On non-Fermi liquid quantum critical points in heavy fermion metals

Heavy electron metals on the verge of a quantum phase transition to magnetism show a number of unusual non-Fermi liquid properties which are poorly understood. This article discusses in a general way various theoretical aspects of this phase transition with an eye toward understanding the non-Fermi liquid phenomena. We suggest that the non-Fermi liquid quantum critical state may have a sharp Fermi surface with power law quasiparticles but with a volume not set by the usual Luttinger rule. We also discuss the possibility that the electronic structure change associated with the possible Fermi surface reconstruction may diverge at a different time/length scale from that associated with magnetic phenomena.     

Antiferromagnetism in metals: from the cuprate superconductors to the heavy fermion materials

The critical theory of the onset of antiferromagnetism in metals, with concomitant Fermi surface reconstruction, has recently been shown to be strongly coupled in two spatial dimensions. The onset of unconventional superconductivity near this critical point is reviewed: it involves a subtle interplay between the breakdown of fermionic quasiparticle excitations on the Fermi surface and the strong pairing glue provided by the antiferromagnetic fluctuations. The net result is a logarithm-squared enhancement of the pairing vertex for generic Fermi surfaces, with a universal dimensionless coefficient independent of the strength of interactions, which is expected to lead to superconductivity at the scale of the Fermi energy. We also discuss the possibility that the antiferromagnetic critical point can be replaced by an intermediate 'fractionalized Fermi liquid' phase, in which there is Fermi surface reconstruction but no long-range antiferromagnetic order. We discuss the relevance of this phase to the underdoped cuprates and the heavy fermion materials.     

Superconductivity in heavy fermion systems

The heavy fermion state in the f-electron systems is due to competition between the RKKY interaction and the Kondo effect. The typical compound is CeCu₆. To understand the electronic state, we studied the Fermi surface properties via the de Haas–van Alphen (dHvA) experiment and energy band calculation for CeSn₃,CeRu₂Si₂,UPt₃, and nowadays, transuranium compounds. Pressure is also an important technique to control the electronic state. The Néel temperature T_N decreases with increasing pressure P and becomes zero at the critical pressure for . The typical compound is an antiferromagnet CeRhIn₅, which we studied from the dHvA experiment under pressure. A change of the 4f-electronic state from localized to itinerant is realized at , revealing the first-order phase transition, together with a divergent tendency of the cyclotron mass at P_c. It is stressed that appearance of superconductivity in CeRhIn₅ is closely related to the heavy fermion state. It is also noted that the parity-mixed novel superconducting state might be realized in a pressure-induced superconductor CeIrSi₃ without inversion symmetry in the crystal structure.     

Temperature- and magnetic field dependence of the Hall effect in the heavy fermion system CeCu6

The temperature dependence of the Hall coefficient and the magnetic field dependence of the Hall resistivity of CeCu₆ have been determined in the temperature range 80 mK<T<10 K and in magnetic fields up to 10T. The Hall coefficientR _H shows a very strong temperature dependence with two extrema and a change of sign, and the Hall resistivity _xy has a strong field dependence with up to two changes of sign. The observed behavior can partially be explained by the field- and temperature dependence of the skew scattering contribution to the Hall coefficient.     

General properties of a magnetic-field-induced landau Fermi liquid in high-temperature superconductors and heavy fermion metals

It has been shown that the magnetic-field-induced transition from a non-Fermi-liquid state to a Fermi liquid state in the Tl₂Ba₂CuO_{6 + x} high-temperature superconductor is similar to a transition observed in heavy fermion metals. This behavior is explained in the theory of the Fermi condensate quantum-phase transition implying the existence of Landau quasiparticles. The Fermi condensate quantum-phase transition can be considered as a universal cause of the strongly correlated behavior observed in various metals and liquids such as high-temperature superconductors, heavy fermion metals, and two-dimensional Fermi systems.     

Superfluid response in heavy fermion superconductors

Motivated by a recent London penetration depth measurement [H. Kim, et al., Phys. Rev. Lett. 114, 027003 (2015)] and novel composite pairing scenario [O. Erten, R. Flint, and P. Coleman, Phys. Rev. Lett. 114, 027002 (2015)] of the Yb-doped heavy fermion superconductor CeCoIn₅, we revisit the issue of superfluid response in the microscopic heavy fermion lattice model. However, from the literature, an explicit expression for the superfluid response function in heavy fermion superconductors is rare. In this paper, we investigate the superfluid density response function in the celebrated Kondo–Heisenberg model. To be specific, we derive the corresponding formalism from an effective fermionic large-N mean-field pairing Hamiltonian whose pairing interaction is assumed to originate from the effective local antiferromagnetic exchange interaction. Interestingly, we find that the physically correct, temperature-dependent superfluid density formula can only be obtained if the external electromagnetic field is directly coupled to the heavy fermion quasi-particle rather than the bare conduction electron or local moment. Such a unique feature emphasizes the key role of the Kondo-screening-renormalized heavy quasi-particle for low-temperature/energy thermodynamics and transport behaviors. As an important application, the theoretical result is compared to an experimental measurement in heavy fermion superconductors CeCoIn5 and Yb-doped Ce_1?xYb_xCoIn₅ with fairly good agreement and the transition of the pairing symmetry in the latter material is explained as a simple doping effect. In addition, the requisite formalism for the commonly encountered nonmagnetic impurity and non-local electrodynamic effect are developed. Inspired by the success in explaining classic 115-series heavy fermion superconductors, we expect the present theory will be applied to understand other heavy fermion superconductors such as CeCu₂Si₂ and more generic multi-band superconductors.     

The low field Hall coefficient of polyvalent metals. II. Ca

The formula by Tsuji for the low field Hall coefficient has been evaluated for Ca in the nearly free electron approximation. The total value consists of the free electron valueR ₀=-1/ne, due to the spherical Fermi surface part (free electrons), and the contributions of the distorted areas in the vicinity of the Bragg planes (Bragg electrons and holes). At room temperature the weight of the (111)-Bragg particles is 15 per cent ofR ₀. The touching of the Fermi surface at the (200)-planes creates Bragg holes, yielding a positive contribution to the Hall coefficient of 0.45R ₀.     

Hybridization gap in heavy fermion compounds

We report the results of optical studies of new heavy fermion compounds YbFe(4)Sb(12) and CeRu(4)Sb(12). We show that these compounds, as well as several other heavy fermion materials with a nonmagnetic ground state, obey a universal scaling relationship between the quasiparticle effective mass m(*) and the magnitude of the energy gap Delta in the excitation spectrum. This result is in accord with the picture of hybridization of localized f-electron and free carrier states.     

Density fluctuations in heavy fermion systems

We show explicitly that the hydrodynamic density modes of a heavy fermion system in the presence of long range Coulomb interactions can be reduced to those of an effective Hamiltonian used previously. Outside the hydrodynamic regime one finds acoustic plasmon (or zero sound) excitations as well as high energy plasmons. When the Fermi level intersects more than one heavy quasiparticle band, a situation which is expected to occur in most cases, then also a low-energy optical plasmon excitation should exist. The latter can be overdamped under special conditions.     

Hydrodynamic fluctuations in heavy fermion systems

A theory of hydrodynamic fluctuations in heavy fermion systems is presented. It is used to compute the attenuation and velocity of longitudinal ultrasound. The attenuation is dominated by the coupling of phonons to electronic density fluctuations. A discrepancy is resolved between theory and experiments on UPt₃, which has been existing with respect to the absolute magnitude of the temperature dependent attenuation. The latter provides direct proof for a large Fermi liquid parameterF ₀ ^s . The phonon Green's function is found to have a four-pole structure, resulting in two diffusive modes. One is the conventional one due to heat diffusion while the other is due to electron density diffusion and is a characteristic feature of heavy fermion systems. The two modes are coupled at finite temperatures. With the help of a model Hamiltonian (slave boson mean-field formulation of the Anderson lattice Hamiltonian) the ultrasound attenuation is calculated for low temperatures.     

Current issues in heavy fermion superconductivity

An attempt is made to summarize our current understanding of the superconductivity occuring in heavy fermion systems. The last three years have seen the discovery of two new superconductors (UNi₂Al₃ and UPd₂Al₃), much more use of directional probes to investigate the anisotropy of the gap structure, further experimental and theoretical inquiry into a possible coupling of magnetic and superconducting order parameters, wider application of pressure and uniaxial stress to examine the onset of ordering and some new indications of unconventional superconductivity. These topics will be reviewed along with others of current interest.     

Asymmetries in heavy fermion production and decay

We calculate the forward-backward asymmetries of heavy fermions and of leptons from their semileptonic decay in e⁺e⁻ annihilation. In particular we study the corrections to the lowest-order prediction resulting from intial-state radiation, kinematic effects in heavy quark decay and the role of quark polarization.     

Dynamic crystal field in heavy fermion systems

The crystal field of valence fluctuating systems is time dependent due to f-electron transfers to the conduction band. We show how these fluctuations influence the line shapes of inelastic neutron scattering spectra. The numerical analysis is performed for CePb₃ and CeAl₂.     

Josephson effects in unconventional heavy fermion superconductors

We discuss the Josephson effect for pairing states which break crystal symmetries in addition to gauge symmetry. We consider theE _1g andE _2u models for the low-temperature phase ofUPt ₃, with order parameters Δ(E _1g )～p _z (p _x +ip _y ) and Δ(E _2u )～p _z (p _x +ip _y )². We report calculations of Josephson critical currents, taking into account the effects of depairing at the interface. For singlet-triplet junctions the critical current is non-zero only for spin-orbit, spin-flip tunneling, and is found to be much smaller than the Ambegaokar-Baratoff value even when the spin-orbit tunneling amplitude is comparable to the spin-independent amplitude.     

Superconducting gap anomaly in heavy fermion systems

The heavy fermion system (HFS) is described by the periodic Anderson model (PAM), treating the Coulomb correlation between the f-electrons in the meanfield Hartree-Fock approximation. Superconductivity is introduced by a BCS-type pairing term among the conduction electrons. Within this approximation the equation for the superconducting gap is derived, which depends on the effective position of the energy level of the f-electrons relative to the Fermi level. The latter in turn depends on the occupation probability n ^f of the f-electrons. The gap equation is solved self-consistently with the equation for n ^f; and their temperature dependences are studied for different positions of the bare f-electron energy level, with respect to the Fermi level. The dependence of the superconducting gap on the hybridization leads to a re-entrant behaviour with increasing strength. The induced pairing between the f-electrons and the pairing of mixed conduction and f-electrons due to hybridization are also determined. The temperature dependence of the hybridization parameter, which characterizes the number of electrons with mixed character and represents the number of heavy electrons is studied. This number is shown to be small. The quasi-particle density of states (DOS) shows the existence of a pseudo-gap due to superconductivity and the signature of a hybridization gap at the Fermi level. For the choice of the model parameters, the DOS shows that the HFS is a metal and undergoes a transition to the gap-less superconducting state.      

London field penetration in heavy fermion superconductors

We present results of a series of high resolution, low fielddc-magnetization measurements on the heavy fermion superconductors UPt₃, UBe₁₃, U_1–x Th _X Be₁₃ and CeCu₂Si₂, from which values of the magnetic penetration depth can be extracted. A study of the temperature variation reveals aT ² power law in all cases. This can not be reconciled with a BCS-like isotropic energy gap but may be explained by the presence of low energy quasiparticle states inside the gap. In the case of very pure superconductors, one such possibility is the assumption of point-nodes in the gap function. We argue, however, that an interpretation in terms of resonant impurity scattering in various anisotropic superconducting states is more likely to explain a broad range of experimental data. The results on differently oriented single crystals of UBe₁₃ and UPt₃ reveal no pronounced anisotropy related to the crystal lattice. Absolute values of atT=0 are deduced by a novel method in which the field is first screened out from the sample by means of an evaporated superconducting film (of lowerT _c ). Above this lowerT _c the sudden penetration of field into the sample can then be measured absolutely.     

Muon spin relaxation in heavy fermion systems

Heavy fermion systems have received a great deal of study by a wide variety of techniques, includingSR. In a number of systems, coexisting superconducting and magnetic states have been reported, leading to speculation of an intimate connection between magnetism and superconductivity in these compounds. We observe a spontaneous magnetic field in the superconducting phase of UPt₃. In addition, the broadening of the transverse field muon precession signal only onsets approximately 60 mK below the superconductingT _c. Our results provide evidence that the lower superconducting phase in theH-T phase diagram of UPt₃ is characterised by broken time-reversal symmetry. Measurements of URu₂Si₂ and CeCu_2.2Si₂ indicate that the magnetically ordered volume fraction is temperature dependent in both systems.