Strongly correlated electron systems at the border of magnetism are of active current interest, particularly because the accompanying quantum criticality provides a route towards both strange-metal non-Fermi liquid behavior and unconventional superconductivity. Among the many important questions is whether the magnetism acts simply as a source of fluctuations in the textbook Landau framework, or instead serves as a proxy for some unexpected new physics. We put into this general context the recent developments on quantum phase transitions in antiferromagnetic (AF) heavy fermion metals. Among these are the extensive recent theoretical and experimental studies on the physics of Kondo destruction in a class of beyond-Landau quantum critical points (QCPs). Also discussed are the theoretical basis for a global phase diagram of AF heavy fermion metals, and the recent surge of materials suitable for studying this phase diagram. Furthermore, we address the generalization of this global phase diagram to the case of Kondo insulators, and consider the future prospect to study the interplay among Kondo coherence, magnetism, and topological states. Finally, we touch upon related issues beyond the AF settings, arising in mixed valent, ferromagnetic, quadrupolar, or spin glass f-electron systems, as well as some general issues on emergent phases near QCPs.

Electron correlations give rise to a variety of novel phenomena, including unconventional superconductivity and non-Fermi liquid behavior. Pertinent materials include cuprate and iron-based superconductors, heavy fermion metals and organic charge-transfer salts. In the presence of correlations, we face the central question of how the electrons are organized and, in particular, whether there are principles that are universal among the various classes of the strongly correlated materials. One such principle, which has come to the forefront in recent years, is quantum criticality 1, 2.

A quantum critical point (QCP) arises at a second-order transition between two ground states. The distinct ground states themselves occur due to competing interactions of a many-body system. The collective fluctuations of the QCP control the physics of the quantum critical regime, which corresponds to a wide parameter range at non-zero temperature. While quantum criticality is of interest in its own right, and is being discussed in a variety of contexts ranging from insulating magnets to designer materials such as quantum dots and cold atom systems, it plays a special role in strongly correlated electron systems. Quantum criticality provides a mechanism for both the non-Fermi liquid behavior and emergent phases such as unconventional superconductivity.

In this paper, we focus primarily on the issues arising in strongly correlated f-electron systems. These systems have been serving as a prototype setting to study the physics of quantum criticality. We will in particular highlight a class of QCPs in the antiferromagnetic (AF) Kondo lattice systems, which features the physics of Kondo destruction. The latter also gives rise to a global phase diagram, involving zero-temperature states that are distinct not only by the AF order but also by the nature of the Fermi surfaces. Experimental probes of such “beyond-Landau” QCPs and emergent phases are discussed. We also consider the analogous issues in Kondo insulators, extend the notion of Kondo destruction to mixed valent or ferromagnetic settings, and briefly discuss quantum criticality in quadrupolar and spin glass cases. Finally, we address several issues concerned with emergent phases around QCPs.

2 Local moments and competing interactions

Heavy fermion systems represent canonical settings in which localized magnetic moments and itinerant conduction electrons coexist 3–6. There are interactions not only among the local moments but also between the local moments and conduction electrons. The competition between these interactions gives rise to distinct ground states. Therefore, heavy fermion systems provide a fertile setting for quantum phase transitions.

The Kondo lattice Hamiltonian reads:

((1))

It describes materials which contain spin-1/2 local moments (S_{i} at every site i) that are physically associated with the localized f electrons, and a band of conduction electrons (c_{iσ}). The parameters t_{ij} and I_{ij} are respectively the hopping matrix of the conduction electrons (with bandwidth W) and the RKKY exchange interaction among the local moments (with a characteristic strength I), while J_{K} is the AF Kondo exchange interaction. We consider the simplest case, corresponding to one local moment per unit cell, and a filling of x conduction electrons per unit cell.

An effective single-ion Kondo energy scale T_{0} signifies the onset of initial Kondo screening: the local moments are essentially decoupled from the conduction electrons at temperatures (T) much larger than T_{0}, but develop dynamical singlet correlations with the conduction electrons as T is lowered towards T_{0}. This provides a means to check the validity of the Kondo lattice Hamiltonian through measurements at temperatures above or around T_{0}. Examples are the observation of a single-ion Kondo peak in the photoemission 7 and STM spectra 8.

As temperature is further lowered, the system may evolve towards different ground states. In one ground state, the local moments form a spin singlet with the conduction electrons. This Kondo entanglement turns each local moment into an electronic Kondo resonance excitation. The excitations occur at low energies, near the bare Fermi energy of the conduction electrons. These resonance excitations hybridize with the conduction electrons, forming renormalized quasiparticle bands that are separated by a hybridization gap 6.

For the generic case of incommensurate fillings with x < 1, the total electron count is 1 + x < 2. Therefore, the Fermi energy will be in the lower hybridized band, located in the peak part of the DOS. This gives rise to a heavy Fermi liquid (FL) state with enhanced quasiparticle mass and a large Fermi surface.

In the special case of commensurate filling, x = 1, the total electron count is 2 per unit cell, corresponding to half-filling. The renormalized Fermi energy must be in the middle of the hybridization gap, leaving the lower hybridized band completely occupied and the upper one completely empty. The result is a Kondo insulator 9–11.

Ground states other than the Kondo entangled state are also possible. The local moments are coupled to each other through an exchange interaction. In Eq. (1), this is the RKKY interaction I; up to Section 7, we will consider it to be AF. This exchange interaction promotes AF order. Correspondingly, the Kondo and RKKY interactions compete against each other 12, 13. This competition is captured by the tuning parameter , where , with ρ_{0} being the density of states of the conduction electrons at the Fermi energy, parameterizes the Kondo interaction.

In experiments, this tuning is accomplished by the application of external parameters. Typical examples are applying pressure, chemical substitution, or magnetic field. Applying pressure increases/decreases δ in Ce/Yb-based heavy fermion compounds. Isoelectronic substitution of the ligands with smaller/larger atoms has the effect of applying positive/negative chemical pressure, although one needs to keep in mind that substitutions also introduce additional disorder. Finally, an external magnetic field influences the tendencies towards Kondo screening and AF order to different degrees (small magnetic fields reduce the Kondo scale in quadratic order but are coupled to the AF order linearly), thereby modifying the ratio δ.

The explicit observation of AF QCPs in several heavy fermion materials provides a setting to study quantum criticality in general. Furthermore, here, quantum criticality becomes intertwined with two of the key notions of correlated electrons: non-Fermi liquid behavior and, in many cases, unconventional superconductivity, occur in the quantum critical regime of heavy fermion metals.

3 Antiferromagnetic quantum critical points

Weak metallic antiferromagnets such as Cr are well-described in terms of a spin-density-wave (SDW) order. The order parameter is the staggered magnetization. Correspondingly, its transition at T = 0 to a paramagnetic metal phase is described by an SDW QCP 14–16. This description falls into the Landau framework, with the order parameter differentiating between the phases and the fluctuations of the order parameter describing the quantum criticality.

Microscopically, this has been studied in terms of a one-band Hubbard model 14. The SDW order is induced by the Coulomb repulsion, and the effective field theory is a quantum Ginzburg-Landau action 14,

((2))

which describes the fluctuations of the order parameter, ϕ, in both space (x) and imaginary time (τ). This yields an effective dimensionality d + z, where d is the spatial dimension and z is the dynamic exponent. The wavevector q and Matsubara frequency ω_{n} are reciprocal to x and τ, respectively.

Theoretical studies on QCPs in AF heavy fermion metals have led to beyond-Landau QCPs, which feature inherently quantum modes besides the order-parameter fluctuations. While this notion extends to general contexts, for heavy fermion metals the emphasis has been on the critical modes associated with the destruction of the Kondo effect. Such critical modes are in addition to the fluctuations of the AF order parameter.

This Kondo destruction has provided insights into early puzzles on dynamical scaling in quantum critical heavy fermion systems, and led to predictions for the nature of Fermi surfaces and energy scales near the QCP that have been extensively observed in magnetotransport and quantum oscillation experiments.

3.1 Kondo destruction and beyond-Landau QCP

Heavy fermion metals are microscopically described in terms of the Kondo lattice Hamiltonian, Eq. (1). Their traditional understanding invokes a ground state with a Kondo singlet between the local moments and conduction electrons. As described earlier, this Kondo entanglement in the ground state gives rise to an excitation spectrum with a Kondo resonance per local moment. The resulting heavy Fermi liquid state has a large Fermi surface, which incorporates 1 + x electrons per unit cell. In this way, the heavy quasiparticles near the Fermi energy can undergo an SDW order, thereby leading to an SDW QCP.

The dynamical spin correlations, however, may invalidate the very process that gives rise to the Kondo singlet ground state and the associated Kondo resonances. Such considerations have led to the notion of a local QCP, characterized by the physics of Kondo destruction. The phenomenon of Kondo destruction was already studied in the earlier renormalization-group (RG) study of the charge analog of the dynamical RKKY versus Kondo competition 17, as well as its generalization to the spinful case 18–20. Motivated by experiments on CeCu_{5.9}Au_{0.1}21, that we discuss in Section 3.3 below, concrete theoretical formulations for the Kondo destruction were advanced 22–24. The theory implied the collapse of the extra energy scale , a jump of the Fermi surface across the QCP, and the critical nature of the quasiparticles on the whole Fermi surface at the QCP (cf. Fig. 1). The Kondo destruction was later also studied in a fermionic slave-particle representation of the Kondo lattice Hamiltonian 25, 26 and in dynamical mean field theory 27. Finally, the notion of non-Fermi liquid quasiparticles on the whole Fermi surface (as opposed to only on the hot spots) has also been emphasized in a recent work based on a self-consistent method 28.

The extended dynamical-mean-field theory (EDMFT) 17, 29, 30 provides a means to study the weakening of the Kondo effect by the dynamical effects of RKKY interactions. The reduction of the Kondo singlet amplitude is described through the decrease of the energy scale .

At the local QCP, the Kondo destruction () occurs at δ_{c}, as shown in Fig. 1. In this case, the local spin susceptibility is

((3))

This form was derived with the aid of an ε-expansion RG approach 22, 24. The exponent α has been found to be close to 0.75 (ranging from 0.72 to 0.83) 31–33.

The Kondo destruction implies that, as the QCP is approached from the paramagnetic side, the quasi-particle residue , where b* is the strength of the pole of the conduction-electron self-energy ∑(k, ω); this pole characterizes the Kondo resonance. In turn, electron excitations have a non-Fermi liquid form on the entire large Fermi surface.

3.2 Dynamical Kondo effect

The breakdown of the large Fermi surface implies that the Fermi surface will be small on the antiferromagnetically ordered side. This is the AF_{S} phase, which will be further discussed in the next section. A dynamical Kondo effect will still generate a mass enhancement in the AF_{S} phase 32–34.

To consider this point further, the static amplitude of the Kondo singlet vanishes in the AF_{S} phase. Thus, the electron distribution function in the ground state displays a discontinuity at the small Fermi surfaces. By contrast, the effective mass is a dynamical quantity, measuring the dispersion of the Landau quasiparticles. The quasiparticle mass enhancement in the AF_{S} phase arises due to the dynamical part of the Kondo-singlet correlations, which persists in this phase. This has been explicitly demonstrated in the calculated local dynamics in the AF_{S} region of the Kondo lattice model 32–34, as illustrated in Fig. 2.

This dynamical Kondo effect ensures the continuity of the divergent effective mass as the QCP is approached from both the AF ordered side and the paramagnetic side.

3.3 Experimental probes of QCPs

Numerous experimental techniques have been used to characterize quantum critical materials. Both temperature-dependent measurements at different fixed control parameter values and isothermal measurements as function of control parameter have been instrumental in advancing the field 4. Temperature dependencies are frequently analyzed by power law fits and then presented in terms of color-coded temperature–tuning parameter phase diagrams, where the quantum critical region appears as a fan emerging from the QCP. Also insightful are scaling analyses. They test in which temperature range and up to which tuning-parameter distance from the QCP the data can be collapsed onto a single universal curve, described by universal critical exponents. These critical exponents are then compared to theoretical predictions.

A milestone was the scaling analysis of the dynamical spin susceptibility of CeCu_{6−x}Au_{x} near the critical concentration x_{c} = 0.1, determined by inelastic neutron scattering at different energy transfers E21. The evidenced E/T scaling and the accompanying anomalous critical exponent are not expected in the order-parameter description of the SDW QCP, and have been interpreted in terms of Kondo destruction 22, 23.

Inelastic neutron scattering experiments on heavy fermion compounds are demanding as the typically small magnetic moments require large single crystalline samples and long data acquisition times. Therefore, such experiments have only been used to study a limited number of quantum critical heavy fermion metals; besides CeCu_{6−x}Au_{x}, these include the earlier results of Aronson et al. 35 on UCu_{5−x}Pd_{x} (which we will come back to in Section 7). Nevertheless, further progress was made by exploring materials with other experimental probes.

An important quantity is the Fermi surface. As it is expected to evolve qualitatively differently across an SDW and a local QCP 23, 24, probes to detect its evolution are of great interest. A simple measurement sensitive to changes of the Fermi surface is the Hall coefficient R_{H}. At T = 0, the Fermi surface is expected to evolve continuously across an SDW QCP but discontinuously, in the form of a jump, across a local QCP. The former is due to the order parameter increasing continuously as the ordered phase gets stabilized with increasing distance δ from the critical tuning parameter value δ_{c}. In the case of strong nesting, R_{H} may show a sizable (but continuous) evolution within the ordered phase. Across a local QCP, on the other hand, the Kondo entanglement is destroyed as the system orders. Thus, the 4f spins contribute to the Fermi surface only in the paramagnetic phase but drop out of the Fermi surface in the ordered phase. This leads to a jump of the Fermi surface. As the destruction of the Kondo entanglement is associated with a temperature scale T*, that is in addition to the ordering temperature T_{N} and collapses with it only at the QCP, experimental signatures of the Kondo destruction are expected to occur away from T_{N} (δ) at finite temperatures. This is again in contrast to the SDW case, where T_{N} (δ) is the only temperature scale and thus the only place in the phase diagram where changes of the Fermi surface can be expected.

Of course, measurements of R_{H} can only be performed at finite temperatures. Thus, it is important to detect both the position (tuning parameter value at a given temperature) where a crossover in R_{H} is observed and the sharpness of the crossover (e.g., in the form of the full width at half maximum, FWHM). To distinguish between the two QCP scenarios, extrapolations of both quantities to T = 0 are needed. So far, this has been reliably done for both YbRh_{2}Si_{2}36, 38 and Ce_{3}Pd_{20}Si_{6}37. In both systems, a crossover in R_{H} was observed across a scale T*, that coincides with T_{N} only at the QCP. The FWHM of the crossover was shown to extrapolate to zero in the T = 0 limit (Fig. 3), thus evidencing the presence of a local QCP in both cases. (In Ce_{3}Pd_{20}Si_{6}, this occurs inside the ordered region of the phase diagram, see Fig. 3. This is further discussed in the next section.) By contrast, a combined hydrostatic and chemical pressure study of the itinerant antiferromagnet Cr_{1−x}V_{x} revealed a continuous evolution of R_{H} across the SDW QCP, with a width of the drop of within the ordered phase below 9.7 GPa that is essentially independent of temperature (Fig. 4).

A second important tool to probe Fermi surface changes are de Haas–van Alphen (dHvA) experiments. These have been used to study the Fermi surface evolution of CeRhIn_{5} as a function of pressure 42 at magnetic fields around 10 T, just above the upper critical field for the suppression of superconductivity H_{c2}.

In this material, the pressure-induced AF to non-magnetic quantum phase transition has been observed to be second order for fields above 3 T 43, 44. The second-order nature of this transition is further evidenced by the tendency of divergence near the critical pressure, p_{c}, for both the cyclotron mass m* 42 and the T^{2}-resistivity coefficient A45. Across such an AF QCP, the dHvA frequencies undergo a sharp jump. In the AF ordered state, at p < p_{c}, they are compatible with the Fermi surface being small, while those in the paramagnetic state, at p > p_{c}, are consistent with the Fermi surface being large. Such a sudden jump of the Fermi surface across the QCP is again to be contrasted with the smooth evolution as indicated by the Hall-coefficient results across the SDW QCP in the Cr:V system.

We close this subsection by remarking on three aspects. First, as we discussed, the dynamical Kondo effect leads to mass enhancement in the AF_{S} phase. Within the local QCP description, this dynamical Kondo effect underlies the mass enhancement observed in the AF phases of YbRh_{2}Si_{2}, CeCu_{6−x}Au_{x} and CeRhIn_{5}. Beyond these, recent experimental observations in the AF state of CeIn_{3} at ambient pressure appear to provide evidence for a dynamical Kondo effect as well 46. CeCu_{2}Ge_{2}, in its AF state, may also belong to AF_{S}, and a similar observation has also been reported recently 47.

Second, the critical destruction of quasiparticles at the QCP has recently been probed by thermal transport measurements in YbRh_{2}Si_{2}, which observed a violation of the Wiedemann-Franz law 48. STM spectroscopy is also poised to probe the nature of the single-electron excitations in the quantum critical regime 49.

Third, recent inelastic neutron scattering measurements on YbRh_{2}Si_{2} have finally observed the AF wavevector in the fluctuation spectrum at T > T_{N}50. This is consistent with the order being AF, which has in the past been inferred from the facts that magnetic field applied in any direction reduces the ordering temperature and that the magnetic susceptibility is reduced below the ordering temperature. Unlike the size of the Fermi surface, the dynamical spin susceptibility per se does not unambiguously address the question of whether f electrons are localized or delocalized in the AF state. Still, the incommensurate wavevector is stable up to quite high energy, extending to as large as 1.25 meV, which is cloe to the (bare) Kondo energy scale; this would not be expected for an SDW order associated with the delocalized f electrons and appears to be more consistent with local-moment magnetism. It would be instructive to see how the AF wavevector may connect to the form of the RKKY interactions.

This neutron study opens the door to future measurements of the dynamical spin susceptibility of YbRh_{2}Si_{2} at low frequencies and temperatures that may be able to address the nature of dynamical scaling in the quantum critical regime. In addition, at magnetic fields that correspond to the field-induced FL regime (Fig. 1, top), a resonance mode has been observed 50, which promises to shed light on the large-Fermi surface regime of the phase diagram.

4 Global phase diagram of antiferromagnetic heavy fermion metals

The notion of Kondo destruction not only provides a means to characterize novel QCPs but also opens up the possibility of new phases at zero temperature.

4.1 Theoretical basis

A zero-temperature phase diagram has been proposed for the AF heavy fermion metals 51–53. Figure 5a, left panel, shows the two-parameter phase diagram. Here the horizontal axis is the Kondo coupling J_{K}, and the vertical axis G marks the quantum fluctuations of the local-moment magnetism; the latter is tuned by frustration and dimensionality.

The starting point of the theoretical basis is the establishment of AF_{S}, an AF phase with destruction of the Kondo effect and the concomitant small Fermi surface. The demonstration of its stability was based on an analysis of the Kondo lattice Hamiltonian in the limit. This limit allows an expansion around a new reference point, corresponding to J_{K} = 0, where the ordered local-moment system is decoupled from the non-interacting conduction electrons. With respect to this reference point, the effect of J_{K} can be systematically studied using an RG method. For the Ising case, the AF-ordered phase of the local-moment magnetism has a spin gap; as a result, J_{K} is irrelevant implying the stability of the AF_{S} phase 51.

For the case with full spin-rotational invariance, corresponding to an SU(2) symmetry, the analysis is more involved. Using a quantum non-linear sigma model (QNLσM) representation, and with the aid of a mixed gapless-boson-and-fermion RG method, the Kondo coupling is shown to be exactly marginal, leading to the stability of the AF_{S} phase 54.

A sufficiently large J_{K} gives rise to the Kondo entanglement, resulting in the standard heavy Fermi liquid state – a paramagnetic phase with a large Fermi surface. Therefore, AF_{S} and P_{L} represent two anchoring phases of the phase diagram in Fig. 5a, left panel. The distinction between these two phases are beyond-Landau: in addition to the presence/absence of AF order, the two are also differentiated by the absence/presence of Kondo entanglement.

The global phase diagram features three types of transition sequences as the system goes from the AF_{S} phase to the P_{L} phase. The type I transition goes directly from AF_{S} to P_{L}, while the type II sequence corresponds to transitions from AF_{S} through AF_{L} to P_{L}. In other words, they differ in terms of whether the collapse of the Kondo effect occurs at the paramagnetic-to-AF QCP, or whether it is delayed until inside the ordered part of the phase diagram. This distinction was inferred from EDMFT studies 22, 24. The AF_{L} phase corresponds to the SDW order of the heavy fermion quasiparticles of the P_{L} phase, and Kondo destruction occurs at the boundary between AF_{L} and AF_{S} phases. Note that, again, the dynamical Kondo effect in the AF_{S} phase ensures the continuity of the quasiparticle mass across its boundary with the AF_{L} phase.

The type III sequence from AF_{S} to P_{L} passes through P_{S}, a paramagnetic phase with Kondo destruction and hence small Fermi surface. The existence of the P_{S} phase can be seen by noting that increasing G of the local-moment magnetism alone (along the vertical axis) suppresses AF order. In the most typical cases, this leads to spin-Peierls order, which contains a spin gap. This spin gap renders a small J_{K} coupling to be irrelevant in the RG sense, thereby yielding the stability of a P_{S} phase. The competition between the P_{L} and P_{S} phase has been analyzed through the effect of the Berry phase in the QNLσM representation 55.

4.2 Materials basis

In Section 3.3, we have discussed the evidence for a quantum phase transition along the trajectory I (AF_{S} directly to P_{L}) in pure YbRh_{2}Si_{2} as a function of the magnetic field, in CeCu_{6−x}Au_{x} as a function of the substitution level x or pressure, and in CeRhIn_{5} as a function of pressure at magnetic fields just above H_{c2}. These three cases are illustrated in Fig. 5b.

Systematic investigations have also been performed on substituted and pressurized YbRh_{2}Si_{2}56–59. The resulting substitution level–magnetic field phase diagram is consistent with the profile of the global phase diagram in Fig. 5b. In particular, compounds with Ni- and slight Ir-substitution for Rh also undergo a type I transition (AF_{S} to P_{L}). For the case of Co-substitution for Rh as well as for the pure YbRh_{2}Si_{2} under sufficient pressure, the transition appears to be divided into two stages, as along the type II trajectory with the Kondo destruction occurring inside the AF order. By contrast, for substitutions with Ge, or Ir concentrations beyond 2.5%, the transition seems to be of type III, a trajectory with the Kondo destruction outside the ordered phase. Direct measurements of the Kondo destruction in substituted and pressurized YbRh_{2}Si_{2} remain, however, to be be carried out.

To explore the global phase diagram, it is essential to broaden both the materials basis and the microscopic parameter variety. In particular, it is instructive to consider materials that lie in different ranges of the parameter G, the degree of quantum fluctuations in the local-moment magnetism. A recent investigation of the cubic compound Ce_{3}Pd_{20}Si_{6}37 probed the lower part of the global phase diagram, where G is small (dimensionality high) because of the absence of crystal anisotropy. A magnetic field-tuned QCP was observed to be accompanied by Kondo destruction. However, the phase above the critial field (cf. Fig. 3) still possess some kind of order. While both the type of order and the nature of the transition to the paramagnetic phase need to be explored in future experiments, these results provide strong evidence for a type II trajectory.

In another cubic material, CeIn_{3}, dHvA experiments revealed a strong enhancement of the effective mass at a critical field within the ordered portion of the temperature–magnetic field phase diagram 60. Thus, one may speculate that a type II trajectory is followed also by this material. In order to further test the global phase diagram, the role of dimensionality might also be studied more systematically, e.g., by the variation of the thickness of thin films of heavy fermion compounds. That such tuning might, in fact, be technically feasible was demonstrated by recent investigations on molecular beam epitaxy grown superlattices 61, 62.

Geometrical frustration is expected to be a means to enhance G and reach the upper portion of the global phase diagram. Earlier studies considered the hexagonal YbAgGe 63. More recently, Yb_{2}Pt_{2}Pb was investigated 64, in which the Yb moments are located on a quasi-two-dimensional Shastry-Sutherland lattice. In both cases, there exists an intermediate magnetic field regime between an AF order and a paramagnetic Fermi liquid regime, as illustrated in Fig. 5b. Other materials in this category might be the fcc YbPtBi 65 and CePdAl with a distorted Kagome lattice 66.

Clearly, this is a subject that will see major developments in the near future, both theoretically and experimentally. These studies may ultimately lead us towards a classification of universality classes in quantum critical heavy fermion systems.

5 From local moment to mixed valent regime

The Kondo lattice Hamiltonian is only valid when the underlying f-electron orbital is half-filled, with n_{f} = 1 per unit cell. Most of the heavy fermion metals displaying AF QCPs, such as YbRh_{2}Si_{2} and CeCu_{6−x}Au_{x}, show a very large mass enhancement and a small Kondo temperature on the order of 10 K, and should therefore be in the Kondo limit. This is also expected to be the case for CeRhIn_{5}, for which the Kondo temperature is similarly about 10 K 67. However, some materials do not appear to be in this limit. For instance, in the quantum critical superconductor β-YbAlB_{4}, the mass enhancement is modest, the Kondo temperature is rather high (on the order of 200 K), and the Yb valence was shown to be 2.75 (distinctly smaller than the integer valence 3) by hard X-ray photoemission spectroscopy 68. While the occurrence of mixed valence is rare in heavy fermion metals, it is more common in Kondo insulators. In fact, these latter were originally referred to as mixed valent semiconductors 69. For the prototypal Kondo insulator SmB_{6}70, mixed valence was detected early on 71. Thus, it is interesting to look at the generalization of the Kondo lattice Hamiltonian to the mixed valent regime, where n_{f} ≠ 1.

5.1 Periodic Anderson model

The underlying microscopic Hamiltonian in the mixed valent regime is the periodic Anderson model

((4))

which contains both spin and charge (valence) degrees of freedom of the f electrons. Here, and .

Away from the local-moment limit, the inter-orbital Coulomb interaction

((5))

that was introduced early on 13, 72, also has to be considered. In the mixed valent regime, it influences the charge Kondo effect and the corresponding mixed valent non-Fermi liquid behavior 73–75.

5.2 Gaussian fixed point: Generalization of the SDW QCP to mixed valence

In the Kondo lattice case, assuming a finite Kondo energy scale at the QCP led to the description of the SDW QCP (Section 3.1). This description has been generalized to the mixed valence setting 76. Microscopically, for U_{fc} below a critical value U_{c}, there is a valence crossover (n_{f} varies continuously with ε_{f}), whereas for U_{fc} > U_{c} there is a first order valence transition (n_{f} shows a discontinuous decrease at some ε_{f}). The quantum-critical end point is a Gaussian fixed point, representing the generalization of the SDW QCP to the co-existing spin and charge sector. Indeed, the effective field theory of Ref. 76 has the form of the Hertz action, Eq. (2), with the damping corresponding to a dynamic exponent z = 3. The QCP retains the generic feature of a Gaussian fixed point, in that ω/T scaling is violated; presumably H/T scaling is correspondingly violated as well.

Like in the SDW QCP, there will be no jump of the Fermi surface across such a QCP. Indeed, in order to realize a jump in the Fermi surface (and the Hall coefficient) in this picture, the transition is taken to be first order 77. Alternatively, it was suggested that a jump in the Fermi surface can arise in a continuous transition due to folding of the large Fermi surface by the AF order 78. This, however, appears incorrect: as the example of Cr:V (Fig. 4) demonstrated, the Hall coefficient must be continuous across such a QCP.

5.3 Interacting fixed point: Generalization of the Kondo destruction to mixed valence

To search for an interacting fixed point, it is important to address whether the Kondo destruction effect can still take place under the mixed valence condition. This is a delicate issue. In the Kondo lattice case, Kondo destruction amounts to the localization of f electrons. This is physically transparent, because localization can readily arise for a commensurate filling of an electronic orbital (one f electron per site). At mixed valence, the f orbital has a fractional per-site occupancy; there is no mechanism known for electron localization in such cases.

This issue has recently been studied in a simplified setting, the pseudo-gapped particle-hole-asymmetric Anderson impurity model. It was shown that Kondo destruction occurs even at mixed valence 79, see Fig. 6. The physics in the spin sector is very similar to that for the local-moment case n_{f} = 1, in that an energy scale collapses as the QCP is approached from the Kondo-screened side. At the QCP, the charge fluctuations are singular along with the spin fluctuations. (This is in contrast to the particle-hole symmetric local-moment case n_{f} = 1, where the singular charge fluctuations are absent.) In line with the interacting nature of the fixed point, both the charge and spin responses display a magnetic field over temperature (H/T) and frequency over temperature (ω/T) scaling. When the results are generalized to the lattice case, the Kondo scale going to zero would imply that the Fermi surface jumps.

Recent measurements on the mixed valent heavy fermion compound β-YbAlB_{4} have shown that the static magnetization exhibits H/T scaling 82, 83, providing evidence for the interacting nature of an underlying QCP. The above theoretical study in a simplified model provides an existence proof of a local QCP at mixed valence with this kind of scaling behavior. If the QCP displays universality, such scaling behavior would also be expected at other mixed valent QCPs with Kondo destruction, irrespective of the microscopic details that give rise to valence fluctuations.

The theoretical results also suggest that, by restraining our further discussion to the Kondo case n_{f} = 1, we do not lose generality.

6 Global phase diagram of Kondo insulators

In the case of half-filling, i.e., x = 1, the competition between the Kondo and RKKY interactions continues to operate. This case presents an opportunity to study a variety of correlated insulator states.

6.1 Global phase diagram

As discussed in Section 2, when Kondo entanglement takes place, the ground state for the case of half-filling is a paramagnetic insulator, the Kondo insulator. Thus, in the global phase diagram, instead of a heavy fermion metal state with a large Fermi surface (P_{L}) we now have a state with no Fermi surface at all (P_{0}). Both the AF_{S} and P_{S} phases are expected to remain stable in the case of half-filling 84 (Fig. 5a, right panel).

There is evidence that transitions between these phases can indeed be realized experimentally. While tuning through a QCP appears to be accomplished in a number of materials as function of magnetic field 85, 86 or doping 87–90, we focus here on the conceptually simpler situation of tuning by external or chemical pressure. As an example, SmB_{6} is tuned by pressure from the Kondo-insulating phase P_{0} to an AF phase 91. The dashed arrow in Fig. 5c suggests that this transition is to the phase AF_{S}. The explicit experimental demonstration that the AF phase of SmB_{6} has a small Fermi surface is, however, still missing.

While most Kondo insulators are cubic, there are also a few representatives with lower crystalline symmetry, the most prominent one being orthorhombic CeNiSn. The isostructural and isoelectronic compounds CePtSn and CePdSn both order antiferromagnetically at low temperatures 92–94. Transitions between the AF phase and the Kondo-insulating phase have been investigated both by chemical and by hydrostatic pressure tuning. The former has been realized by isoelectronic substitutions, i.e., in Ce(Pt_{1−x}Ni_{x})Sn 88, 95, 96 or Ce(Pd_{1−x}Ni_{x})Sn 97, and by hydrogenation 98. Tuning by hydrostatic pressure appears more challenging. The magnetic order in CePdSn was shown to be suppressed only by hydrostatic pressure above 60 kbar 99. A pressure of 20 kbar applied to CePtSn reduces the Néel temperature only slightly 100. The ensemble of these experiments is represented schematically in Fig. 5c by the dashed arrow through CePtSn, CePdSn, and CeNiSn. As for SmB_{6}, an explicit demonstration of a small Fermi surface AF_{S} in the antiferromagnetically ordered state still needs to be done. The lower symmetry of CeNiSn compared to SmB_{6} is represented in Fig. 5c by placing CeNiSn and its relatives at a higher G value than pressure tuned SmB_{6}.

It would be instructive to consider other Kondo insulator systems in the context of the global phase diagram. For instance, CeRu_{2}Al_{10} and CeFe_{2}Al_{10}101 might fit into this framework as well.

6.2 Correlated topological insulators and their interplay with Kondo coherence and magnetism

In recent years, spin–orbit coupling in band insulators has been extensively studied as giving rise to topological insulators 102, 103. These bulk insulators have gapless helical surface states. In Kondo insulators, topological insulating states have been studied either through the spin–orbit coupling as manifested in the hybridization matrix 104 or in conjunction with the spin–orbit coupling of the conduction electrons 105. As such, Kondo insulators present an opportune system to study the interplay between topological and Kondo/magnetic effects, and even unconventional superconductivity near such boundaries. Recent measurements of several groups in SmB_{6} have provided tentative evidence for the surface states of a topological insulator 106, 107. The evolution of such surface states under pressure (cf. Fig. 5c) represents an intriguing issue for future studies.

From the perspective of topological states 105, it is instructive to note that the 5d electrons of CePtSn will contain a large spin–orbit coupling, which may influence the topological nature of its expected insulating state at high pressure.

7 QCPs in other degrees of freedom

7.1 Weak ferromagnets and ferromagnetic heavy fermion metals

Based on the field theory of Hertz for the ferromagnetic case, Eq. (2), the coupling to the fermions near the Fermi surface has been shown to yield non-analyticity in the coupling constants 108. This is believed to turn the T = 0 transition first order, creating a “fish-tail” structure in the temperature–pressure–magnetic field phase diagram 109.

In Kondo lattice systems, the effect of local moments must be considered. For the ferromagnetic Kondo lattice, Ref. 110 studied the ordered phase using the QNLσM basis (generalizing the study for the AF ordered phase mentioned in Section 4.1). Inside the ferromagnetic phase, the Fermi surface is small. The existence of this F_{S} phase raises the prospect that the T = 0 ferromagnetic to paramagnetic transition may also come in different varieties, in a way that is analogous to the global phase diagram of the AF heavy fermion metals described earlier. In particular, this raises the intriguing possibility for a QCP in ferromagnetic heavy fermion metals.

The early dHvA measurements 111 in the ferromagnetic compound CeRu_{2}Ge_{2} (with a Curie temperature T_{c} = 8 K) can be interpreted in terms of the F_{S} phase. Other ferromagnetic heavy fermion compounds of interest include the Re-doped URu_{2}Si_{2}112, CeRu_{2}Al_{2}B 113, and YbNi_{3}Al_{9}114. Recent experiments have provided evidence for a ferromagnetic QCP, both in YbNi_{4}P_{2}115 and in the iron pnictide Ce(Ru_{1−x}Fe_{x})PO 116.

7.2 Other degrees of freedom

Quadrupolar order has also been discussed in the context of quantum criticality in a number of heavy fermion compounds. In the filled skutterudite PrOs_{4}Sb_{12}, antiferroquadrupolar order is believed to characterize a high-field ordered phase (HFOP), that is adjacent to a superconducting phase 117. Thus, the possibility arises that the superconductivity in PrOs_{4}Sb_{12} is driven by quantum critical fluctuations of quadrupolar degrees of freedom. In UPd_{3} antiferroquadrupolar order can be weakened by small amounts of Np. At 5% doping, where the order is just suppressed, unconventional properties are observed, implicating a potential QCP 118.

UCu_{5−x}Pd_{x} is the earliest system in which a dynamical spin susceptibility with an anomalous critical exponent and E/T scaling was discovered 35. There have been theoretical studies which attributed this behavior to the effects of disorder, including a distribution of Kondo temperature scales 119 or a Griffiths phase 120. However, it could also result from an interacting spin-glass QCP with Kondo destruction 51 that appears to occur in Sc_{1−x}U_{x}Pd_{3}.

8 Novel phases near quantum critical points

QCPs tend to nucleate new phases, as illustrated by the observation of superconductivity near heavy fermion QCPs 121. From a macroscopic perspective, general scaling considerations show that entropy is maximized near a QCP 80, 81. This is illustrated in Fig. 7. Such an accumulation of entropy has been explicitly demonstrated across the field-induced metamagnetic QCP in Sr_{3}Ru_{2}O_{7}122. An enhanced entropy suggests that the system is particularly prone to developing new order. In the global phase diagram, we may consider AF_{L} and P_{S} as new phases that emerge in the vicinity of QCPs. Another instructive case is provided by CeAgSb_{2}, which possesses ferromagnetic order below about 10 K at ambient pressure. A pressure of about 3 GPa reduces the Curie temperature to about 4 K. Before the ferromagnetic order is completely suppressed, however, it succumbs to an AF state 123. In other words, the AF state emerges at the border of a ferromagnetically ordered state.

Of course, the most striking example is superconductivity developing near AF QCPs. We have discussed the strong evidence that the AF quantum phase transition in CeRhIn_{5} is described by a local QCP. A superconducting state of very large T_{c} (when measured against the Kondo temperature) in CeRhIn_{5}67 occurs in the vicinity of its AF QCP 43, 44. Hence, this material provides a striking example for superconductivity that is driven by Kondo destruction, developing from a normal state with fluctuating Fermi surfaces and critical quasiparticles.

In the case of CeCu_{2}Si_{2}, there is evidence that the asymptotic low-energy quantum critical behavior is of the SDW type. Nonetheless, the exchange energy gain is much larger than the condensation energy, which has been interpreted 124, 125 in terms of a Kondo-destruction energy scale that is nonzero but relatively small compared to the Kondo scale. This suggests that Kondo destruction physics is part of the dynamics that drives superconductivity in CeCu_{2}Si_{2}.

9 Summary and outlook

We close with some perspectives and outlook.

The study of Kondo destruction and local quantum criticality has broad implications. The notion that quasiparticles break down over the entire Fermi surface represents a drastic departure from the traditional SDW description of an AF QCP. In the SDW case, quasiparticles disintegrate only near hot spots, the parts of the Fermi surface that are connected by the AF wavevector. We have discussed how the non-Fermi liquid behavior intertwines with the additional modes of quantum criticality, and how this quantum critical behavior has been experimentally probed through observations that pertain to dynamical scaling, Fermi surface jump, extra energy scale, and mass divergence.

The notion of Kondo destruction has also provided a means to characterize new quantum phases in AF heavy fermion metals. The resulting global phase diagram contains novel phases near the QCPs, that are not expected when the phases are only distinguished in terms of spontaneously broken symmetries as in the Landau framework. Experimental evidence is growing for the phases that are differentiated by both Fermi surfaces and magnetic order parameter. Furthermore, the global phase diagram is fueling the recent efforts to extend the basis of heavy fermion materials with varying dimensionality or frustration-enhanced quantum fluctuations.

These insights on the normal state will undoubtedly impact on our understanding of superconductivity in heavy fermion metals, which tends to occur near QCPs. The SDW quantum critical excitations provide the basis for the spin-fluctuation theory of heavy fermion superconductivity, which assumes that a large Fermi surface is fully formed and quasiparticles near the hot spots are coupled to the critical spin fluctuations. Given the considerable evidence for local quantum criticality, it will be important to theoretically address how superconducting pairing arises out of such a drastic non-Fermi liquid state.

The insights gained on these issues, regarding non-Fermi liquid behavior and unconventional superconductivity, will likely be relevant to the understanding of other strongly correlated electron systems. Several additional aspects of heavy fermion physics also promise to have implications in broader contexts. The development of local quantum criticality has taken place through the Kondo destruction physics in heavy fermion metals. As an interacting critical theory involving inherently quantum modes, it promises to connect with the physics of quantum phase transitions in other contexts. It has already influenced the development of quantum criticality in insulating magnets. More recently, tantalizing connections have been drawn between local quantum criticality in Kondo lattice systems and criticality emerging from holographic models in weakly curved anti-de Sitter (AdS) spacetime through the string/field theory duality 84, 126, 127; “semi-local” critical points with closely related form of correlation functions have been constructed from the gravity side 128, 129. New insights are likely to come from exploring this connection further.

Recent efforts, both theoretical and experimental, were devoted to spin–orbit coupling and topological states in heavy fermion systems. When spin–orbit couplings are explicitly taken into account, the tuning among Kondo insulator and AF states we discussed for remove Kondo insulator systems would be generalized to the transformation among the topological and Kondo coherent/magnetic states. It is therefore likely that Kondo insulator systems will serve as a promising setting to study correlation effects on topological insulating states.

Acknowledgements

We would like to thank many colleagues for collaborations and discussions, including E. Abrahams, P. Coleman, J. Custers, J. Dai, S. Friedemann, P. Gegenwart, C. Geibel, P. Goswami, K. Ingersent, S. Kirchner, C. Krellner, H. Liu, A. H. Nevidomskyy, E. Nica, J. Pixley, A. J. Schofield, F. Steglich, O. Stockert, A. M. Strydom, S. Wirth, J. Wu, S. Yamamoto, R. Yu, H. Q. Yuan, J.-X. Zhu, and L. Zhu. This work has been supported in part by NSF, the Robert A. Welch Foundation Grant No. C-1411, and the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 227378.

Biographical Information

Qimiao Si obtained his B.S. (1986) degree from University of Science and Technology of China and Ph.D. (1991) degree from the University of Chicago. He did postdoctoral works (1991–1995) at Rutgers University and University of Illinois at Urbana-Champaign. In 1995 he joined the faculty of Rice University, where he is the Harry C. and Olga K. Wiess Professor of Physics. He is a theorist working on condensed matter and statistical physics, with an emphasis on strongly correlated electron systems. Specific research subjects have included quantum criticality, non-Fermi liquid physics, heavy fermion phenomena, high temperature iron-pnictide and cuprate superconductivity, and mesoscopic and disordered electronic systems.

Biographical Information

Silke Paschen, experimental condensed matter physicist, graduated in 1992 from Graz University of Technology and received her Ph.D. in 1995 from the Ecole Polytechnique in Lausanne. She held a PostDoc position at ETH Zurich, an associated professorship awarded by the Excellence program of the Max Planck Society at the Max Planck Institute for Chemical Physics of Solids in Dresden, and a visiting professorship at Nagoya University. From 2005, she is full professor at the Institute of Solid State Physics at Vienna University of Technology, which she heads since 2007. Her research focuses on the synthesis and broad experimental characterization of strongly correlated electron systems, with focus on heavy fermion compounds and Kondo insulators, quantum criticality and thermoelectricity.