Topological basis for understanding the behavior of the heavy-fermion metal $\rm {\beta-YbAlB_4}$ under application of magnetic field and pressure

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by  V. R. Shaginyan

Topological basis for understanding the behavior of the heavy-fermion metal
β YbAlB under application of magnetic field and pressure
4
−
V. R. Shaginyan,1,2,∗ A. Z. Msezane,2 K. G. Popov,3,4 J. W. Clark,5,6 V. A. Khodel,7,5 and M. V. Zverev7,8
1Petersburg Nuclear Physics Institute, NRC Kurchatov Institute, Gatchina, 188300, Russia
2Clark Atlanta University, Atlanta, GA 30314, USA
3Komi Science Center, Ural Division, RAS, Syktyvkar, 167982, Russia
4Department of Physics, St.Petersburg State University, Russia
5McDonnell Center for the Space Sciences & Department of Physics,
6 Washington University, St. Louis, MO 63130, USA
1 6Centro de Ciˆencias Matema´ticas, Universidade de Madeira, 9000-390 Funchal, Madeira, Portugal
0 7NRC Kurchatov Institute, Moscow, 123182, Russia
2 8Moscow Institute of Physics and Technology, Dolgoprudny, Moscow District 141700, Russia

y
a Informativerecentmeasurementsontheheavy-fermionmetalβ−YbAlB4performedwithapplied
magneticfieldandpressureascontrolparametersareanalyzedwiththegoalofestablishingasound
M
theoretical explanation for the inferred scaling laws and non-Fermi-liquid (NFL) behavior, which
demonstrate some unexpected features. Most notably, the robustness of the NFL behavior of the
0
thermodynamicproperties and of theanomalous T3/2 temperaturedependenceof theelectrical re-
1
sistivityunderappliedpressureP inzeromagneticfieldBisatvariancewiththefragilityoftheNFL

phaseunderapplication ofafield. Weshowthataconsistenttopological basisforthiscombination
]
l ofobservations,aswellastheempiricalscalinglaws,maybefoundwithinfermion-condensationthe-
e
oryintheemergenceanddestructionofaflatband,andexplain thattheparamagneticNFLphase
-
r takes place without magnetic criticality, thus not from quantum critical fluctuations. Schematic
st T −B and T −P phase diagrams are presented toilluminate this scenario.
.
t
a PACSnumbers: 71.27.+a,71.10.Hf,72.15.Eb
m
-
d I. INTRODUCTION inzeromagneticfield; Howtoexplainthatthe paramag-
n neticNFLphasetakesplacewithoutmagneticcriticality,
o thus not from quantum critical fluctuations.
c Recent measurements on the heavy-fermion (HF)
[
metal β YbAlB4 have been performed under the ap-
−
3 plication of both a magnetic field B and hydrostatic II. SCALING BEHAVIOR
v pressure P, with results that have received considerable
2 theoretical analysis1–10. Measurements of the magne-
8 To address these challenges within a topological sce-
tization M(B) at different temperatures T reveal that
1 the magnetic susceptibility χ = M/B T−1/2 demon- nario based on the emergence of a fermion conden-
1 ∝ sate (FC), we begin with an examination of the scal-
strates non-Fermiliquid (NFL) behaviorand divergesas
0 ing behavior of the thermodynamic functions of this HF
1. mT∗→div0e,rgimespalysinmg∗thaBt−t1h/e2 quTas−ip1/a2rtaictleaeqffueacnttiuvemmcraists- compound considered as homogeneous HF liquid11–13.
0 ical point (QCP).2∝This kin∝d of quantum criticality is We note that the existence of FC has been convinc-
6 ingly demonstrated by purely theoretical and experi-
commonly attributed to scattering of electrons off quan-
1 mental arguments, see e.g.14–18. The Landau functional
: tum critical fluctuations related to a magnetic instabil-
v E(n) representing the ground-state energy depends on
ity; yet in a single crystal of β YbAlB , the QCP in
Xi question is located well away fr−om a pos4sible magnetic the quasiparticle momentum distribution nσ(p). Near
thefermion-condensationquantumphasetransition(FC-
r instability, making the NFL phase take place without QPT), the effective mass m∗ is governed by the Landau
a magneticcriticality.2 Additionally,itisobservedthatthe
equation11,12,20
QCP is robust under application of pressure P, in that
the divergent T and B dependencies of χ are conserved 1 1
and are accompanied by an anomalous T3/2 dependence m∗(T,B) = m∗(T =0,B =0) (1)
of the electrical resistivity ρ.6 In contrast to resilience
oafttihneysemdaigvneergtiecncfieesldunBdeirspsrueffisscuireen,tatpoplsiucaptpiroenssoftheevmen, + p12F Xσ1 Z ppFFp1Fσ,σ1(pF,p1)∂δnσ1∂(Tp,1B,p1)(d2pπ1)3,
leading to Landau Fermi liquid (LFL) behavior at low
temperatures1,2. Thus, among other unusual features, here written in terms of the deviation δn (p)
σ
≡
the metal β YbAlB presents challenging theoretical n (p,T,B) n (p,T = 0,B = 0) of the quasiparticle
4 σ σ
− −
problems: How to reconcile the frailty of its NFL be- distribution from its field-free value under zero pressure.
haviorunderapplicationofamagneticfield,withthe ro- The Landau interaction F(p1,p2)=δ2E/δn(p1)δn(p2)
bustnessoftheNFLphaseagainstapplicationofpressure serves to bring the system to the FCQPT point where
2
m∗ at T = 0. As this occurs, the topology withc aconstant.11–13 Eqs.(4),(5),(6),and(7)willbe
3
→ ∞
of the Fermi surface is altered, with the effective mass used along with Eq. (1) to account for the experimental
m∗ acquiring temperature and field dependencies such observations on β YbAlB . We note that the scaling
4
that the proportionalities C/T χ m∗(T,B) relat- behavior at issue r−efers to temperatures T . T , where
f
∼ ∼
ing the specific heat ratio C/T and the magnetic sus- T is the temperatureat whichthe influence ofthe QCP
f
ceptibility χ to m∗ persist. Approaching the FCQPT, becomes negligible.11,12.
m∗(T =0,B =0) andthusEq.(1)becomeshomo-
geneous, i.e., m∗(T→=∞0,B) B−z and m∗(T,B = 0)
T−z, with z depending on ∝the analytical properties o∝f
F.11–13,21 On the ordered side of the FCQPT at T = 0, 1 B||c crossover
the single-particle spectrum ε(p) becomes flat in some
β-YbAlB
interval pi < pF < pf surrounding the Fermi surface at N 0.1 LFL 4
)
p , coinciding there with the chemical potential µ, T
F M/d 0.01 00..3612 mmTT NFL
ε(p)=µ. (2) d 3.1 mT Theory
2
1/ 6.2 mT 44 mT
B1E-3
At the FCQPT the flat interval shrinks, so that pi ( 12 mT 0.1 T
→
p p , and ε(p) acquires an inflection point at p , 19 mT 0.2 T
F f F
with→ε(p pF) µ (p pF)3. Anotherinflectionpoint 1E-4 22 mT 0.3 T
≃ − ≃ − 25 mT 0.5 T
emerges in the case of a non-analytical Landau interac-
31 mT 2 T
tion F, instead with
1E-5
0.1 1 10 100 1000
ε µ (pF p)2,p<pF (3) (T/B)
− ≃ − − N
ε µ (p p )2,p>p
F F
− ≃ −
at which the effective mass diverges as m∗(T 0)
T−1/2. Such specific features of ε can be used→to iden∝-
FIG.1: (coloronline). Scalingbehaviorofdimensionlessnor-
tify the solutions of Eq. (1) corresponding to different malized magnetization (B1/2dM(T,B)/dT) versus dimen-
N
experimental situations. In particular, the experimental
sionless normalized (T/B) at magnetic field values B given
N
results obtained for β YbAlB4 show that near QCP inthelegend. Dataareextractedfromthemeasurementsde-
at B 0, the magnetiz−ation obeys1–10 M(B) B−1/2. scribed in Ref. 6. Regions of LFL behavior, crossover, and
≃ ∝
Thisbehaviorcorrespondstothespectrumε(p)givenby NFL behavior are indicated by arrows. The theoretical pre-
Eq.(3)with(p p )/p 1. AtfiniteB andT nearthe diction is represented by a single scaling function.
f i F
− ≪
FCQPT, the solutions of Eq. (1) determining the T and
B dependenciesofm∗(T,B)canbewellapproximatedby BasedonEq.(7),weconcludethemagnetizationM as
asimpleuniversalinterpolatingfunction11–13. Theinter- described within the topological setting of fermion con-
polation occurs between the LFL (m∗ a+bT2) and densationdoesexhibittheempiricalscalingbehavior,be-
NFL (m∗ T−1/2) regimes separated b∝y the crossover ing given by
region at w∝hich m∗ reaches its maximum value m∗ at
N ∗
m (T/B )
temperatureTM,andrepresentstheuniversalscalingbe- M(T,B)= χ(T,B )dB N 1 dB . (8)
1 1 1
havior of Z ∝Z √B1
m∗(T,B) 1+c 1+c T2 At T < B the system is predicted to show LFL behav-
m∗N(TN)= m∗M = 1+c211+c21TN5N/2. (4) hioarsweinttherMed(Bth)e∝NFBL−1r/e2g,iownhearnedasMat(TT)> BT,−t1h/2e.sMysoterme-
∝
over, dM(T,B)/dT again exhibits the observed scal-
Herec andc arefitting parameters,T =T/T isthe
1 2 N M ing behavior, with dM(T,B)/dT T at T < B and
normalized temperature, and dM(T,B)/dT T−3/2 at T > B.∝Thus our analytical
m∗ B−1/2, (5) resultsareinac∝cordanceexperiment,2,4,5freefromfitting
M ∝ parameters and empirical functions.
In confirmation of the analysis of the scaling
while
behavior, Fig. 1 displays our calculations of the
T B1/2 and T B. (6) dimensionless normalized magnetization measure
M ∝ M ∝ (B1/2dM(T,B)/dT) versus the dimensionless normal-
N
It follows from Eqs. (4), (5), and (6) that the effective ized ratio (T/B)N. The normalization is implemented
mass exhibits the universal scaling behavior by dividing B1/2dM(T,B)/dT and T/B respectively by
the maximum value of (B1/2dM(T,B)/dT) and by
M
m∗(T,B)=c 1 m∗ (T/B), (7) the value of (B/T)M value the maximum occurs. It is
3√B N seen that the calculated single scaling function of the
3
ratio (T/B) tracks the data over four decades of the where A and D are fitting parameters.11,12 We rewrite
N 0
normalized quantity (B1/2dM(T,B)/dT) , while the Eq. (9) in terms of the reduced variable A/A ,
N 0
ratio itself varies over five decades. It also follows from
Eq. (8) that (B1/2dM(T,B)/dT) exhibits the proper A(B) D1
N 1+ , (10)
scaling behavior as a function of (B/T)N. Figure 2 A0 ≃ B
illustrates the scaling behavior (B1/2dM(T,B)/dT)
N where D =D/A is a constant, thereby reducing A(B)
of the archetypal HF metal YbRhSi . The solid curve 1 0
2 to a function of the single variable B. Figure 3 presents
representing the theoretical calculations is taken from
the fit of A(B) to the experimental data.1 The theoret-
Fig. 1. Thus, we find that the scaling behavior of
β YbAlB , as extracted from measurements22,23 and
4
−
shown in Fig. 1, is not unique, as Fig. 2 demonstrates
the same crossover under application of the magnetic β-YbAlB
field in the wide range of the applied pressure. 4
3
Theory
A/A=1+D/B
0 1
0
A
1 Theory YbRh Si A/ 2
2 2
N T=0.08 K
T) T=0.33 K
d T=0.75 K
M/ T=1.5 K 1
d B||c
2
1/
B
(
0.64 GPa, 0.26 K
0.1 0.64 GPa, 0.31K 0
0.64 GPa, 0.42K 1 2 3 4 5
B
0.64 GPa, 0.77K
1.28 GPa, 0.77K
1.28 GPa, 1.26K
1.28 GPa, 1.5K
0.01 0.1 1 10
(B/T) FIG. 3: (color online). Experimental data for normalized co-
N
efficient A(B)/A0 as represented by Eq. (10), plotted as a
function of magnetic field B (solid circles). Measured val-
ues of A(B) are taken from Ref. 1, with D1 the only fitting
parameter. The solid curveis thetheoretical prediction.
FIG. 2: (color online). Scaling behavior of the archetypal
HF metal YbRhSi2. Data for (B1/2dM(T,B)/dT)N versus ical dependence (10) agrees well with experiment over a
(B/T) are extracted from measurements of dM/dT versus
N
Batfixedtemperatures.22,23Thesolidcurverepresentingthe substantial range in B. This concurrence suggests that
the physics underlying the field-induced re-entranceinto
theoreticalcalculationisadaptedfromthatofFig.1. Applied
pressures and temperatures are shown in the legends. LFL behavior is the same for classes of HF metals. It is
important to note here that deviations of the theoretical
curvefromthe experimentalpointsatB >2.5Taredue
to violation of the scaling at the QCP.5
Fig. 4 compares our calculations of χ(B) m∗
and C/T = γ(B) m∗ with the experim∝ental
III. THE KADOWAKI-WOODS RATIO measurements.5 Appea∝ling to Eq. (5), the behavior
A(B) (m∗)2, and the good agreement of theory with
∝
experiment shown in this figure, we verify Eq. (10)
Under application of magnetic fields B > Bc2 30 and conclude that the Kadowaki-Woods ratio A/γ2
cmaTn baenddraivtensutffioctiehnetlLyFlLowstatetemhpaevraintugrreess,isβtiv−itYybo≃Af ltBhe4 mA/uχch2 a≃s cinonostth.eirshceoanvsye-rfveerdmiinonthcoemcpasoeunodfsβ1,3−,11Y,2b6A. lB∝4,
form ρ(T) = ρ + AT2. Measurements of the coeffi-
0
cient A of the T2 dependence have providedinformation
on its B-field dependence1. Being proportional to the IV. THE PHASE DIAGRAMS
quasiparticle-quasiparticlescatteringcrosssection,A(B)
is found to obey24,25 A (m∗(B))2. In accordance with The results of the above analysis of the scaling prop-
∝
Eq. (5), this implies that erties of this HF system based on a topological scenario
allowustoconstructtheschematicT B phasediagram
−
D ofβ YbAlB4presentedinFig.5,withthemagneticfield
A(B) A0+ , (9) B as−control parameter. At B = 0, the system acquires
≃ B
4
B||c β-YbAlB NFL
4
) γ 100 crossover
ol0.02 γ T
m ( e
u/ m ur
em χ Theory J/K2 erat NFL
χ ( 50m mp LFL
0.01 o
l) Te T (B)
B||c M
SC QCP
1 B (T) 10

Magnetic field B
FIG. 4: (color online). Measurements5 of magnetic suscepti- FIG.5: SchematicT −B phasediagram. Verticaland hori-
∗
bilityχ=dM/dB=a1mN (leftaxis,squaredatapoints)and zontalarrowshighlightLFL-NFLandNFL-LFLtransitionsat
∗
electronic specific heat coefficient C/T = γ = a2mN (right fixedB andT,respectively. HatchedareaseparatestheNFL
axis,stars),plottedversusmagneticfieldB. Solidcurvetrac-
phasefromtheweaklypolarizedLFLphaseandidentifiesthe
ing scaling behavior of m∗ : theoretical results from present
N transition region. Dashed line in hatched area representsthe
study with fittingparameters a1 and a2. functionT ∝B (seeEq.(6)). TheQCP,located at theori-
M
gin andindicated bythearrow, isthequantumcritical point
at which the effective mass m∗ diverges. It is surrounded by
a flat band satisfying Eq. (2), implying the presence of thesuperconducting phaselabeled SC.
a fermion condensate in a strongly degenerate state of
matter that becomes susceptible to transition into a su-
perconducting state11,27. This NFL fermion-condensate
regime exists at elevated temperatures and fixed mag- NFL
netic field. QCP indicated by the arrow in Fig. 5 is lo-
catedattheoriginofthephasediagram,sinceapplication T
ofanymagneticfielddestroystheflatbandandshiftsthe e QCP
r
system into the LFL state, provided that the supercon- u NFL
t
ducting state is not in play11,12,28. The hatched area in a System Location
r
e
thefiguredenotesthecrossoverregionthatseparatesthe mp FC & LFL
NFL state from the LFL state, also indicated in Fig. 1.
Significantly, the heavy-fermion metal β YbAlB is Te Flat band
4
−
in fact a superconductor on the ordered side of the cor-
responding phase transition. When analyzing the NFL
behavior of ρ(T) on the disordered side of this transi-
0 1 2
tion, it should be kept in mind that several bands si-
Control parameters: P/P , x/x
multaneously intersect the Fermi surface, so that the
c c
HF band never covers the entire Fermi surface. Ac-
cordingly, it turns out that quasiparticles that do not
belong to the HF band make the main contribution to
FIG.6: SchematicT−xphasediagramofHFsystemexhibit-
theconductivity. Theresistivitythereforetakestheform
ρ(T)=m∗ γ(T), where m∗ is the averageeffective ingafermioncondensate. PressureP/Pc andnumber-density
norm norm indexx/x aretaken ascontrol parameters, with x thecrit-
mass of normal quasiparticles and γ(T) describes their c c
ical doping. At P/P > 1 and sufficiently low temperatures,
c
damping. The main contribution to γ(T) can be esti-
thesystemislocatedintheLFLstate(shadowedarea). Mov-
mated as29–32 γ T2m∗(m∗ )2. Based on Eqs. (3)
and (6), we obtai∝n32 ρ(T) noTr3m/2. On the other hand, isnygstpemastdtehveelQopCsPapfloainttbtaonPd/tPhcat<is1tihnteostighneaNtuFrLeroefgfioernm,tiohne
∝
one would expect that at T 0 the flat band (2) condensation(FC).Theupwardverticalarrowtracksthesys-
→
comes into play, producing the behavior ρ(T) A T, temmovingintheLFL-NFLdirectionalongT atfixedcontrol
1
∝
with the factor A proportional to the flat-band range parameters. Not shown is the low-temperature stable phase
1
(p p )/p 1. However, such behavior is not seen, satisfyingtheNernsttheorem(superconductinginthecaseof
befca−useithisFa≪reaofthephasediagramiscapturedbysu- β−YbAlB4) that must exist for P/Pc or x/xc below unity.
perconductivity, as already indicated in Fig. 5. The low-
T resistivity ρ(T,P = 0) T3/2 found experimentally6
∝
5
for the normal state of β YbAlB is consistent with cated by arrows in Fig. 6), the system enters the region
4
−
this analysis. When the pressure P is raised to a crit- P/P > 1, where it is situated prior to the onset of the
c
ical value P , there is a crossover to Landau-like be- FCQPT and demonstrates LFL behavior at sufficiently
c
havior ρ(T) = ρ + A T2. Assuming that P x, low temperatures (shaded area in the figure). The tem-
0 2
where x is the doping or the HF number density,2∝3 we perature range of this region shrinks when P/P 1,
c
observe that such behavior closely resembles the NFL and m∗ diverges as described by Eq. (11). These o→bser-
behavior ρ(T) T1.5±0.1 revealed in measurements of vations are in accord with the experimental evidence.6
∝
the resistivity in the electron-doped high-T supercon-
c
ductors La2−xCexCuO4.33,34 In that case the effective
massm∗(x)divergesasx x orP P 33,34 according
to11,12,32 → c → c V. SUMMARY
2
(m∗(x))2 A a + a2 . (11) To summarize, we have analyzed the thermodynamic
∝ ≃(cid:18) 1 x/xc−1(cid:19) properties of the heavy-fermion metal β −YbAlB4 and
explainedtheirenigmaticscalingbehaviorwithinatopo-
Herea1 anda2 areconstants,whilexc isthecriticaldop- logicalscenarioinwhichFCphaseplaysanessentialrole.
ing at which the NFL behavior changes to LFL behav- WehaveexplainedwhytheobservedNFLbehaviorisex-
ior, the FC having decayed at xc and the system having tremely sensitive to a magnetic field, and how the ther-
moved to the disordered side of the FCQPT. modynamic properties and anomalous T3/2 dependence
InFig.6wedisplaytheschematicT xphasediagram oftheelectricalresistivityremainintactunderthe appli-
−
exhibited by β YbAlB4 when the system is tuned by cation of a pressure.
−
pressure P or by number density x. At P/P < 1 (or
c
x/x < 1) the system is located on the ordered side of
c
topological phase transition FCQPT and demonstrates
VI. ACKNOWLEDGMENTS
NFL behavior at T . T . Thus, the NFL behavior in-
f
duced by the FC that persists at P <P is robust under
c
applicationofpressureP/P <1.12,28(Wenotethatsuch VRS acknowledges support from the Russian Science
c
behavior is also observed in quasicrystals.13,35). At low Foundation, Grant No. 14-22-00281. This research was
temperatures the FC state possessing a flat band, high- also supported by RFBR Grants#14-02-00044 and 15-
lighted in the figure, is strongly degenerate. This degen- 02-06261, and by grant NS-932.2014.2 from the Russian
eracy stimulates the onset of certain phase transitions Ministry of Sciences (MVZ and VAK). VAK and JWC
and is thereby lifted before reaching T = 0, as required thank the McDonnell Center for the Space Sciences for
by the Nernst theorem.27,36 With rising pressure (indi- timely support.
∗ Electronic address: [email protected] 10 S.WatanabeandK.Miyake,J.Phys.Soc.Jpn.83,103708
1 S.Nakatsuji,K.Kuga,Y.Machida,T.Tayama,T.Sakak- (2014).
ibara, Y.Karaki,H.Ishimoto, S.Yonezawa,Y.Maeno, E. 11 V. R. Shaginyan, M. Ya. Amusia, A. Z. Msezane, and K.
Pearson, G.G.Lonzarich, L.Balicas, H.Lee,andZ.Fisk, G. Popov, Phys. Rep.492, 31 (2010).
Nat. Phys. 4, 603 (2008). 12 M. Ya. Amusia, K. G. Popov, V. R. Shaginyan, V.
2 Y.Matsumoto,S.Nakatsuji,K.Kuga,Y.Karaki,N.Horie, A. Stephanovich, Theory of Heavy-Fermion Compounds,
Y. Shimura, T. Sakakibara, A. H. Nevidomskyy, and P. SpringerSeriesinSolid-StateSciences182,Cham,Heidel-
Coleman, Science331, 316 (2011). berg, New York,Dordrecht, London (2014).
3 Y. Matsumoto, K. Kuga, T. Tomita, Y. Karaki, and S. 13 V. R. Shaginyan, A. Z. Msezane, K. G. Popov, G. S.
Nakatsuji, Phys. Rev.B 84, 125126 (2011). Japaridze, and V. A. Khodel, Phys. Rev. B 87, 245122
4 Y. Matsumoto, S. Nakatsuji, K. Kuga, Y. Karaki, Y. (2013).
Shimura,T.Sakakibara,A.H.Nevidomskyy,andP.Cole- 14 D. Lidsky, J. Shiraishi, Y. Hatsugai, and M. Kohmoto,
man, J. Phys.: Conf. Ser. 391, 012041 (2012). Phys. Rev.B 57, 1340 (1998).
5 Y.Matsumoto,K.Kuga,Y.Karaki,Y.Shimura,T.Sakak- 15 S.-S.Lee, Phys. Rev.D 79, 086006 (2009).
ibara, M. Tokunaga, K.Kindo, and S. Nakatsuji, J. Phys. 16 D. Yudin,D. Hirschmeier, H. Hafermann, O.Eriksson, A.
Soc. Jpn. 84, 024710 (2015). I.Lichtenstein,andM.I.Katsnelson,Phys.Rev.Lett.112,
6 T. Tomita, K. Kuga, Y. Uwatoko, P. Coleman, S. Nakat- 070403 (2014).
suji, Science 349, 506 (2015). 17 G. E. Volovik, Phys.Scr. T164, 014014 (2015).
7 A.H.NevidomskyyandP.Coleman,Phys.Rev.Lett.102, 18 A.A. Shashkin, V.T. Dolgopolov, J.W. Clark, V.R.
077202 (2009). Shaginyan, M.V. Zverev, and V.A. Khodel, Phys. Rev.
8 A. Ramires, P. Coleman, A. H. Nevidomskyy, and A. M. Lett. 112, 186402 (2014).
Tsvelik, Phys.Rev.Lett. 109, 176404 (2012). 19 M. Yu. Melnikov, A. A. Shashkin, V. T. Dolgopolov,
9 S.WatanabeandK.Miyake,J.Phys.Soc.Jpn.82,083704 S.-H. Huang, C. W. Liu, and S. V. Kravchenko,
(2013). arXiv:1604.08527.
6
20 L. D. Landau,Sov. Phys.JETP 3, 920 (1956). 29 V. A. Khodel and M. V. Zverev, JETP Lett. 85, 404
21 V. A. Khodel, V. R. Shaginyan, and V. V. Khodel, Phys. (2007).
Rep. 249, 1 (1994). 30 V.R.Shaginyan,A.Z.Msezane,K.G.Popov,J.W.Clark,
22 Y.Tokiwa,T.Radu,C.Geibel,F.Steglich,andP.Gegen- M.V.Zverev,andV.A.Khodel,Phys.Rev.B86,085147
wart, Phys.Rev.Lett. 102, 066401 (2009). (2012).
23 Y. Tokiwa, P. Gegenwart, C. Geibel, and F. Steglich, J. 31 V.A.Khodel,V.R.Shaginyan,andP.Schuck,JETPLett.
Phys. Soc. Jpn. 78, 123708 (2009). 63, 752 (1996).
24 V.A. Khodel and P. Schuck,Z. Phys.B: Condens. Matter 32 V. A. Khodel, J. W. Clark, K. G. Popov, and V. R.
104, 505 (1997). Shaginyan,JETP Lett. 101, 413 (2015).
25 P. Gegenwart, J. Custers, C. Geibel, K. Neumaier, T. 33 N. P. Armitage, P. Furnier, and R. L. Greene, Rev. Mod.
Tayama, K. Tenya, O. Trovarelli, and F. Steglich Phys. Phys. 82, 2421 (2010).
Rev.Lett. 89, 056402 (2002). 34 K.Jin, N.P. Butch,K.Kirshenbaum,J. Paglione, and R.
26 K. Kadowaki and S. B. Woods, Solid State Commun. 58, L. Greene, Nature, 476, 73 (2011).
507 (1986). 35 K. Deguchi, S. Matsukawa, N. K. Sato, T. Hattori, K.
27 V.A.Khodel,J.W. Clark, and M.V.Zverev,Phys.Rev.B Ishida, H. Takakura, and T. Ishimasa, Nature Materials
78, 075120 (2008). 11, 1013 (2012).
28 V. R. Shaginyan, A. Z. Msezane, K. G. Popov, G. S. 36 J.W.Clark, M.V.Zverev,andV.A.Khodel,Ann.Phys.
Japaridze, and V. A. Khodel, Europhys. Lett. 106, 37001 327, 3063 (2012).
(2014).