transition (Coleman, Marston and Schofield, 2005b). d 1 (d 2)α/2 χ (T ) d q T − (185) loc ∼ (q Q)2 T α ∼ % − + 5.3.3 Schwinger bosons requiring that d 2. ≤ Afinalapproachtoquantumcriticality,currentlyunder In my judgement, the validity of the original scaling development, attempts to forge a kind of ‘spherical model’ by Schroeder et al still stands and that these difficulties for the antiferromagnetic QCP through the use of a large stem from a misidentification of the critical local modes N expansion in which the spin is described by Schwinger driving the scaling seen by neutrons. One possibility, for bosons, rather than fermions (Arovas and Auerbach, 1988; example, is that the right soft variables are not spin per se, Parcollet and Georges, 1997), but the fluctuations of the phase shift associated with the Kondo effect. This might open up the way to an alternative † nb Sab b bb δab (187) formulation of local criticality. = a − N where the spin S of the moment is determined by the 5.3.2 Quasiparticle fractionalization and deconfined constraint nb 2S on the total number of bosons per = criticality site. Schwinger bosons are well suited to describe low- dimensional magnetism (Arovas and Auerbach, 1988). How- One of the competing sets of ideas under consideration at ever, unlike fermions, only one boson can enter a Kondo present is the idea that, in the process of localizing into an ordered magnetic moment, the composite heavy electron breaks up into constituent spin and charge components. In x general, 2

x e− sσ h− (186) 1 σ ! + where sσ represents a neutral spin-1/2 excitation or ‘spinon’. Quantum critical This has led to proposals (Coleman, Pepin,´ Si and Ramaza- Spinon shvili, 2001; Senthil, Vojta, Sachdev and Vojta, 2003; Pepin,´ Spin liquid 2005) that gapless spinons develop at the QCP. This idea is faced with a conundrum, for, even if free neutral spin-1/2 Valence bond excitations can exist at the QCP, they must surely be con- solid fined as one tunes away from this point, back into the Fermi liquid. According to the model of ‘deconfined criticality’ pro- Figure 35. ‘Deconfined criticality’ (Senthil et al., 2004). The quan- posed by Senthil et al.(2004),thespinonconfinementscale tum critical droplet is defined by two divergent length scales - ξ 1 ξ introduces a second diverging length scale to the phase governing the spin correlation length, ξ 2 on which the spinons 2 confine, in the case of the Heisenberg model, to form a valence transition, where ξ 2 diverges more rapidly to infinity than ξ 1. bond solid. (Adapted using data from T. Senthil, A. Vishwanath, One possible realization of this proposal is the quantum melt- L. Balents, S. Sachdev and M.P.A. Fisher, Science 303 (2004) ing of two-dimensional S 1/2HeisenbergAFM,wherethe 1490.) = Heavy fermions: electrons at the edge of magnetism 141 singlet. To obtain an energy that grows with N,Parcolletand Recently, it has proved possible to find the Fermi liquid Georges proposed a new class of large N expansion based large N solutions to the fully screened Kondo impurity around the multichannel Kondo model with K channels (Cox model, where the number of channels is commensurate with and Ruckenstein, 1993; Parcollet and Georges, 1997), where the number of bosons (K 2S)(Rech,Coleman,Parcollet = k K/N is kept fixed. The Kondo interaction takes the form and Zarand, 2006; Lebanon and Coleman, 2007). One of = the intriguing features of these solutions is the presence of JK † agapforspinonexcitations,roughlycomparablewiththe Hint Sαβc cνα (188) = N νβµ ν 1,K,α,β Kondo temperature. Once antiferromagnetic interactions are = ) introduced, the spinons pair-condense, forming a state with where the channel index ν runs from one to K.When a large Fermi surface, but one that coexists with gapped written in terms of Schwinger bosons, this interaction can spinon (and holon) excitations (Coleman, Paul and Rech, be factorized in terms of a charged, but spinless exchange 2005a). fermion χ ν (‘holon’) as follows: The gauge symmetry associated with these particles guar- antees that, if the gap for the spinon goes to zero con- 1 χ †χ tinuously, then the gap for the holon must also go to H (c† b )χ † H.c ν ν (189) int να α ν J zero. This raises the possibility that gapless charge degrees → να √N + + ν K ) * + ) of freedom may develop at the very same time as mag- Parcollet and Georges originally used this method to study netism (Figure 36). In the two impurity model, Rech et al. the overscreened Kondo model (Parcollet and Georges, have recently shown that the large N solution contains 1997), where K>2S. a‘Jones–Varma’QCPwhereastaticvalencebondforms

T

J H

0.6 〈sbsb−s〉 ≠ 0

0.4

0.2 2-ch FL F T K 0.1 0.2 0.3 0.4 0.5 0.6 J H H

J

/ ~

T 1 / S Spin liquid 〈sbsb−s〉 ≠ 0

Heavy Fermi liquid ∆g AFM

〈b〉 ≠ 0

TK / J H

Figure 36. Proposed phase diagram for the large N limit of the two impurity and Kondo latticemodels.Background–thetwoimpurity model, showing contours of constant entropy as afunctionoftemperatureandtheratiooftheKondotemperaturetoHeisenbergcoupling constant. (Reproduced from Rech, J., P. Coleman, O. Parcollet, and G. Zarand, 2006, Phys. Rev. Lett 96,016601.)Foreground–proposed phase diagram of the fully screened,multichannelKondolattice,whereS is the spin of the impurity. At small S,thereisaphase ˜ ˜ transition between a spin liquid and heavy-electron phase. At large S,aphasetransitionbetweentheAFMandheavy-electronphase.If ˜ this phase transition is continuous in the large N limit, then both the spinon and holon gap are likely to close at the QCP. (Reproduced from Lebanon, E., and P. Coleman, 2007, Fermi liquididentitiesfortheInfinite U Anderson Model, Phys. Rev. B (submitted), URL http://arxiv.org/abs/cond-mat/0610027.) 142 Strongly correlated electronic systems

2 † 1 1 2 † between the Kondo impurities. At this point, the holon then c Vc fσ f c √2V and f Vf cσ 2 | σ σ | 0= 2 | σ σ and spinon excitations become gapless. On the basis of f 1c1 √2V | 0=& & this result, Lebanon, Rech, Coleman and Parcollet (2006) [2] The f-sum rule is a statement about the instantaneous, or have recently proposed that the holon spectrum may short-time diamagnetic response of the metal. At short become gapless at the magnetic QCP (Figure 36) in three 2 times dj/dt (nce /m)E,sothehigh-frequencylimit = 2 dimensions. of the conductivity is σ(ω) ne 1 .Butusingthe = m δ iω Kramer’s Kronig¨ relation −

dx σ(x) 6CONCLUSIONSANDOPENQUESTIONS σ(ω) = iπ x ω iδ % − − Ishallendthischapterwithabrieflistofopenquestionsin at large frequencies, the theory of heavy fermions. 1 dx 1. To what extent does the mass enhancement in heavy- ω(ω) σ(x) = δ iω π electron materials owe its size to the vicinity to a nearby − % quantum phase transitions? 2. What is the microscopic origin of heavy-fermion super- so that the short-time diamagnetic response implies the f-sum rule. conductivity and in theextreme cases UBe13 and PuCoGa5 how does the pairing relate to both spin quenching and [3] To prove this identity, first note that any two-dimensional matrix, M,canbeexpandedasM m0σ 2 m σ 2σ, the Kondo effect? 1 = 1 +$· $ (b (1, 3))wherem0 Tr[Mσ2]andm Tr[Mσσ2], 3. What is the origin of the linear resistivity and the = = 2 $ = 2 $ logarithmic divergence of the specific heat at a ‘hard’ so that in index notation heavy-electron QCP? 1 4. What happens to magnetic interactions in a Kondo Mαγ Tr[Mσ2](σ 2)αγ = 2 insulator, and why do they appear to vanish? 1 5. In what new ways can the physics of heavy-electron Tr[Mσσ2] (σ 2σ)αγ systems be interfaced with the tremendous current devel- +2 $ · $ opments in mesoscopics? The Kondo effect is by now a Now, if we apply this relationship to the αγ components well-established feature of Coulomb blockaded quantum of σ αβ σ γδ,weobtain dots (Kouwenhoven and Glazman, 2001), but there may $ ·$ be many other ways in which we can learn about local 1 T moment physics from mesoscopic experiments. Is it pos- σ αβ σ γδ σ σ 2σ (σ 2)αγ $ ·$ = 2 $ $ δβ sible, for example, to observe voltage-driven quantum # $ 1 T phase transitions in a mesoscopic heavy-electron wire? σ σ 2σ b σ (σ 2σ b)αγ +2 $ $ δβ This is an area grown with potential. b 1,3 )= # $ It should be evident that I believe there is tremendous T If we now use the relation σ σ 2 σ 2σ,togetherwith prospect for concrete progress on many of these issues in $ =− $ σ σ 3andσσbσ σ b,weobtain the near future. I hope that, in some ways, I have whet your $ ·$= $ $ =− appetite enough to encourage you also to try your hand at 3 1 σ αβ σ γδ (σ 2)αγ (σ 2)δβ (σσ2)αγ (σ 2σ) their future solution. $ ·$ =−2 + 2 $ · $ δβ

NOTES ACKNOWLEDGMENTS [1] To calculate the matrix elements associated with valence fluctuations, take This research was supported by the National Science Founda- 1 tion grant DMR-0312495. I would like to thank E. Lebanon f 1c1 (f †c† c† f †) 0 , and T. Senthil for discussions related to this work. I would | 0=√ − | 0 2 ↑ ↓ ↑ ↓ also like to thank the Aspen Center for Physics, where part f 2 f †f † 0 and c2 c† c† 0 of the work for this chapter was carried out. | 0= ↑ ↓ | 0 | 0= ↑ ↓| 0 Heavy fermions: electrons at the edge of magnetism 143

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