Critical Quasiparticle Theory Applied to Heavy Fermion Metals Near an Antiferromagnetic Quantum Phase Transition

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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 quantum 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∕v QÞ χðq;ωÞ¼ 0 F ; [1] nonzero quasiparticle weight ZðωÞ for jωj not too small, although, Im 2 2 2 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 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 m∕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 equation 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 ðHÞ −1∕2 −3∕2 1∕4 in below N . The crossover (dot-dashed line) is discussed in the text, Z ¼ 3 ðkFξ0Þ ðω∕vFQÞ : [3] below Eq. 3). Z Zðω;TÞ ∝ At nonzero temperature, is approximately given by Scaling ½ω2 þðπTÞ21∕8 . As discussed in ref. 2, the critical regime is characterized by fluc- In the following, we describe the phase diagram obtained with- tuations of dimensionality d ¼ 3. We can determine ν and z from in our theory as consisting of the ordered phase bounded by the the critical quasiparticle theory as follows: It can be shown that Néel temperature TN ðHÞ, the critical regime (“critical cone”)at Z 1∕2 T >T>T ðHÞ when the frequency dependence of is a power less than , the x FL and the Landau Fermi liquid regime at Z 1 T T ðHÞ frequency-dependent factors appearing in Eq. have the argu- < FL (see Fig. 1). In addition to these phase boundaries ment ω. Then from Eq. 1 and the result ZðωÞ ∝ ωα, where we and crossover lines, Hall effect and other measurements in YRS found α ¼ 1∕4, we can determine the dynamical exponent z ¼ 4 (9–11) show anomalies at a temperature TðHÞ in the critical re- because at the critical point gion. The origin of the T ðHÞ line is not clear at present. Recent experiments on YRS under pressure (12) or YRS doped with Imχðq;ωÞ−1 ∼ Z2ðωÞq4 þ ω2∕Z2ðωÞ: [4] Co or Ir (13) have shown that while the critical field for the mag- T netic transition is found to shift, the line stays unchanged. This q suggests that the T feature is only weakly or not at all tied to the Here and in the following, denotes the deviation from the ordering wave vector Q. The scaling exponent of the spatial AFM transition. The present theory ascribes the critical behavior ξ ν ¼ 1∕3 r ðHÞ ∼ Zq2∼ to interaction of renormalized quasiparticles with renormalized correlation length is found as from 0 ðqzÞ1∕4q2 ∼ q3 ξ ∼ r−1∕3 T ¼ 0 ξ ∝ antiferromagnetic spin fluctuations, rather than with fluctuations so that 0 .At we thus find ðH − H Þ−1∕3 associated with T ðHÞ, as has been suggested elsewhere (11, c .The characteristic correlation time is given by τ ∼ ξz ∼ ðH − H Þ−4∕3 14, 15). c c . H ¼ H r ¼ 0 T ¼ 0 In this paper, we elaborate the critical quasiparticle theory When c (i.e., 0 ), the approach to the critical point is described by a temperature dependence of the control further by deriving the scaling behavior of quantities near the 1∕zν QCP, and we add the critical behavior of the Grüneisen coeffi- parameter, which can be written rðHc;TÞ ∼ T . The condition rðH ;TÞ ≈ rðH;T ¼ 0Þ T ¼ T ðHÞ ∝ jH − H jzν cient, thermopower, thermal expansion, magnetization, and sus- c or FL c defines the ceptibility to the experimental consequences of the theory. Again, boundary of the “critical cone” of quantum criticality. Then, with- we compare these results of the theory to experimental informa- in the quantum critical regime, the temperature dependence of −1∕z tion on YbRh2Si2, which has a magnetically tuned QCP. the correlation length is found as ξ ∝ T . It may be shown that The deduction of the critical behavior of several thermo- the above critical temperature dependent correction to r dynamic quantities from the scaling form of the free energy has is caused by the interaction of spin fluctuations treated at the been discussed by Zhu et al. (16). The corresponding exponents Hartree level, giving rise to rðHc;TÞ ∼ T∕ZðTÞ. for these quantities are determined by the correlation length ex- From these results (ν ¼ 1∕3, z ¼ 4), we may write Imχðq;ωÞ,or ponent ν, the dynamical exponent z, and the dimensionality d. equivalently the structure factor Sðq;ωÞ, in the scaling form

2of5 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1200346109 Abrahams and Wölfle Downloaded by guest on September 24, 2021 x1∕2 ρðT;HÞ − ρð0Þ ∝ ξðαþ1ÞzT2: [10] Sðq;ωÞ¼ξ4Φðωξ4;qξÞ; Φðx;yÞ ∝ : [5] ð1 þ x1∕4y2Þ2 þ x3∕2 Using the values α ¼ 1∕4, z ¼ 4, which we found in the self- Note that at the critical wave vector (at q ¼ 0), ω∕T scaling holds. consistent critical quasiparticle theory (2), we find that Eqs. 6, The scaling form for the free energy density may be deduced by 7, and 8 become dimensional analysis (3): 3∕4 ρðT;HcÞ − ρð0Þ ∝ T [11] dþz r0 −ðdþzÞ 1∕ν z z fðH;TÞ¼ξ Φf ðr0ξ ;Tξ Þ → T Ψ 1 [6] Tzν ρðT;HÞ − ρð0Þ ∝ ξ5T2 ∝ jH − H j−5∕3T2: [12] T c rνðdþzÞΨ~ : [7] or 0 rzν 0 The available data agree very well with Eq. 11 as found in ref. 2 by direct calculation. At the time of this writing, for the field depen- In our case, d ¼ 3, ν ¼ 1∕3, z ¼ 4. 2 dence of the coefficient of T in Eq. 12, there is insufficient data We may generalize these results to arbitrary dimension d. First, close enough to Hc to allow a quantitative comparison. we note (from the derivation of the quasiparticle weight factor Z d ¼ 4 presented in ref. 1) that is the upper critical dimension, so Thermopower that the spin fluctuations do not destroy the Fermi liquid state for At very low temperatures, when impurity scattering dominates, d>4 d 2 .At < , we find that the critical quasiparticles are no the thermopower S may be expressed in terms of the electrical longer well defined. We therefore consider here only dimensions conductivity σ ðμÞ, depending on the chemical potential μ,by within the interval 2 ≤ d ≤ 4. imp the Mott formula Using the exponent α to describe the frequency power law of the quasiparticle weight Z, we find, from Eq. 4, that the dynamical π2 T ∂ ln σ critical exponent is given by z ¼ 2∕ð1 − 2αÞ and the correlation S ¼ imp : [13] length exponent is ν ¼ 1∕ð2 þ zαÞ. 3 e ∂μ PHYSICS Specific heat ℓ σ ðμÞ ∝ k2 ℓ Because the mean free path is not renormalized, imp F , The critical part of the specific heat at the critical tuning para- where kF is the Fermi wave vector, σ ðμÞ, is independent of the r ¼ 0 6 C ∝ Td∕z ¼ T3∕4 imp meter is found from Eq. : c as we found effective mass. The derivative with respect to μ may be expressed before (2). We can obtain the behavior of the specific heat near by the one with respect to the Fermi wave vector as ∂∕∂μ ¼ T ¼ 0 as a function of r from Eq. 7 as follows (16): Near x ¼ 0, μ ð1∕v Þð∂∕∂kFÞ and ∂σ ∕∂kF is independent of m . Therefore, Ψ~ ðxÞ behaves as Ψ~ ðx → 0Þ¼Ψ~ ð0Þþbx . Therefore F imp we find S ∝ TmðTÞ. In the Gaussian fluctuation regime, we Cðr;TÞ ∝ rðdþzÞν−μzνμðμ − 1ÞTμ−1 ¼ μðμ − 1ÞTμ−1r−1∕3ð4μ−7Þ: [8] get S ∝ T lnðTÞ, which has been observed (18). In the critical regime proper, at the critical field H ¼ 0.064 T we predict S ∝ T3∕4. This behavior is valid in the region of the T, H phase diagram The comparison of our result to the experiment reported in T∕rzν 1 T T ðHÞ ∝ jH − H jzν defined by < , which is < FL c . ref. 18 is shown in Fig. 2. Unfortunately, the data was taken at As we showed in ref. 2, the theoretical behavior of γ ¼ Cc∕T at H ¼ 0.06 T, below the critical field of 0.064 T. At the measure- criticality found above agrees very well with the data. For the low ment field, the AFM Néel temperature TN is about 20 mK (18), T dependence of γ on r ¼jH − Hcj, we note that in YRS on so we do not expect good agreement at the very lowest tempera- either side of the QCP, Fermi liquid behavior obtains (17) so that tures, as seen in the figure. −1∕3 μ ¼ 2. Then, from Eq. 8, γ ¼ C∕T varies as jH − Hcj . This is precisely the behavior observed in experiment on YRS (17, 18). Magnetization and Susceptibility The temperature dependence of the magnetization may be found Resistivity from the derivative of the free energy in Eq. 6 with respect to The scaling analysis may be applied to determine the temperature field: and magnetic-field power-law dependences of the resistivity, which (assuming impurity and umklapp scattering) is given by

ρðT;HÞ − ρð0Þ ∝ ΓðT;HÞ∕ZðT;HÞ;

where Γ is the quasiparticle relaxation rate, which we found (2) at 3∕2 2 d ¼ 3 and H ¼ Hc, to be proportional to ω ∕Z ðωÞ. Because Γ has dimensions of energy, its scaling form is ΓðT;HÞ¼ −z z α −zα ξ ΦΓðTξ Þ as in Eq. 7. With ZðωÞ ∝ ω ∝ ξ , we have

ðα−1Þz z ρðT;HÞ − ρð0Þ ∝ ξ ΦΓðTξ Þ:

To insure that ρ is finite at H ¼ Hc and T>0, we require ΦΓðxÞ ∝ x1−α as x → ∞, so that

1−α ρðT;HcÞ − ρð0Þ ∝ T : [9]

As we discussed above for the specific heat, away from criticality, at low temperature outside either side of the critical cone defined Fig. 2. Thermopower: Comparison of theory and data of ref. 18 near the T ¼ T ρðTÞ − ρð0Þ ∝ T2 by FL, Fermi liquid behavior, obtains, so critical magnetic field and below the critical temperature for quantum critical 2 that ΦΓðxÞ ∝ x . Then scaling.

Abrahams and Wölfle PNAS Early Edition ∣ 3of5 Downloaded by guest on September 24, 2021 ∂f 1 ∂f 1 dþz 0 r0 − 1 M ¼ − ¼ − ¼ − T z Ψ T zν: crit 1 ∂H T Hc ∂r0 T Hc Tzν [14] 0 Assuming M is regular at r0 ¼ 0 so that Ψ ð0Þ¼const, we get

dþz−1∕ν MðTÞ − Mð0Þ ∝−T z ¼ −T: [15]

This result agrees fairly well with the experimental data at low T (below 400 mK) as reported in ref. 19. A very good fit is obtained if the critical T dependence of Eq. 15 is augmented by the subleading T2 contribution due to Fermi liquid effects. This is shown on Fig. 3, where the theoretical behavior at H ¼ 0.06 T is given by

MðTÞ − Mð0Þ¼−0.68T þ 0.63T2: [16] Fig. 4. Magnetic susceptibility: Comparison of theory, Eq. 17 and data of For the field dependence at small T, we use Eq. 7: ref. 21 near the critical magnetic field and below the critical temperature for quantum critical scaling. Similarly to Eq. 14 and Fig. 3, we have added 2 ∂f 1 a T Fermi-liquid contribution. M ¼ − ¼ − νðd þ zÞrνðdþzÞ−1Ψ~ ð0Þ; crit ∂H H 0 T¼0 c Grüuneisen Ratio In the case of a magnetic field tuned QCP, as in YRS, the mag- d ¼ 3 z ¼ 4 ν ¼ 1∕3 r ¼ H∕H − 1 which for , , , 0 c is netic Grüuneisen ratio (16) is studied: 7 1 H 4∕3 MðHÞ − MðH Þ¼− − 1 Ψ~ ð0Þ: [17] ð∂M∕∂TÞH ð∂S∕∂HÞT c 3 H H ΓM ¼ − ¼ − : [20] c c CH CH

Therefore, the critical field dependence of the susceptibility is The critical part of the Grüuneisen ratio has been discussed in ref. 16 and measured in YRS by Tokiwa et al. (19). As in ref. 16, ∂M χ ¼ ∝−ðH − H Þ1∕3: [18] the behavior of ΓM may be determined from the scaling analysis. crit c 0 ∂H We find, for z ¼ 4, ν ¼ 1∕3, and μ ¼ 2, that ½∂S∕∂Hr0¼0 ∝ T . Then The available data (20) indicate a cusp-like behavior of χðHÞ H −3∕4 when approaching c but do not extend close enough to allow ΓM ðr ¼ 0;TÞ ∝ T [21] for a quantitative comparison T The critical dependence of the susceptibility is found from −1∕3 Eq. 14 as ΓM ðr;T → 0Þ¼ : [22] H − Hc

∂M 1 ∂M dþz−2∕ν 1∕4 χ ¼ ¼ ∝ T z ∝ T ; [19] The factor −1∕3 ¼ νðd − zÞ in Eq. 22 is universal as the scaling ∂H Hc ∂r0 functions in the ratio defining ΓM cancel. 00 These results as well as those for the thermodynamic Grünei- where we assumed Ψ ð0Þ¼const. We can compare this result to sen parameter can also be obtained by direct calculation from the experiment reported by Gegenwart et al. (21), where χ was ac derivatives of the self-energy expressions given in ref. 2. measured below 0.2 K at a field of 0.05 T. Because TN is about Just as for the specific heat and the resistivity (2), there is ex- 30 mK at this field, we do not include data below 40 mK in the cellent agreement between the critical quasiparticle theory and comparison shown on Fig. 4.

Fig. 5. Magnetic Grüneisen ratio: Comparison of theory, Eq. 21 and data of Fig. 3. Magnetization: Comparison of theory and data of ref. 19 near the ref. 19 at the critical magnetic field and below the critical temperature for critical magnetic field. A T 2 Fermi-liquid contribution has been added to the quantum critical scaling. The inset shows the comparison of the data and critical behavior. Eq. 20 when the previously fit MðTÞ and CðTÞ are used.

4of5 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1200346109 Abrahams and Wölfle Downloaded by guest on September 24, 2021 the experiment (19) for the magnetic Grüuneisen ratio. The analysis of the theory to derive the critical behavior of a number factor νðd − zÞ¼−1∕3 in Eq. 22 is measured as −0.30 0.01 of experimentally observed quantities in the heavy fermion com- −3∕4 and the comparison of our result of Eq. 21 of the T depen- pound YbRh2Si2. The good agreement with all these observed dence at H ¼ Hc to the experiment is shown on Fig. 5. quantities indicates that the critical quasiparticle theory captures We may evaluate the consistency of our results by using the essential features of the critical behavior in this compound. We theory fits for MðTÞ, Eq. 16 and for the specific heat shown note that YbRh2Si2 is especially suitable for application of the 3∕4 on Fig. 1 of ref. 2 (C ¼ 0.034 þ 0.454T ) in Eq. 20. There theory because it has the wide region of 2D critical behavior that are no further adjustable parameters and the agreement is shown is necessary to ultimately access the strong coupling regime of 3D in the inset to Fig. 5. fluctuations at lower temperatures, as discussed in ref. 2, where The volume thermal expansion coefficient βðTÞ enters the the critical quasiparticle theory was introduced. conventional thermodynamic Grüuneisen ratio ΓT ¼ βðr0;TÞ∕ Besides YRS, other candidate systems have been studied with Cðr ;TÞ P 0 . The critical behavior here is tuned by pressure , so that different degrees of detail. For CeCu6−xAux it has been found (for r0 ∝ ðPc − PÞ∕Pc. The critical part of β is determined by deriva- a review see ref. 3) that some of the low-temperature properties tives of the free energy as like the specific heat and the resistivity indicate quasi-2D antiferromagnetic fluctuations in the Gaussian regime. Neutron scatter- 1 ∂2f 0 ing studies at somewhat higher energies have been interpreted as βcðr0;TÞ¼ ∝ T ; Pv ∂T∂r0 showing critical scaling behavior incompatible with the Gaussian theory. It is conceivable that the behavior is different depending where v is the specific volume and we have used Eq. 6. To this we on the energy scale. Further candidate systems like CeCoIn5 also may add the conventional Fermi-liquid contribution ∝T so that appear to show quasi-2D antiferromagnetic Gaussian fluctua- β ¼ a þ bT in the critical region. tions (for an overview, see ref. (3). In all of these systems we expect a crossover to 3D fluctuations at lower temperatures, in Conclusion which case our theory should apply. The critical quasiparticle theory describes a renormalized Gaus- sian picture of critical fluctuations, where the fluctuations are as- ACKNOWLEDGMENTS. We thank Jörg Schmalian for valuable suggestions sumed to be noninteracting beyond effects that renormalize the and Alexander Balatsky, Andrey Chubukov, Max Metlitski, Subir Sachdev, quantities entering the fluctuation spectrum. The renormaliza- and Qimiao Si for useful discussions. This work was supported in part by PHYSICS tions of these parameters, the staggered static susceptibility and the Deutsche Forschungsgemeinschaft research unit 960 “Quantum phase transitions.” Part of this work was carried out at the Aspen Center for Physics the Landau damping term, are, however, of a strong-coupling (National Science Foundation Grant 1066293), Los Alamos National Labora- nature. It remains to be shown that fluctuation interactions are tory and as a Carl Schurz Memorial Professor at the University of Wisconsin, irrelevant within a more complete theory. We have used a scaling Madison (P.W.).

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