Universal low-temperature behavior of the CePd_{1-x}Rh_x ferromagnet
aa r X i v : . [ c ond - m a t . s t r- e l ] J u l epl draft Universal low-temperature behavior of the CePd − x Rh x ferromagnet V.R. Shaginyan , K.G. Popov and V. A. Stephanovich
Petersburg Nuclear Physics Institute, Gatchina, 188300, Russia Komi Science Center, Ural Division, RAS, Syktyvkar, 167982, Russia Opole University, Institute of Mathematics and Informatics, Opole, 45-052, Poland
PACS – Strongly correlated electron systems; heavy fermions
PACS – Electronic structure
Abstract. - The heavy-fermion metal
CePd − x Rh x evolves from ferromagnetism at x = 0 to anon-magnetic state at some critical concentration x c . Utilizing the quasiparticle picture and theconcept of fermion condensation quantum phase transition (FCQPT), we address the questionabout non-Fermi liquid (NFL) behavior of ferromagnet CePd − x Rh x and show that it coincideswith that of both antiferromagnet YbRh ( Si . Ge . ) and paramagnet CeRu Si and CeNi Ge .We conclude that the NFL behavior being independent of the peculiarities of specific alloy, isuniversal, while numerous quantum critical points assumed to be responsible for the NFL behaviorof different HF metals can be well reduced to the only quantum critical point related to FCQPT. The nature of quantum criticality determining the non-Fermi liquid (NFL) behavior observed in heavy-fermion(HF) metals is everyday topic of the physics of correlatedelectrons. A quantum critical point (QCP) can arise bysuppressing the transition temperature T c of a ferromag-netic (FM) (or antiferromagnetic (AFM)) phase to zeroby tuning some control parameter ζ other than temper-ature, such as pressure P , magnetic field B , or doping x as it takes place in the case of the HF ferromagnetCePd − x Rh x [1, 2] or the HF metal CeIn − x Sn x [3]. TheNFL behavior around QCPs manifests itself in variousanomalies. One of them is power in T variations of thespecific heat C ( T ), thermal expansion α ( T ), magnetic sus-ceptibility χ ( T ) etc.It is widely believed that the NFL behavior is deter-mined by quantum phase transitions which occur at thecorresponding QCP’s. According to this concept, NFL be-havior in this case is due to the presence of thermal andquantum fluctuations suppressing quasiparticles [4–6] sothat the quantum criticality in these systems can be de-scribed by conventional theory related to a spin-density-wave instability [7] or scenarios where the heavy electronslocalize at magnetic QCP’s, for example, due to a destruc- (a) Email: [email protected] (b)
Email: [email protected] and Homepage: http: //cs.uni.opole.pl / ∼ stef tion of the Kondo resonance [8]. Unfortunately, up to nowit was not possible to describe all available experimentalfacts related to the NFL behavior within a single theorybased on the above scenarios.Measurements performed on the three dimensional FMCePd − x Rh x show that around some concentration x = x c ≃ . − . x = 0 to a non-magnetic state at QCP with the crit-ical concentration x c [1, 2]. At x = x c , measurementson CePd − x Rh x show that the electronic contribution tothe specific heat C ( T ) and the thermal expansion coeffi-cient α ( T ) behave as C ( T ) ∝ α ( T ) ∝ √ T [1, 9]. At theconcentrations x < x c , C ( T ) /T shows a peak at sometemperature T max , while under the application of mag-netic field T max shifts to higher values [2]. Above dis-cussed scenarios for NFL behavior [6–8] imply that itsdetails would in particular depend on system’s magneticground state. Namely, within these scenarios, one can as-sume that the NFL peculiarities of CePd − x Rh x are tobe different from those of either CeNi Ge and CeRu Si exhibiting a paramagnetic ground state [10, 11] or fromthose of AFM cubic HF metal CeIn − x Sn x [3] and HFmetal YbRh (Si . Ge . ) exhibiting (in measurementsof C ( T ) /T ) a weak AFM ordering at T <
20 mK [12].On the other hand, the measurements of χ ( T ) have shownthat the quantum critical fluctuations in this metal have ap-1.R. Shaginyan et al. strong FM component and thus are unique among all otherquantum critical HF systems [13]. Obviously the criticalfluctuations taking place at QCPs in the different HF met-als are different so that it may seem that we cannot have auniversal behavior in these metals. Also, the above tradi-tional scenarios have no grounds to consider these QCPsas different manifestations of some single QCP. Moreover,the behavior of C ( T ) /T in YbRh (Si . Ge . ) is formedby AFM fluctuations while that of χ ( T ) is determinedby FM ones. The distinctive features of FM, AFM andparamagnetic systems suggest the intrinsic differences intheir QCPs resulting in the diversity of their thermody-namic properties. Existing theories corroborate this pointof view, they predict that magnetic and thermal proper-ties of CePd − x Rh x [1,2,4–6,14] should differ from those ofYbRh (Si . Ge . ) since the latter substance is supposeto combine FM and AFM orders.Below we shall see that NFL properties of the function C ( T ) /T in CePd − x Rh x coincide with those of χ ( T ) inCeRu Si and YbRh (Si . Ge . ) as well as with thoseof C ( T ) /T in YbRh (Si . Ge . ) . Also, the NFL be-havior of α ( T ) in CePd − x Rh x coincides with that of α ( T )in HF metals CeNi Ge and CeIn − x Sn x . The observedpower laws and universal behavior of C ( T ) and α ( T ) inCePd − x Rh x can be hardly accounted for within the abovescenarios when quasiparticles are suppressed, for there isno reason to expect that C ( T ), χ ( T ), α ( T ) and other ther-modynamic quantities are affected by the fluctuations orlocalization in a correlated fashion.It might be possible to explain this universal behavior byLandau Fermi liquid (LFL) theory based on the existenceof quasiparticles since C ( T ) /T ∝ α ( T ) ∝ χ ( T ) ∝ M ∗ where M ∗ is the effective mass. Unfortunately, the effec-tive mass of conventional Landau quasiparticles is tem-perature, magnetic field, pressure etc. independent [15]and this fact contradicts to the measurements on HF met-als. On the other hand, when the electronic system ofHF metals undergoes the fermion condensation quantumphase transition (FCQPT), the fluctuations are stronglysuppressed and cannot destroy the quasiparticles whichsurvive down to the lowest temperatures [16–19]. In con-trast to the conventional M ∗ , the effective mass of thesequasiparticles strongly depends on T , x , B etc. so that wehave every reason to suggest that they are indeed respon-sible for the universal behavior observed in HF metals.We note that the direct observations of quasiparticles inCeCoIn have been reported recently [20].In this Letter we show that the NFL properties of HFmetals coincide regardless of their magnetic ground stateproperties. Namely, the NFL features observed in FMCePd − x Rh x , in cubic AFM CeIn − x Sn x , in paramagnetsCeNi Ge and CeRu Si and in YbRh (Si . Ge . ) dis-playing both AFM and FM fluctuations, coincide. Ourmain conclusion is that observed universal behavior is in-dependent of the peculiarities of the given alloy such asits lattice structure, magnetic ground state, dimensional-ity etc. so that numerous previously introduced QCPs can be substituted by the only QCP related to FCQPT.The schematic phase diagram of the HF metals underconsideration is reported fig. 1. We show two LFL re-gions (left one being paramagnet (PM) or having long-range magnetic order and right one corresponds to reen-trant LFL phase induced by a magnetic field), separatedby NFL one. The control parameter ζ (see also above)can be pressure P , magnetic field B , or doping x . Thevariation of ζ drives the system from LFL region to NFLone and then again to LFL. The caption ”Magnetic fieldinduced LFL” means that only magnetic field can gener-ate the reentrant LFL phase. If ζ is not a magnetic field,the right LFL-NFL boundary lies on the abscissa axis.To study the universal low temperature features of HFmetals, we use the model of homogeneous heavy-electronliquid with the effective mass M ∗ ( T, B, ρ ), where the num-ber density ρ = p F / π , and p F is the Fermi momen-tum [15]. This permits to avoid complications associatedwith the crystalline anisotropy of solids [17]. To describethe effective mass M ∗ ( T, B ) as a function of temperatureand applied magnetic field B , when the heavy-electron sys-tem evolves from the LFL state, we use the Landau equa-tion relating the effective mass M ∗ ( T, B ) to the bare mass M and Landau interaction amplitude F ( p , p , ρ ) [15]1 M = 1 M ∗ ( T, R ) + Z p F p F ∂F ( p F , p , ρ ) ∂ p F n ( p , T, R ) d p (2 π ) , (1)where n ( p , T, R ) is the quasiparticle distribution function n ( p , T, R ) = n ( ξ + R ) + n ( ξ − R )2 , (2) n ( ξ ± R ) = (cid:26) (cid:20) ξT ± R (cid:21)(cid:27) − , (3) R = µ B B/T . Here ξ = ε ( p , T ) − µ , µ B is the Bohr mag-neton, ε ( p , T ) is the single-particle energy and µ standsfor a chemical potential.We first consider the case when at T → ε ( p = p F ) = µ at B →
0, we see from eq. (3) that n ( p , T, B ) → θ ( p F − p ), θ ( p ) is the step function. In thiscase eq. (1) reads [15, 21] M ∗ ( ρ ) = M − N F ( p F , p F , ρ ) / . (4)Here N is the density of states of a free electron gas, F ( p F , p F , ρ ) is the p -wave component of Landau ampli-tude. LFL theory implies that the amplitude can be rep-resented as a function of ρ only, F ( p F , p F , ρ ) = F ( ρ ).We assume that at ρ → ρ FC , F ( ρ ) achieves some valuewhere the denominator tends to zero and find from eq. (4)that the effective mass diverges as [22, 23] M ∗ ( ρ ) ≃ A + A ρ FC − ρ , (5)p-2niversal low-temperature behavior of the CePd − x Rh x ferromagnetwhere A , A are constants and ρ FC is QCP of FCQPT.Assuming that the control parameter ζ is represented by x and x c corresponds to ρ FC we obtain ( ζ FC − ζ ) /ζ FC =( x c − x ) /x c ≃ ( ρ FC − ρ ) /ρ FC , while at ζ > ζ FC the systemis on the fermion condensation (FC) side of FCQPT [18].Now we consider the temperature behavior of the ef-fective mass M ∗ ( T ) in a zero magnetic field. Upon us-ing eq. (4) and introducing the function δn ( p , T ) = n ( p , T ) − θ ( p F − p ), eq. (1) takes the form1 M ∗ ( T ) = 1 M ∗ ( ρ ) − Z p F p F ∂F ( p F , p , ρ ) ∂ p F δn ( p , T ) d p (2 π ) . (6)We integrate the second term on the right hand side of eq.(6) over the angular variable Ω, use the notation F ( p F , p, ρ ) = M p F Z p F ∂F ( p F , p , ρ ) ∂ p F d Ω(2 π ) , (7)and substitute the variable p by z = ξ ( p ) /T . Since in HFmetals the band is flat and narrow, we use the approxi-mation ξ ( p ) ≃ p F ( p − p F ) /M ∗ ( T ) and with respect to eq.(6) finally obtain MM ∗ ( T ) = MM ∗ ( ρ ) − β ∞ Z f (1 + βz )1 + e z dz + β /β Z f (1 − βz )1 + e z dz, (8)Here β = T M ∗ ( T ) /p F and f ( z ) = F ( p F , z, ρ ). The mo-mentum p F is defined from the relation ε ( p F ) = µ .To investigate the low temperature behavior of M ∗ ( T ),we evaluate the integral (8). Going beyond the usual ap-proximation [24], we may obtain following final result MM ∗ ( T ) = MM ∗ ( ρ ) + βf (0) ln { − /β ) } + λ β + λ β + ..., (9)where λ and λ are constants of order unity. Herethe logarithmic term is the result of an effective summa-tion of the main nonanalytic (at T →
0) contributions,proportional to exp( − /β ). To analyze eq. (9), we firstassume that β ≪
1. Then, omitting terms of the orderof exp( − /β ), we obtain that at T ≪ T F ∼ p F /M ∗ ( ρ )the sum on the right hand side represents a T -correctionto M ∗ ( ρ ) and the system demonstrates the LFL behav-ior [25]. At higher temperatures, the system enters atransition regime when the effective mass reaches its max-imal value M ∗ M at some temperature T M . It can be easilychecked that the terms proportional to β and β in eq.(9) are ”responsible” for the maximum. The normalizedeffective mass M ∗ N ( T ) = M ∗ ( T ) /M ∗ M as a function of nor-malized temperature T N = T /T M is reported in the insetto fig. 1, showing several regimes. At T N ≪
1, the LFLregime with almost constant effective mass, occurs. At T N ∼ M/M ∗ ( ρ ) ≪ β , eq. (9) reads M/M ∗ ( T ) ∝ T M ∗ ( T ) , giving [25, 26] M ∗ ( T ) ∝ T − / . (10) ~ T -1/2 ~ T -2/3 (LFL) N o r m a li z e d m ass Normalized temperature
Transition region
NFL
LFL
Control parameter,
PM,FM T e m p e r a t u r e , a r b . un i t s NFL FC or AFM state Magnetic field induced LFL
Fig. 1: Schematic phase diagram of the systems under consid-eration. Control parameter ζ represents doping x , magneticfield B , pressure P etc. ζ FC denotes point at which the ef-fective mass diverges. If ζ is not a magnetic field, then theright boundary line NFL-LFL lies on the abscissa axis. Inset- normalized effective mass M ∗ N ( T ) = M ∗ ( T ) /M ∗ M ( M ∗ M is itsmaximal value at T = T M ) versus the normalized temperature T N = T /T M . Several regions are shown. First goes the LFLregime ( M ∗ N ( T ) ∼ const) at T N ≪
1, then transition regime(shaded area) where M ∗ N ( T ) reaches its maximum. At elevatedtemperatures T − / regime given by eq. (10) occurs followedby T − / behavior, see eq. (11). Numerical calculations based on eqs. (8) and (9) showthat at rising temperatures the linear term ∝ β gives themain contribution and leads to new regime when eq. (9)reads M/M ∗ ( T ) ∝ β yielding M ∗ ( T ) ∝ T − / . (11)Note, that ”rising temperatures” are still sufficiently lowfor the expansion of integrals in eq. (8) in powers of β to bevalid. In the inset to fig. 1 both T − / and T − / regimesare marked as NFL ones since the effective mass dependsstrongly on temperature, which is not the case for thetransition region. If the system is located at the FCQPTcritical point, it follows from eq. (5) that M ∗ ( ρ FC ) → ∞ and T F → T = 0 the effective massdepends on B as [26, 28] M ∗ ( B ) ∝ ( B − B c ) − / , (12)where B c is the critical magnetic field driving both HFmetal to its magnetic field tuned QCP and correspondingN´eel temperature toward T = 0. In some cases B c =0. For example, the HF metal CeRu Si is characterizedby B c = 0 and shows neither evidence of the magneticordering or superconductivity nor the LFL behavior downto the lowest temperatures [11]. In our simple model B c isp-3.R. Shaginyan et al. taken as a parameter. At elevated temperatures and fixedmagnetic field, the effective mass depends on temperatureas in the case when the system is placed on the FL sidein accordance with eqs. (10) and (11) [25, 29]. Since themagnetic field enters eq. (1) as the ratio R = µ B B/T , at T N . M ∗ ( B, T ) M ∗ ( B ) ≈ c R c R / , (13)which represents an approximation to solutions of eq. (1)that agrees with eqs. (10) and (12). Here R = T / [( B − B c ) µ B ], c and c are fitting parameters. As we have seenthe effective mass reaches its maximal value M ∗ M at some R = R M and we again define a normalized effective massas M ∗ N ( T, B ) = M ∗ ( T, B ) /M ∗ M . Taking into account eq.(13) and introducing the variable y = R / R M we obtainthe function M ∗ N ( y ) ≈ M ∗ ( B ) M ∗ M c y c y / , (14)which describes a universal behavior of the effective mass M ∗ N ( y ) when the system transits from LFL regime to thatdescribed by eq. (11). At ρ < ρ FC , M ∗ ( ρ ) is finite, seeeq. (5). In this case the eq. (14) is valid at T N . M ∗ ( T, B ) /M ∗ ( ρ ) ≪ /M ∗ ( ρ ) onthe right hand side of eq. (6) is small and can be safelyomitted [27]. As a result, the behavior of M ∗ N ( y ) has tocoincide with that of the normalized effective mass M ∗ N ( T )displayed in the inset to fig. 1.The effective mass M ∗ ( T, B ) can be measured in ex-periments on HF metals. For example, M ∗ ( T, B ) ∝ C ( T ) /T ∝ α ( T ) /T and M ∗ ( T, B ) ∝ χ AC ( T ) where χ AC ( T ) is ac magnetic susceptibility. If the correspond-ing measurements are carried out at fixed magnetic field B (or at fixed both the concentration x and B ) then, as itfollows from eq. (13), the effective mass reaches the max-imum at some temperature T M . Upon normalizing boththe effective mass by its peak value at each field B and thetemperature by T M , we observe that all the curves mergeinto single one, given by eq. (14) thus demonstrating ascaling behavior.As it is seen from fig. 2, the behavior of the normalizedac susceptibility χ NAC ( y ) = χ AC ( T /T M , B ) /χ AC (1 , B ) = M ∗ N ( T N ) obtained in measurements on the HF param-agnet CeRu Si [11] agrees with both the approxima-tion given by eq. (14) and the normalized specific heat( C ( T N ) /T N ) /C (1) = M ∗ N ( T N ) obtained in measurementson the HF FM CePd − x Rh x [2]. It is also seen from fig.2, that at temperatures T N ≤ Si whose electronic system is placed at FCQPT [29], thatis in fig. 1 at ζ FC . As to the normalized specific heat(shown by downright triangles in fig. 2) measured onCePd − x Rh x with x = 0 . C/TCePd Rh x Normalized temperature N o r m a li z ed m a ss AC susceptibilityCeRu Si Fig. 2: Normalized magnetic susceptibility χ N ( T N , B ) = χ AC ( T /T M , B ) /χ AC (1 , B ) = M ∗ N ( T N ) for CeRu Si in mag-netic fields 0.20 mT (squares), 0.39 mT (upright triangles) and0.94 mT (circles) against normalized temperature T N = T /T M [11]. The susceptibility reaches its maximum χ AC ( T M , B ) at T = T M . The normalized specific heat ( C ( T N ) /T N ) /C (1) ofthe HF ferromagnet CePd − x Rh x with x = 0 . T N isshown by downright triangles [2]. Here T M is the temperatureat the peak of C ( T ) /T . The solid curve traces the universalbehavior of the normalized effective mass determined by eq.(14), it is also shown in figs. 3, 4, 5 and 6. Parameters c and c are adjusted for χ N ( T N , B ) at B = 0 .
94 mT. tronic system is located on the FL side and the deflec-tion ( x c − x ) /x c ≃ ( ρ − ρ FC ) /ρ FC at x = 0 . x c ≃ . T − / region [27]. On the other hand, at diminish-ing temperatures the scaling is ceased at relatively hightemperatures as soon as the LFL behavior related to thedeflection from x c sets in. Normalized temperature
C/T CePd Rh B=0 T B=0.5 T B=1 T B=2 T B=3 T N o r m a li z ed m a ss Fig. 3: The normalized effective mass M ∗ N ( T N , B ) extractedfrom the measurements of the specific heat on CePd − x Rh x with x = 0 . B ≥ M ∗ N ( T N ) coincides with that ofCeRu Si (solid curve, see the caption to fig. 2). Now we consider the behavior of M ∗ N ( T ), extracted frommeasurements of the specific heat on CePd − x Rh x underp-4niversal low-temperature behavior of the CePd − x Rh x ferromagnetthe application of magnetic field [2] and shown in fig. 3.It is seen from fig. 3 that at B ≥
1T the value M ∗ N de-scribes the normalized specific heat almost perfectly, co-incides with that of CeRu Si and is in accord with theuniversal behavior of the normalized effective mass givenby eq. (14). Thus, we conclude that the thermodynamicproperties of CePd − x Rh x with x = 0 . x → x c so thatthe behavior of the normalized effective mass would devi-ate from that given by eq. (14). C/T CePd Rh B=0 T B=0.5 T B=1 T B=2 T B=3 T
Normalized temperature N o r m a li z ed m a ss Fig. 4: Same as in Fig. 3 but x = 0 .
85 [2]. At B ≥ M ∗ N ( T N ) demonstrates the universal behavior (solid curve, seethe caption to fig. 2). In fig. 4, the effective mass M ∗ N ( T N ) at fixed B ’s isshown. Since the curve shown by circles and extractedfrom measurements at B = 0 does not exhibit any maxi-mum down to 0.08 K [2], we conclude that in this case x is very close to x c and function M ∗ N ( T N ) is approximatelydescribed by eq. (11), while the maximum is shifted tovery low temperatures or even absent. As seen from fig.4, the application of magnetic field restores the universalbehavior given by eq. (14). Again, this permits us to con-clude that thermodynamic properties of CePd − x Rh x with x = 0 .
85 are determined by quasiparticles rather than bythe critical magnetic fluctuations.The thermal expansion coefficient α ( T ) is given by [24] α ( T ) ≃ M ∗ T / ( p F K ( ρ )). The compressibility K ( ρ ) is notexpected to be singular at FCQPT and is approximatelyconstant [30]. Taking into account eq. (11), we find that α ( T ) ∝ √ T and the specific heat C ( T ) = T M ∗ ∝ √ T .Measurements of the specific heat C ( T ) on CePd − x Rh x with x = 0 . C ( T ) /T = AT − q with q ≃ . A =const [1]. Hence, we conclude that thebehavior of the effective mass given by eq. (11) agreeswith experimental facts. Measurements of α ( T ) /T onboth CePd − x Rh x with x = 0 . Ge [10]are shown in fig. 5. It is seen that the approxima- CePd Rh /T Normalized temperature N o r m a li z ed m a ss CeNi Ge /T B=0 B=2 B=4 B=6 B=8 Fig. 5: The normalized thermal expansion coefficient( α ( T N ) /T N ) /α (1) = M ∗ N ( T N ) for CeNi Ge [10] and forCePd − x Rh x with x = 0 .
90 [2] versus T N = T /T M . Dataobtained in measurements on CePd − x Rh x at B = 0 are mul-tiplied by some factor to adjust them in one point to the datafor CeNi Ge . Dashed line is a fit for the data shown by thecircles and pentagons at B = 0 and represented by the func-tion α ( T ) = c √ T with c being a fitting parameter. The solidcurve traces the universal behavior of the normalized effectivemass determined by eq. (14), see the caption to fig. 2. tion α ( T ) = c √ T is in good agreement with the resultsof measurements of α ( T ) in CePd − x Rh x and CeNi Ge over two decades in T N . We note that measurements onCeIn − x Sn x with x = 0 .
65 [3] demonstrate the same be-havior α ( T ) ∝ √ T (not shown in fig. 5). As a result,we suggest that CeIn − x Sn x with x = 0 .
65, CePd − x Rh x with x ≃ .
9, and CeNi Ge are located at FCQPT (infig. 1 at ζ FC ) and recollect that CePd − x Rh x is a threedimensional FM [1, 2], CeNi Ge exhibits a paramagneticground state [10] and CeIn − x Sn x is AFM cubic metal [3]. M ∗ N ( T N ) extracted from measurements on the HF met-als YbRh (Si . Ge . ) , CeRu Si , CePd − x Rh x andCeNi Ge is reported in fig. 6. It is seen that the uni-versal behavior of the effective mass given by eq. (14) isin accord with experimental facts. YbRh (Si . Ge . ) is located on the FC side where the system demonstratesthe NFL behavior down to lowest temperatures [27]. Inthat case, ζ (see fig. 1) is represented by B and ζ FC = B c .In the LFL regime induced by the magnetic field, the ef-fective mass M ∗ ( B ) ∝ ( B − B c ) − / and does not fol-low eq. (12) [27, 29]. As a result, the range of the scal-ing behavior in temperature shrinks to the transition and T − / regions, see inset to fig. 1. It is seen from fig. 6that M ∗ N ( T N ) shown by downright triangles and collectedon the AFM phase of YbRh (Si . Ge . ) [12] coincideswith that collected on the FM phase (shown by uprighttriangles) of YbRh (Si . Ge . ) [13]. We note that inthe case of LFL theory the corresponding normalized ef-fective mass M ∗ NL ≃ T and B .The peak temperatures T max , where the maxima of C ( T ) /T , χ AC ( T ) and α ( T ) /T occur, shift to higher val-p-5.R. Shaginyan et al. /T CeNi Ge B=2 T B=8 T
Normalized temperature N o r m a li z ed m a ss C/T B=0.5 T C/T B=2 T CePd Rh AC susceptibilityCeRu Si AC susceptibility C/TYbRh (Si Ge ) B=0.1 T
Fig. 6: The universal behavior of M ∗ N ( T N ), extractedfrom χ AC ( T, B ) /χ AC ( T M , B ) for both YbRh (Si . Ge . ) and CeRu Si [11, 13], ( C ( T ) /T ) / ( C ( T M ) /T M ) for bothYbRh (Si . Ge . ) and CePd − x Rh x with x = 0 .
80 [2, 12],and ( α ( T ) /T ) / ( α ( T M ) /T M ) for CeNi Ge [10]. All the mea-surements are performed under the application of magneticfield as shown in the insets. The solid curve gives the universalbehavior of M ∗ N determined by eq.(14), see the caption to fig.2. C/TAC susceptibilityYbRh (Si Ge ) T m ax ( K ) B (T)
Fig. 7: The peak temperatures T max ( B ), extracted from mea-surements of C/T and χ AC on YbRh (Si . Ge . ) [12, 13]and approximated by straight lines. The lines intersect at B ≃ .
03 T. ues with increase of the applied magnetic field. It fol-lows from eq. (14) that T M ∝ ( B − B c ) µ B . In fig.7, T max ( B ) are shown for C/T and χ AC , measured onYbRh (Si . Ge . ) . It is seen that both functions canbe represented by straight lines intersecting at B ≃ .
03 T.This observation [12, 13] as well as the measurements onCePd − x Rh x , CeNi Ge and CeRu Si demonstrate thesame behavior [2, 10, 11].In summary, we have shown, that bringing the differ-ent experimental data (like C ( T ) /T , χ ac ( T ), α ( T ) /T etc)collected on different HF metals (YbRh (Si . Ge . ) ,CeRu Si , CePd − x Rh x , CeIn − x Sn x and CeNi Ge ) tothe above normalized form immediately reveals their uni-versal scaling behavior. This is because all above experi- mental quantities are indeed proportional to the normal-ized effective mass. Since the effective mass determines thethermodynamic properties, we conclude that above alloysdemonstrate the universal NFL thermodynamic behavior,independent of the details of the HF metals such as theirlattice structure, magnetic ground state, dimensionalityetc. This conclusion implies also that numerous QCPs as-sumed earlier to be responsible for the NFL behavior ofdifferent HF metals can be well reduced to a single QCPrelated to FCQPT.This work was supported in part by RFBR, project No.05-02-16085. REFERENCES[1]
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