Critical quasiparticle theory applied to heavy fermion metals near an antiferromagnetic quantum phase transition Elihu Abrahamsa,1 and Peter Wölfleb,1 aDepartment of Physics and Astronomy, University of California, Los Angeles, CA 90095; and bInstitute for Theory of Condensed Matter and Institute for Nanotechnology, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany Contributed by Elihu Abrahams, January 10, 2012 (sent for review December 9, 2011) We use the recently developed critical quasiparticle theory to sics can be found in ref. 3. Below Tx, however, the critical fluc- derive the scaling behavior associated with a quantum critical point tuations begin to interact with each other and consequently the in a correlated metal. This is applied to the magnetic-field induced effects on the quasiparticles change, e.g., the effective mass, or Z quantum critical point observed in YbRh2Si2, for which we also factor, acquires a singular power-law frequency dependence. This derive the critical behavior of the specific heat, resistivity, thermo- is the region described by the critical quasiparticle theory. Fig. 1 power, magnetization and susceptibility, the Grüneisen coefficient, illustrates these regions in the phase diagram for the heavy- and the thermal expansion coefficient. The theory accounts very fermion metal YRS. In YRS, an AFM quantum critical point is well for the available experimental results. accessed by tuning a magnetic field H, but the theory is appro- priate whatever the nature of the ordered phase and whatever the dynamical scaling ∣ non-Fermi-liquid properties ∣ spin excitation spectrum ∣ tuning parameter. non-Gaussian fluctuations Fig. 1 shows the phase diagram for YbRh2Si2 near the quan- tum critical point. Five different regions may be seen: (1) the ecent advances in low-temperature experimental techniques antiferromagnetically ordered region at temperatures below PHYSICS Rhave stimulated much interest in quantum critical phenom- the Néel temperature TN ðHÞ; (2) the Landau Fermi liquid regime T ≲ T ðHÞ ena, which comprise phase transitions at zero temperature at FL ; (3) a high temperature local-moment regime “ ” T ≫ T T ( quantum critical point ) and associated effects due to quantum at KL, where KL is the characteristic temperature of the fluctuations at very low temperatures. Ref. 1 gives an introductory lattice Kondo effect, below which coherent heavy quasiparticles review of the subject. form by hybridization of f electrons and conduction electrons; These developments have generated a variety of difficult the- (4) a regime of quantum critical quasi-2D antiferromagnetic T ≲ T ≲ T oretical questions; among them is the issue of how to treat the Gaussian fluctuations at x KL, characterized by moder- regime of strongly interacting quantum fluctuations. In a recent ate non-Fermi liquid behavior; (5) the true critical regime, “QC,” paper (2), we developed an extension of the quasiparticle concept for T ≲ Tx governed by 3D antiferromagnetic fluctuations inter- of Fermi liquid theory to the non-Fermi-liquid regime near a acting strongly with the heavy quasiparticles. quantum critical point (QCP). In essence, the theory goes beyond The basic input for the theory is a phenomenological form for the Gaussian regime of critical fluctuations by introducing inter- the critical antiferromagnetic (AFM) spin fluctuation correlator actions among the quantum fluctuations into the correlation at low temperature below Tx, which reflects that it is generated function of the fluctuations. Central to the analysis is the concept from quasiparticles having a nonzero frequency-dependent Z of critical quasiparticles, which is based on the recognition that the factor single-particle spectral function can display a quasiparticle peak at nonzero excitation energy or temperature. This is expressed as a ðN ∕ZÞðω∕và QÞ χðq;ωÞ¼ 0 F ; [1] nonzero quasiparticle weight ZðωÞ for jωj not too small, although, Im 2 2 2 à 2 ½r0ðHÞþZðq − QÞ ξ þðω∕v QÞ as in a non-Fermi liquid, at the Fermi surface Zðω ¼ 0Þ¼0. 0 F We realized the critical quasiparticle theory for the case of an where N0 is the bare density of states at the Fermi surface, antiferromagnetic quantum critical point and applied it to several à à vF ¼ðmb∕m ÞvF is the renormalized quasiparticle Fermi velocity, quantities, principally resistivity and specific heat, for successful −1 ξ0 ≃ k is the microscopic AFM correlation length, and r0ðHÞ¼ comparison to experimental results on the heavy-fermion metal F 1 þ FðQ;HÞ ∝ H∕Hc − 1. Here FðQ;HÞ is a dimensionless gener- YbRh2Si2 (YRS), thereby showing that the theory, which de- alized Landau parameter, which → −1 at the critical point. For scribes a physically transparent scenario, is capable of accounting convenience, we denote the underlying tuning parameter by H, for experimental results on a quantum critical metal. which for YRS is the magnetic field. The interaction between The implementation of the theory for an antiferromagnetic I FðQ;HÞ (AFM) quantum critical point for a heavy-fermion compound quasiparticles and spin fluctuations is related to by “ I ¼ ZFðQ;HÞ∕N0. The “bare” quantities mb, N0, vF here are to is based on the recognition that below a lattice Kondo tempera- T ” T s p d be understood as those quantities already renormalized at KL by ture KL, hybridization between conduction ( , , ) electrons and local magnetic moments (f orbitals) produces a heavy- hybridization and the lattice Kondo effect, as discussed above. electron liquid with an associated mass enhancement due to the The additional (frequency-dependent) mass enhancement caused f f by interaction with critical spin fluctuations is denoted as originally localized character of the electrons. However, the mÃ∕m ¼ 1∕Z electrons are also responsible for the antiferromagnetism in a re- b . We do not consider questions relevant to the T ≫ T gion of the phase diagram of the material. Near the AFM critical crossover (4) from the local-moment phase at KL to the point, critical spin fluctuations are enhanced and interact with the heavy quasiparticles. This produces further mass enhancement Author contributions: E.A. and P.W. designed research; E.A. and P.W. performed research; and occurs in two stages. Above a certain temperature Tx but be- E.A. and P.W. analyzed data; and E.A. and P.W. wrote the paper. T low KL, the critical fluctuations are Gaussian (i.e., noninteract- The authors declare no conflict of interest. ing); those of 2D AFM character (or 3D ferromagnetic character) 1To whom correspondence may be addressed. E-mail: [email protected] or produce a logarithmic mass enhancement. A review of this phy- [email protected]. www.pnas.org/cgi/doi/10.1073/pnas.1200346109 PNAS Early Edition ∣ 1of5 Downloaded by guest on September 24, 2021 T ≲ T T (K) heavy-fermion phase at KL. The experimentally determined T ≈ 25 T ≈ 0 3 values in YRS are KL K and x . K. The justification for the form of Eq. 1 is discussed in detail in ref. 2. We neglect the momentum dependence of the Z factor by 10.00 lattice Kondo effect assuming that the critical behavior at the hot spots is spread over the whole Fermi surface by impurity scattering or other interac- 5.00 tion effects. The Z factor is determined by the quasiparticle self energy, which in turn is determined by the interaction with 2 d the spin fluctuations. Briefly, the quasiparticle width Γ, related Gaussian to the self energy Σ by ΓðωÞ¼ZðωÞΣðωÞ, is found from 1.00 −3 ω 3 2 γ Γ ≃ cikF ðN0∕ZÞ∫ 0 dν∫ d qI Imχðq;νÞ ∝ ω . Then by Kramers– Kronig, ReΣ ∝ ωγ∕ZðωÞ. With Z−1 ¼ 1 −∂ReΣðωÞ∕∂ω, this leads 0.50 (2) to a self-consistency relation in the form of a differential equa- tion for ZðωÞ: 3 d non-Gaussian d 0.10 Z−1 ¼ 1 þ λ ðZ−3ω3∕2Þ: [2] QC dω 0.05 This equation has two different physically meaningful solutions. The first is a weak coupling solution, valid provided the second term on the r.h.s. of Eq. 2 is ≪1 in the energy range considered. 0.01 AFM FL This corresponds to the conventional spin-density wave scenario H (T) as discussed in the works of Hertz (5), Millis (6), and Moriya (7). 0.04 0.05 0.06 0.07 0.08 0.09 There exists, however, a second solution in the strong-coupling domain; it is accessible provided the initial Z−1 at the scale when the 3D antiferromagnetic fluctuation regime is entered (in the T ≈ 0 3 case of YRS at . K) is sufficiently large such that the sec- Fig. 1. Phase diagram for YbRh2Si2 in the neighborhood of the critical mag- 2 ond term on the r.h.s. of Eq. dominates. We conjecture that in netic field Hc ≃ 0.06 T: The dashed lines represent crossovers. As the tempera- T YRS, 2D antiferromagnetic (assisted by 3D ferromagnetic) spin ture is lowered below KL, the lattice Kondo effect and heavy quasiparticles fluctuations above T ≈ 0.3 K can provide the necessary growth of develop and weakly interacting (“Gaussian”) 2D AFM fluctuations associated T the effective mass everywhere on the Fermi surface, even in the with the quantum critical point cause non-Fermi liquid behavior. The x line absence of impurity scattering. (In the case of quasi-2D antifer- represents the crossover to strongly interacting 3D fluctuations, which are dominant within the cone of quantum criticality (QC), bounded on the right romagnetic fluctuations in a 3D metal this has been noticed first T ðHÞ in ref. 8). Then the strong-coupling solution is by the crossover at FL into the heavy Fermi-liquid state and on the left by the curved dashed line within the antiferromagnetic ordered state that sets T ðHÞ T ÃðHÞ −1∕2 −3∕2 1∕4 in below N .