Valence Fluctuations Revealed by Magnetic Field Scan: Comparison with Experiments in YbXCu_4 (X=In, Ag, Cd) and CeYIn_5 (Y=Ir, Rh)
Shinji Watanabe, Atsushi Tsuruta, Kazumasa Miyake, Jacques Flouquet
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Valence Fluctuations Revealed by Magnetic Field and Pressure Scans: Comparisonwith Experiments in YbXCu (X=In, Ag, Cd) and CeYIn (Y=Ir, Rh) Shinji
Watanabe , Atsushi
Tsuruta , Kazumasa Miyake , and Jacques Flouquet Department of Applied Physics, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo, 113-8656, JapanDivision of Materials Physics, Department of Materials Engineering Science, Graduate School of Engineering Science,Osaka University, Toyonaka, Osaka 560-8531, Japan D´epartement de la Recherche Fondamentale sur la Mati`ere Condense´e, SPSMS, CEA Grenoble, 17 rue des Martyrs, 38054Grenoble Cedex 9, France The mechanism of how critical end points of the first-order valence transition (FOVT) arecontrolled by a magnetic field is discussed. We demonstrate that critical temperature is sup-pressed to be a quantum critical point (QCP) by a magnetic field. This results explain thefield dependence of the isostructural FOVT observed in Ce metal and YbInCu . Magnetic fieldscan can make the system reenter in a critical valence fluctuation region. Even in intermediate-valence materials, the QCP is induced by applying a magnetic field, at which magnetic suscep-tibility also diverges. The driving force of the field-induced QCP is shown to be a cooperativephenomenon of the Zeeman effect and the Kondo effect, which creates a distinct energy scalefrom the Kondo temperature. The key concept is that the closeness to the QCP of the FOVT isvital in understanding Ce- and Yb-based heavy-fermions. This explains the peculiar magneticand transport responses in CeYIn (Y=Ir, Rh) and metamagnetic transition in YbXCu forX=In as well as the sharp contrast between X=Ag and Cd. KEYWORDS: quantum critical point, first-order valence transition, valence fluctuations, CeIrIn , CeRhIn ,YbInCu , YbAgCu , YbCdCu
1. Introduction
Quantum critical phenomena in itinerant fermion sys-tems with strong correlations have attracted much at-tention in condensed matter physics. When the contin-uous transition temperature of the magnetically orderedphase is suppressed by controlling material parametersand reaches absolute zero, the quantum critical point(QCP) emerges. In the paramagnetic metal phase nearthe QCP, enhanced spin fluctuations cause non-Fermi liq-uid behaviors in physical quantities exhibiting quantumcriticalities, and even trigger other instabilities suchas unconventional superconductivity. So far, the mag-netic QCP and the role of spin fluctuations have beenextensively discussed from both theoretical and ex-perimental sides. Recently, critical phenomena associated with chargedegrees of freedom have attracted attention. In particu-lar, valence instability and its critical fluctuations haveattracted much attention as a possible origin of anoma-lies in Ce- and Yb-based heavy-fermion systems.
5, 6
Va-lence transition phenomena were closely studied fourdecades ago under the label of intermediate valence. Ev-idence of its occurrence comes from the γ - α transition ofCe metal six decades ago (see Fig. 1(a)) with first-ordervalence transition (FOVT), which in the (temperature T , pressure P ) plane starts at T v ∼
120 K and termi-nates at the critical end point (CEP) ( T CEP ≈
600 K, P CEP ≈ As the intercept of the T v ( P ) line ofthe FOVT is rather high at P = 0, no quantum criticalitywas discovered. For many anomalous Ce compounds, noFOVT has been detected despite the fact that their va-lence deviates from three where the occupation number (¯ n f ) of the 4f shell is unity; the conditions are such thatthe system is always in a valence crossover regime, i.e.,the system escapes from the CEP but, as will be stressedlater, can feel its proximity. PT600400200 20 (GPa)(K) a phaseg phase(n large) f (n small) f f (n small) f T v (GPa)(a) Ce (b) YbInCu CEP
Fig. 1. (color online) Temperature-pressure phase diagram of (a)Ce metal and (b) YbInCu .
13, 14 (a) The first-order valencetransition between the γ -phase and the α -phase (solid line) ter-minates at the critical end point (CEP) (solid circle) in the fcclattice. The shaded area around P = 0 represents the β -phase.(b) The first-order valence transition (solid line) is suppressedunder pressure in the cubic AuBe (C15b type) lattice. Notethat the T axis is shown on the logarithmic scale. The shadedarea represents the magnetically ordered phase. ¯ n f denotes thenumber of electrons per Ce in (a) and the number of holes perYb in (b) (see text). An excellent example of FOVT for Yb systems was
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Author Name reported for YbInCu (see Fig. 1(b)), and the Ybcase can be regarded as the hole analog of Ce; ¯ n f beingthe hole occupation number of the 4f shell (14 for Yb and 13 for Yb where ¯ n f = 1). Although YbInCu aswell as Ce metal is a prototypical example that shows theFOVT, most Ce- and Yb-based heavy-fermions seem tobe in a valence-crossover regime. When the CEP is sup-pressed by tuning material parameters and enters theFermi-degeneracy regime, diverging valence fluctuationsare considered to be coupled with Fermi-surface instabil-ity. This multiple instability seems to give the key mech-anism that dominates the low-temperature properties ofmaterials including valence-fluctuating ions such as Ceand Yb. It may play a dominant role in heavy-fermionquantum instabilities.In Fig. 1(b), with increasing P , T v ( P ) decreases andeven becomes sufficiently low such that FOVT reachesthe magnetic boundary
13, 14 clearly in a narrow P win-dow. The interplay between valence transition and mag-netic transition can be strong. Looking carefully into thedisappearance under pressure of long range magnetismin Ce- and Yb-based heavy-fermion systems, this inter-play is often strong. There are only a few material serieslike the CeCu Si series and CeRu Si series wherethe magnetic QCP at P c is not coupled with valence fluc-tuation. Thus, discriminating between valence and spinquantum criticality is often difficult.The proof of valence fluctuations in the quantum de-generacy regime seems to be supported by evidence ofthe two-superconductivity mechanism in the CeCu Si -CeCu Ge series where P v − P c ∼ Amarked increase in the superconducting transition tem-perature T SC is observed at a pressure where the valenceof Ce changes sharply in CeCu Ge , CeCu Si ,
16, 17 and CeCu (Ge x Si − x ) .
18, 19
The importance of quan-tum criticality is shown above T SC by the observationof non-Fermi liquid T -linear resistivity in wide tempera-ture region. The T -linear resistivity has been observed ina variety of Ce- and Yb-based heavy-fermion systems. Theoretically, the possibility of the valence-fluctuation-mediated superconductivity in the P - T phase diagram of CeCu Ge was pointed out in ref. 5.It was shown that the T -linear resistivity emergesin a wide temperature range near the QCP of thevalence transition. Residual resistivity was also shownto be markedly enhanced when the system is tunedto approach the QCP by controlling P and/or theconcentration of the chemical doping. Near the QCP,the superconducting transition temperature was shownto be enhanced by valence fluctuations on the basisof the slave-boson mean-field theory taking account ofits Gaussian fluctuations. The stability and lattice-structure dependence of density-fluctuation-mediatedsuperconductivity were argued phenomenologically. Recent numerical studies have revealed the significanceof valence fluctuations near the QCP of the FOVT andclarified its new aspects: The emergence of unconven-tional superconductivity due to an anomalous increasein the coherence of quasiparticles near the QCP, and theabsence of phase separation as well as non diverging totalcharge compressibility even at the QCP at least in elec- tronic origin due to the non conserving order parameterof the valence transition. In (
T, P ) phase diagrams of heavy-fermion systems,magnetic, valence, and superconductivity boundaries canseriously cause interference. This suggests the idea thatthis interplay also occurs in the ( T , magnetic field h )plane for P close to P v near the QEP. Valence fluctua-tions are essentially relative charge fluctuations betweenf and conduction electrons. Hence, it is highly nontrivialhow magnetic field affects the valence QCP as well asQEP. To resolve these fundamental issues, we have the-oretically studied the magnetic field dependence of thecritical points of the FOVT. In this paper, we report the mechanism of how theQCP as well as the CEP of the FOVT is controlled by amagnetic field in great detail. We discuss how this newlyclarified mechanism gives an explanation of unresolvedobservations in Ce- and Yb-based systems. First, weshow how critical end temperature is suppressed to ab-solute zero by applying a magnetic field, which explainsthe field dependence of the isostructural FOVT tempera-ture observed in Ce metal and YbInCu . Our results alsoexplain the peculiar magnetic response in CeIrIn , wherethe first-order transition line emerges in the temperature-magnetic field phase diagram, giving rise to the increasein residual resistivity as well as the appearance of the T -linear resistivity. The differences in the location of thematerial with respect to CEP explains the sharp contrastbetween YbAgCu and YbCdCu in their magnetizationcurves in spite of the fact that both have nearly the sameKondo temperatures. Our results indicate that the QCPas well as the FOVT is induced even in moderately inter-mediate valence materials by applying a magnetic field,which causes various anomalies such as non-Fermi liq-uid behavior in the resistivity, the increase in the resid-ual resisitivity, and diverging magnetic susceptibility. Wediscuss the significance of the proximity to the criticalpoints of the FOVT to understand unresolved phenom-ena in Ce- and Yb-based heavy-fermions. The key con-cept is the closeness to the QCP of the FOVT.
2. First-Order Valence Transition under Mag-netic Field at Finite Temperature
To give a quantitative outlook of the field dependenceof the valence transition, let us consider the Claudius-Clapeyron relation for the FOVT temperature T v : dT v dh = − m K − m MV S K − S MV , (1)where m and S denote the magnetization and entropy,respectively, and h denotes the magnetic field. Here, Kindicates the Kondo regime where the f-electron (hole)density per site ¯ n f is close to 1 in the Ce (Yb)-basedsystem, i.e., Ce (4f ) and Yb (4f ), and MV indi-cates the “mixed-valence” regime with ¯ n f < Sincethe magnetization as well as the entropy in the Kondoregime is larger than those in the MV regime, as ob-served in the specific heat and the uniform susceptibil-ity, it turns out that T v is suppressed by applying h (seeFig. 2(a)). Then, the critical end point is eventually sup-pressed to T = 0 K by h . . Phys. Soc. Jpn. Full Paper
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Furthermore, the field dependence of T v in the zero-temperature limit is also derived using the above rela-tion: For T →
0, the entropy shows the T -linear behaviorin both the Kondo and MV regimes so that S K − S MV is approximated to be proportional to T v in the casewhere T v is smaller than the characteristic energy scalesin the Kondo and MV regimes. Noting that m K − m MV istemperature-independent for T →
0, we have δT v /δh = − C /T v , leading to T v = √ C √ h v − h with constants C , which explains well the observed behavior in the Cemetal and YbInCu (see Fig. 2(b)). We stress herethat our analysis not only provides a firm ground forsmall- T v behavior by considering the coherence of elec-trons essential for low temperature, but also interpolatesthe high T v satisfying the relation ( h/h v ) + ( T /T v ) =1 to zero temperature, since this relation was derivedby assuming an isolated atomic entropy, which is jus-tified only in the high-temperature regime. Although theabove discussion is about the FOVT T v temperature, itturns out that the critical end temperature T CEP is alsosuppressed, as shown in Figs. 2(a) and 2(b).
Ph T
QCP(a) (b)
T hT h v v
CEP 0
Fig. 2. (color online) (a) Schematic phase diagram, showing theFOVT surface in the T - P - h space, where P represents a con-trol parameter (e.g., pressure and chemical concentration). Thecritical end points (CEPs) form a continuous transition line thatreaches T = 0 as the QCP. (b) FOVT line ( h/h v ) + ( T/T v ) =1 in the T - h plane for a fixed P , corresponding to the dashedline in (a). According to the Maxwell relation, the volume vari-ation V o ( h ) with h is related to the pressure derivativeof the magnetization m ( ∂V o /∂h = − ∂m/∂P ) propor-tional to the Pauli susceptibility in this paramagneticstate, and thus inversely proportional to its Kondo tem-perature. For Ce, ∂m/∂P decreases under P as ¯ n f ; thus, ∂V o /∂h is positive and the system enters in the trivalentstate upon increasing h . The same occurs for Yb be-cause, now, ∂m/∂P increases with P ; thus, n f : ∂V o /∂h is negative. The phenomenological dependence of T v ( h )as reported in Fig. 2(a) was derived in agreement withthis picture.
3. Extended Periodic Anderson Model
Although we have shown that the h dependence of thecritical temperature T CEP can be understood from theviewpoint of the free-energy gain induced by the largerentropy in the Kondo regime, it is highly nontrivial howthe QCP of the FOVT is controlled by h at T = 0. Toproceed with our analysis, we introduce a minimal modelthat describes the essential part of Ce- and Yb-based systems: H = X k σ ε k c † k σ c k σ + ε f X iσ n f iσ + V X iσ (cid:16) f † iσ c iσ + c † iσ f iσ (cid:17) + U N X i =1 n f i ↑ n f i ↓ + U fc N X i =1 n f i n c i − h X i ( S f zi + S c zi ) , (2)where c iσ ( c † iσ ) is the annihilation (creation) operator ofthe conduction electron at the i -th site with a spin σ , and f iσ ( f † iσ ) is that of the f electron. The number operator isdefined by n a iσ = a † iσ a iσ and n a i = n a i ↑ + n a i ↓ for a = f andc. Here, ε k denotes the energy dispersion for conductionelectrons. ε f is the f level and V is the hybridizationbetween the f and conduction electrons. The effect ofapplying pressure is to increase both the hybridization V and the f-level ε f relatively to the Fermi level, thelatter of which plays a more crucial role in the valencetransition than the former in Ce and Yb compounds. Inother words, the increase in pressure is parameterizedessentially by that in ε f .The on-site Coulomb repulsion for f electrons is givenby the U term. The U fc term is the Coulomb repulsionbetween the f and conduction electrons, which is con-sidered to play an important role in the valence tran-sition.
21, 23, 30–36
Namely, the periodic Anderson modelwithout U fc cannot explain a sharp or discontinuous va-lence transition as discussed in refs. 21 and 23. For ex-ample, in the case of Ce metal that exhibits the γ - α tran-sition, the 4f- and 5d-electron bands are located at theFermi level. Since both 4f and 5d orbitals are locatedon the same Ce site, this U fc term cannot be neglected.In Yb systems, f iσ ( f † iσ ) is regarded as the annihilation(creation) operator of the f hole and hence ε f denotes thef-hole level. For YbInCu , a considerable magnitude ofthe In 5p and Yb 4f hybridization was pointed out by theband-structure calculation and recent high-resolutionphotoemission spectra have detected a remarkable in-crease in the magnitude of the p-f hybridization at theFOVT. These results suggest the importance of both V k and U fc .The reason why the critical-end temperature T CEP isso high, that is as much as 600 K, in Ce metal in con-trast to that in YbInCu can be understood in terms ofthe difference in the magnitude of U fc . In YbInCu , U fc originates from the intersite interaction, which should besmaller than that of Ce metal. This view also gives thereason why most Ce and Yb compounds only show va-lence crossover, but not FOVT. Namely, most of the com-pounds seem to have a moderate U fc owing to its intersiteorigin, which is smaller than the critical value for caus-ing a discontinuous jump of the valence. However, evenin the valence-crossover regime, underlying effect of va-lence instability causes intriguing phenomena, as shownbelow. It should be noted that the importance of the U fc term in playing a crucial role in the isostructural FOVTin YbInCu in the hole picture of eq. (2) was pointedout in refs. 35 and 36. The last term in eq. (2) is the J. Phys. Soc. Jpn.
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Zeeman term with h being the magnetic field includingthe g factors. We apply the slave-boson-mean-field theory
21, 40 to theHamiltonian eq. (2) at T = 0 where the slave-boson-mean-field theory is a reasonable approximation. To de-scribe the state for U = ∞ , we consider V f † iσ b i c iσ insteadof V f † iσ c iσ in eq. (2) by introducing the slave-boson op-erator b i at the i -th site to describe the f state and re-quire the constraint P σ n f iσ + b † i b i = 1 by the methodof the Lagrange multiplier P i λ i (cid:16)P σ n f iσ + b † i b i − (cid:17) . For H U fc in eq. (2), we employ mean-field decouplingas n f i n ci ≃ n f n ci + n c n f i − n f n c . By approximating meanfields as uniform ones, i.e., b = h b i i and ¯ λ = λ i , the setof mean-field equations is obtained by ∂ h H i /∂λ = 0 and ∂ h H i /∂b = 0 as follows:¯ λ = V N X k σ f ( E − k σ ) − f ( E + k σ ) p (¯ ε f σ − ¯ ε k σ ) + 4 ¯ V , (3)1 − | ¯ b | = 12 N X k σ, ± " ± ¯ ε f σ − ¯ ε k σ p (¯ ε f σ − ¯ ε k σ ) + 4 ¯ V × f ( E ± k σ ) , (4)and the following equation holds for the total electronnumber:¯ n f + ¯ n c = 1 N X k σ (cid:2) f ( E − k σ ) + f ( E + k σ ) (cid:3) . (5)Here, f ( E ) is the Fermi distribution function and E ± k σ are the lower ( − ) and upper (+) hybridized bands for aquasiparticle with spin σ , respectively: E ± k σ = 12 (cid:20) ¯ ε f σ + ¯ ε k σ ± q (¯ ε f σ − ¯ ε k σ ) + 4 ¯ V (cid:21) , (6)where ¯ ε k σ , ¯ ε f σ , and ¯ V are defined by ¯ ε k σ ≡ ε k + U fc ¯ n f − hσ , ¯ ε f σ ≡ ε f +¯ λ + U fc ¯ n c − hσ and ¯ V ≡ V | ¯ b | . The dispersionof the conduction electrons is taken as ε k = k / (2 m ) − D with − D being set as the bottom of the conductionband and the density of states N ( ε ) is set to satisfy thenormalization condition, R D − D dεN ( ε ) = 1 per spin inthree dimension (see inset of Fig. 3). We take D as theenergy unit and show the results for V = 0 . n = (¯ n f + ¯ n c ) / / When ε f is deep, the Kondo state with ¯ n f = 1 is real-ized. As ε f increases, electrons move from the f level intothe conduction band via hybridization, giving rise to theMV state. Hence, ¯ n f decreases gradually for U fc = 0 asshown in the inset of Fig. 3, as calculated using the slave-boson mean-field theory in model (2). As U fc increases,¯ n f decreases sharply as a function of ε f . For large U fc ,¯ n f = ∂ h H i /∂ε f shows a discontinuous jump, which in-dicates the level crossing of the ground states betweenthe Kondo state and the MV state.
21, 23, 24 (see U fc = 1 . U fc , since a large U fc forces the electrons topour into either the f level or the conduction band.
23, 24 -0.6 -0.4 -0.2 0 0.201.02.0 f -1 -0.5 000.20.40.60.81 f n f n f e f e U f c KondoMixed Valence U fc QCP e Fig. 3. (color online) Ground-state phase diagram in the plane of U fc and ε f for D = 1, V = 0 . n = 7 /
8. The FOVT line (solidline with open triangles) terminates at a QCP (a solid circle) for h = 0 .
00. The dashed line represents the valence-crossover pointsat which χ v has a maximum as a function of ε f for each U fc . Theinset shows the energy band of conduction electrons, and ¯ n f vs ε f for U fc = 0 .
0, 0.4, 0.8, 1.2, 1.6, and 2.0 under h = 0. The ground-state phase diagram at a zero magneticfield determined by the slave-boson mean-field theoryis shown in Fig. 3. The FOVT line represented by thesolid line with open triangles in Fig. 3 satisfies the re-lation ε f + U fc ¯ n c ∼ µ with µ being the chemical po-tential
6, 21, 23 in the mean-field framework. This im-plies that the f state with the energy ε f + U fc ¯ n c andthe f state with a conduction electron at the Fermilevel with the energy µ are degenerate, giving rise to thevalence transition. The FOVT line terminates at theQCP. The QCP in the ε f - U fc plane is identified to be( ε QCPf , U
QCPfc ) = (0 . , . n f disappears and the valence susceptibility χ v ≡ − ∂ h H i ∂ε = − ∂ ¯ n f ∂ε f (7)diverges. Namely, valence fluctuations diverge at theQCP. Even for U fc < U QCPfc , enhanced valence fluctua-tions remain,
24, 42 as shown by the dashed line in Fig. 3,where χ v has a maximum as a function of ε f for each U fc (see Fig. 4). The valence-crossover line with enhanced χ v regarded as a straight extension of the FOVT line to the U fc < U QCPfc regime implies that the valence fluctuationsare a result of the degeneracy of the f and f states,as mentioned above. The characteristic energy scale ofthe system, the so-called Kondo temperature, which isdefined as T K ≡ ¯ ε f σ − µ , is estimated to be T K = 0 . This is consis-tent with the existence of the QCP in the ground-state . Phys. Soc. Jpn.
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Author Name 5 e f c v fc -0.2-0.4 -0.31680 U =1.4 fc Fig. 4. (color online) ε f dependence of valence susceptibility χ v for U fc from 0.0 to 1.4 at D = 1, V = 0 . n = 7 / h = 0.Inset shows χ v vs. ε f for U fc = 1 . phase diagram, since by detouring around the QCP, theKondo and MV states can be adiabatically connected,with Luttinger’s sum rule satisfied.Note that when the hybridization V is decreased (in-creased), the QCP is shifted to a larger (smaller) U fc and | ε f | position in Fig. 3, as confirmed by the DMRG cal-culation as well as the slave-boson mean-field theory. When pressure is applied to the Ce-based compoundssuch as CeCu Ge and CeIrIn , the anion approaches the4f electron at the Ce site, which makes the f-electron level ε f increase. Since the interorbital Coulomb repulsion U fc and the hybridization V also increase under pressure,applying pressure is considered to draw a trajectory linefrom the left and bottom position to the right and topposition in Fig. 3. On the other hand, when pressure isapplied to Ce metal, it is expected that the increase inthe hybridization V will be prominent rather than ε f ,because of the monoelemental constitution of the metal.Hence, applying pressure is considered to draw a trajec-tory line from the left and bottom position to the rightand top position in the V - U fc plane, instead of in the ε f - U fc plane in Fig. 3. This is consistent with the esti-mation of model parameters for the γ - α transition in Cemetal based on the analysis of photoemission spectra us-ing the single-impurity Anderson model. The surface ofFOVT exists in the parameter space of ε f , U fc , and V forthe ground state. A trajectory line is drawn in the spacefor the corresponding experimental parameter, such aspressure in Fig. 2.In Ce metal, the X-ray L III edge absorption spectraled to the conclusion that the Ce valence jumps betweenCe +3 . ( γ phase) and Ce +3 . ( α phase) at T = 300 K. One may think that the valence change seems to be toosmall in comparison with our theoretical result shown inthe inset of Fig. 3. First, we should note that the abovemeasurement was performed at a rather high tempera-ture (see Fig. 1(a)), and hence the magnitude of the ¯ n f jump at T = 0 in the inset of Fig. 3 should be markedlyreduced by thermal fluctuation effects. While, for qual-itatively accurate comparison, it is necessary to use re- alistic band structures for the f and conduction bands,the momentum dependence of the hybridization, and theCoulomb interactions U fc and U in the model (2), itshould be noted that when the symmetry of the wavefunction of hybridized conduction electrons is the sameas that of f electrons, the X-ray absorption measurementdetects the spectra as it comes from f electrons. Hence,there is a tendency that this type of measurement un-derestimates the magnitude of the valence jump.A key parameter for describing valence instabilitiesis the interorbital Coulomb repulsion U fc , as mentionedabove. For Ce metal, the onsite U fc has a considerablevalue giving U fc > U QCPfc 45 and hence the FOVT is con-sidered to occur at a very high critical end temperature T CEP of about 600 K. In the Ce-based compounds such asCeCu Ge , CeCu Si , and CeCu (Ge x Si − x ) , U fc origi-nates from its intersite Coulomb repulsion and hence U fc is reduced from the onsite value, which seems to compa-rable to U QCPfc . Namely, these compounds seem to be lo-cated in the valence crossover regime (although the sharppeak of the residual resistivity and the sharp drop of the T coefficient in the resistivity under pressure suggestthat these are rather close to the QCP
15, 16, 18, 19 ).Here, we should also comment on the magnetically or-dered phase, which can appear in the ground-state phasediagram in Fig. 3 depending on the strength of V . Al-though we focus on the nature of the FOVT line withthe QCP and hence the magnetically ordered phase isnot shown in Fig. 3, the magnetically ordered phase isconsidered to be realized in the Kondo regime, which isbasically located in the small- ε f region in Fig. 3. Then,in the valence-crossover regime for U fc < U QCPfc , as ε f in-creases, the magnetic order is suppressed and the param-agnetic metal phase appears. In the paramagnetic metalphase, as ε f further increases, the Kondo state is changedto the MV state at the valence-crossover point repre-sented by the dashed line in Fig. 3. We note that inthe Kondo regime near the QCP in Fig. 3, the super-conducting correlation is enhanced, which was shownby the slave-boson mean field theory taking into ac-count the Gaussian fluctuations and the DMRG cal-culation applied to eq. (2). This seems to correspondto the T - P phase diagrams of CeCu Ge , CeCu Si , and CeCu (Ge x Si − x ) ,
18, 19 where with P application,the antiferromagnetic (AF) order is suppressed and inthe narrow pressure range just before a sharp valenceincrease of Ce, the superconducting transition tempera-ture is enhanced. We also note that the reason why thesuperconducting correlation is enhanced was clarified bythe unbiased calculation: The coherence of electronswith large valence fluctuations is enhanced in the Kondoregime near the QCP, giving rise to an enhanced pairingcorrelation (see ref. 23 for details).We also note that the nontrivial result has been ob-tained by the DMRG calculation on the model (2): Totalcharge compressibility is defined by κ ≡ n ∂ (2 n ) ∂µ , (8)with 2 n being the total filling and 2 n = ¯ n f + ¯ n c notdiverging even at the QCP.
23, 42
This is in sharp con-
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Author Name trast to the mean-field result where the phase separationis accompanied by the FOVT in the ground-state phasediagram, giving rise to diverging κ . In the mean fieldframework, the valence fluctuation, i.e., relative chargefluctuation diverges at the valence QCP, which triggersthe total charge instability as well. However, when quan-tum fluctuations and electron correlations are taken intoaccount correctly, χ v diverges at the valence QCP, but κ remains finite.
23, 42
This has been clarified to be due tothe fact that the order parameter of the valence tran-sition n f is not a conserving quantity; [ n f , H ] = 0. The system can be unstable with respect to the relativecharge fluctuation while keeping the total charge stable(see ref. 23 for details). Hence, it is predicted that, whenthe material parameters could be experimentally tunedclose to the valence QCP, the compressibility is κ = − V o (cid:18) ∂V o ∂P (cid:19) N e (9)with N e being the total electron number not showingdivergence at least in electronic origin.When we add a temperature axis to Fig. 3 at h = 0, thephase diagram of the T - ε f - U fc space is shown schemati-cally in Fig. 5(a). The first-order transition surface con-tinues to the valence crossover surface. The boundarybetween the two surfaces forms a critical end line, whichreaches T = 0 K, forming the QCP. At the critical endline as well as at the QCP, the valence susceptibility (7)diverges, i.e., χ v = ∞ . Note that, even at the valence-crossover surface, valence fluctuations develop well asshown in Fig. 4. Figure 5(b) shows a two-dimensional cutof Fig. 5(a) for a certain U fc > U QCPfc : The FOVT line(a solid line) terminates at the critical end point (a filledcircle), which continues to the valence crossover line (adashed line). The reason why T v is an increasing functionof ε f can be understood from the Claudius-Clapeyron re-lation dT v dε f = n fK − n fMV S K − S MV , (10)where n f and S denote the number of f electrons (or fholes) per site and the entropy, respectively. Note herethat the increase in ε f parameterizes that of pressure inour model. We have dT v /dε f >
0, since n fK > n fMV and S K > S MV are satisfied at least in the deep first-ordertransition region for U fc > U QCPfc . Namely, to achievethe free-energy gain caused by the larger entropy, theKondo phase is realized on the higher- T side. Owing tothe thermodynamic third law, the first-order transitiontemperature T v should be perpendicular to ε f for T v → However, note that the Kondo-volume-collapse scenario assumes a special volume de-pendence on the Kondo coupling to realize the first-ordertransition, whose validity should be carefully examined. ε U T
QCP(a) ffc ε T (b) f T v T v* CEP
Fig. 5. (Color online) (a) Schematic phase diagram in the ε f - U fc - T space for a certain V and U ( > U fc ). The FOVT surface (graysurface) with the critical end line (bold line) reaching on T = 0at the QCP continues to the valence-crossover surface (light-gray surface). (b) A two-dimensional cut of (a) for a certain U fc > U QCPfc . The FOVT line (solid line) T v ( ε f ) terminates atthe CEP (filled circle), which continues to the valence-crossoverline (dashed line) T ∗ v ( ε f ). Actually, it has been pointed out that the Kondo-volume-collapse scenario is not consistent with the isostructuralFOVT in YbInCu by Sarrao: The measured Gr¨uneisenparameter Γ = − d ln T K /d ln V o = 43 leads to the conclu-sion that the difference in T K at the first-order transitionshould be ∆ T K = 10 K for the measured volume change,∆ V o /V o = 0 . T K ∼
400 K ( T K and V o denote the Kondo tempera-ture and volume, respectively). Thus, the volume changeis too small to explain the T K change. In our approach(2), the parameters ( U fc , ε f ) for each material determinewhether they show the FOVT, or the valence crossover,when T or pressure (or chemical composition) is changed.Our scenario does not need to assume the special volumedependence on Kondo coupling to cause an FOVT differ-ently from that in the Kondo-volume-collapse scenario. When we apply a magnetic field to the Hamilto-nian eq. (2) we find a remarkable result in the valence-crossover regime for U fc < U QCPfc . Figure 6(a) showsthe relation of the magnetization m = P i h S f zi + S c zi i /N vs h for ( ε f , U fc )= ( − . , . − . , . χ = ∂m/∂h = ∞ ) emerges at h = h m = 0 .
01 and0.02, respectively. To clarify its origin, we determine theFOVT line as well as the QCP under the magnetic field.The result is shown in Fig. 7. It is found that the FOVTline extends to the MV regime and the location of theQCP shifts to a smaller- U fc and smaller- | ε f | direction,when h is applied. This low- h behavior of the FOVT lineagrees with the low-temperature limit of T v discussed in §
2, in which the FOVT line extends up to the higherpressure region as h is increased as shown in Fig. 2(a).In Fig. 6(b) we show the m - h curve at U fc = 1 . ε f values ranging from − .
32 to − .
36. The Kondotemperature T K at h = 0 is estimated as 0.0353, 0.0873,0.1346, 0.1611, and 0.1823 for ε f = − . − . − . − .
33, and − .
32, respectively. From these results, themechanism is understood as follows: At h = 0, T K is . Phys. Soc. Jpn. Full Paper
Author Name 7 h m (a) 0 0.02 0.06-0.36 e f =-0.32-0.33-0.34-0.35 h (b)0.080.04 Fig. 6. (color online) m - h curve (a) for ( ε f , U fc ) = ( − . , . − . , . ε f rangingfrom − .
36 to − .
32 at U fc = 1 .
42. In both cases, D = 1, V = 0 . n = 7 / originally large for ε f = − .
32 and − .
33, since the sys-tem is in the MV regime. However, by applying h , theQCP is induced, which makes reduces T K , since the sys-tem is forced to be closer to the Kondo regime by h .At a magnetic field h = h m where the QCP is reached,metamagnetism occurs with the singularity δm ∼ δh / as shown by Millis et al . (see Fig. 6(a)). The uniformsusceptibility diverges at the QCP of the FOVT. Thephenomena lead to the emergence of strong ferromag-netic fluctuations. Namely, valence fluctuations divergethere, which are essentially charge fluctuations. On theother hand, for ε f = − .
35 and − .
36, the QCP is notreached, so that no metamagnetism appears.Note that this mechanism is different from the ordi-nary metamagnetism emerging when the magnetic fieldis applied to the Kondo state, which has been dis-cussed as the origin of the metamagnetism observed inCeRu Si . Namely, the present metamagnetism iscaused by the field-induced QCP in the valence-crossoverregime at h = 0 (for moderate ε f and not large U fc
1, see Fig. 7).It corresponds to the collapse of antiferromagnetic cor-relations and the emergence of ferromagnetic fluctuationin a sharp h window.An interesting result is shown in Fig. 7, which ex-hibits a nonmonotonic h dependence of the QCP: As h increases, the QCP shows an upturn at approximately h = 0 .
04, which is comparable to T K at the QCP for h = 0. The upturn of the QCP has also been confirmedfor a constant density of states N ( ε ) = 1 / (2 D ). Thisnontrivial field dependence of the QCP appears in thevalence-crossover regime for U fc < U QCPfc in Fig. 7 insharp contrast to the regime for U fc > U QCPfc , where T v is monotonically suppressed by h as shown in Fig. 2(b). The nonmonotonic behavior can be understood fromthe structure of the valence susceptibility χ v , which isgiven essentially by the RPA, as discussed in ref. 6. -0.37 -0.36 -0.35 -0.34 -0.33 -0.321.401.451.50 f U f c h e Kondo Mixed Valence
Fig. 7. (color online) Ground-state phase diagram in the planeof U fc and ε f for D = 1 and V = 0 . n = 7 /
8. The FOVTline with a QCP for h = 0 .
00 (open triangle), h = 0 .
01 (filledtriangle) h = 0 .
02 (filled inverse triangle), h = 0 .
03 (filled star), h = 0 .
04 (filled diamond), h = 0 .
05 (filled square), and h = 0 . h , whichis shown as a visual guide. The dashed line represents the valence-crossover points at which χ v has a maximum as a function of ε f for each U fc at h = 0 . Namely, it is given as χ v ( q ) ≈ χ (0)fc ( q )1 − U fc χ (0)fc ( q ) , (11)where χ (0)fc is the bubble diagram composed of f and con-duction electrons. In the Kondo regime ( h < ∼ T K ), wheref electrons have a predominant spectral weight at approx-imately ǫ ∼ ε f with width ∆ ≃ πV N ( ε F ), χ (0)fc is esti-mated as χ (0)fc ≈ / | ε f | and is shown to be an increasingfunction of h . Therefore, U QCPfc decreases as h is applieduntil it reaches around h ∼ T K , and | ε QCPf | ≈ U QCPfc also decreases. For h > ∼ T K , mass enhancement ( ∼ /z )is quickly suppressed and the MV regime is reached.Then, χ (0)fc is given as χ (0)fc ≈ / ∆ < / | ε f | with us-ing the help of shift of the f level towards the Fermilevel, i.e., ε f → ε f + U fc δ ¯ n c ( δ ¯ n c being the change inthe number of conduction electrons per site due to entryinto the MV regime), so that U QCPfc becomes larger than U QCPfc ( h ∼ T K ). The magnetic field h m at QCP when CEP collapsesin a magnetic field corresponds to the difference in T K between h = 0 and h = h m : h m ∼ ∆ T QCPK = T QCPK ( h = 0) − T QCPK ( h = 0) . (12)A new energy scale distinct from T K reproduces the close-ness to the valence QCP. Under a magnetic field, theproximity of the intermediate-valence crossover regime toQCP can lead to the emergence of metamagnetism witha jump of m without initially showing the temperature-driven FOVT at h = 0. Thus, it is a field-reentrantFOVT. J. Phys. Soc. Jpn.
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Author Name m (a) h / t n f n c (b) -6 -4 -2 0 200.20.40.60.81 n f ε f cf Fig. 8. (color online) Magnetization process for ε k = − k ), V = 1, U = 10 , ε f = − U fc = 1 at n = 7 / T = 0 DMRG method: (a) m - h curve (filled circle). An arrowindicates the metamagnetic transition. Inset: ε f dependence of ¯ n f extrapolated to the bulk limit for U fc = 0 (open circle), U fc = 1(filled triangle) and U fc = 2 (open triangle) at h = 0. An arrowindicates ε f = −
1. (b) ¯ n f (filled triangle) and ¯ n c (open square).Inset: Lattice structure used in the calculation. In (a) and (b),dashed lines represent h = h m . To examine the mechanism more precisely, we ap-ply the density-matrix-renormalization-group (DMRG)method
57, 58 to eq. (2) in one dimension. Since valencefluctuations are basically ascribed to be of atomic ori-gin, the fundamental properties are expected to be cap-tured even in one dimension. We show here the re-sults for V = 1 and U = 10 in eq. (2) at n = 7 / N = 40 sites (open-boundary condi-tion), as illustrated in the inset of Fig. 8(b). Here, thetransfer term for conduction electrons is expressed as − P N − i =1 ,σ ( c † i,σ c i +1 ,σ +H . C . ). This lattice may be regardedas a one-dimensional mimic of CeIrIn and YbXCu ,which will be discussed in detail in § h = 0, the relation of ¯ n f vs ε f for U fc = 0 .
0, 1.0, and2.0 is shown in the inset of Fig. 8(a). As U fc increases, thechange in ¯ n f as a function of ε f becomes sharp. We showin Fig. 8(a) the magnetization m = P i h S f zi + S c zi i /N inthe MV state for ¯ n f indicated by an arrow in the inset ofFig. 8(a), which is obtained at h = 0. A plateau appearsat m = 1 − n = 1 /
8, which is expected to disappear ifwe take a more realistic choice of parameters, e.g., themomentum dependences of V and ε f . The main result isthat metamagnetism emerges, as indicated by an arrow.The increase in ¯ n f with a simultaneous decrease in ¯ n c at h = h m is shown in Fig. 8(b). It is caused by the field-induced extension of the QCP to the MV regime. Namely,these results indicate that the mean-field conclusion isnot altered even after properly taking into account thequantum fluctuations and electron correlations.To further explore the nature of this metamagnetism, we calculate P i h S f zi i /N and P i h S c zi i /N , and we findthat h S c zi i decreases slightly at h = h m , while h S f zi i increases considerably. Since the Kondo cloud is stillformed even at h = h m , i.e., h S f i · S c i i <
0, the decrease of h S c zi i is ascribed to the field-induced Kondo effect , whichis a consequence of the energy benefit by both the Kondoeffect and the Zeeman effect. Although this mechanismitself has been known to work in the Kondo regime, this result shows that such a mechanism works in theMV regime as a driving force of the field-induced valenceQCP. The DMRG calculation has shown that the magneticsusceptibility diverges at the QCP of the FOVT under amagnetic field, as shown in Fig. 8(a). It is consistent withthe slave-boson mean-field theory shown in Fig. 6(a). Thesimultaneous divergence of the magnetic and valence sus-ceptibilities at the QCP has been confirmed by the unbi-ased calculation; in addition the one-dimensional calcu-lation has been shown to be not special. It captures theessential physics of valence transition. The main reason isthe locality of the valence transition. Namely, the valencetransition has a local atomic origin. Because of this localnature, its basic properties and the ground-state phasediagram of the valence transition do not depend on spa-tial dimensions. Actually, the DMRG calculation in onedimension has been known to give essentially the samephase diagram determined by the slave boson mean fieldtheory (see ref. 23). Recently, the same phase diagramhas also been obtained by the dynamical mean field the-ory in infinite dimension. The mean-field result supported by the DMRG resultimplies that the RPA approach described in § z d = 3, and the con-dition d + z d ≥ d = 3-dimensional systems. This is a condition for the third-order term in the free-energy expansion to be irrelevant,as discussed by Hertz for the fourth-order term. Then,the universality class of the valence QCP essentially be-longs to the Gaussian fixed point, which justifies the RPAapproach.
As shown in §
2, the FOVT temperature T v ( h ) issuppressed by applying h . We note, however, that thetemperature dependence of T v ( h ) can change accordingto the location in the ground-state phase diagram (seeFig. 7). When the system is located at the deep first-ordertransition side, i.e., at U fc ≫ U QCPfc in Fig. 7, T v ( h ) is adecreasing function of h , as shown in Fig. 2(b). Actually,the proof by the Claudius-Clapeyron relation is basedon the fact that m K > m MV and S K > S MV at T v ( h ) ineq. (1); this is justified at the deep first-order transitionside. Ce metal and YbInCu correspond to this case.On the other hand, near the QCP ( U fc ∼ U QCPfc ) as . Phys. Soc. Jpn.
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Author Name 9 well as in the valence-crossover regime at U fc < U QCPfc ,a different situation can arise: In the case that ¯ n f at theQCP is not very close to 1 (as is certainly the case forYb systems ), the above relations on magnetization andentropy can change. In addition, details of the trajectoryline of the QCP under h may severely affect the h de-pendence of T v ( h ). For example, CeIrIn , which will bediscussed in detail in the §
4, is considered to be locatedin the valence-crossover regime for U fc < U QCPfc and thefield-induced T v ( h ) is considered to increase under h , incontrast to those in the cases of Ce metal and YbInCu .We have already stressed that a valence-crossover sur-face exists in the T - ε f - U fc space as shown in Fig. 5(a).Hence, even for U fc ≪ U QCPfc , the valence crossover sur-face will be induced by applying a magnetic field, givingrise to an increase in magnetization. The enhancementof the magnetic susceptibility at a certain magnetic field(see Figs. 7 and 5(a)) will lead to a pseudo-metamagneticeffect. Thus, the valence-crossover temperature T ∗ v ( h )also forms a line in the T - h phase diagram.When T ∗ v ( h ) is induced by applying h , if the systemis close to the QCP (namely, is located closely to thefilled circles in Fig. 7 under h ≥ T - h phase diagram: (a) The T -linear resistivity appears in the wide- T range when h approaches h ∗ v at which T ∗ v ( h ) becomes zero. (b) Residualresistivity is enhanced toward h ∗ v and has a maximum at h ∗ v . (c) Magnetic susceptibility has a peak at T = T ∗ v ( h ).(d) NQR frequency changes sharply at T = T ∗ v ( h ), sincethe charge distribution at the Ce or Yb site and its sur-rounding ions changes owing to the valence change of Ceor Yb, leading to the change in their electric-field gra-dient. (e) The lattice constant shows a sharp change at T ∗ v ( h ) and hence the magnetostriction changes sharply.When the FOVT T v ( h ) is induced by h , the abovephysical quantities show discontinuous jumps at T = T v ( h ). If the system is close to the QCP, valence-fluctuation-induced anomalies such as the T -linear re-sistivity will also be observed even in this case. Let usconduct a test with experiments.In a series of Yb- and Ce-based compounds, the abovepredictions have been actually observed, which will bediscussed in detail in the next section.
4. Explanation for YbXCu and CeYIn We here discuss the potentiality of our theory to re-solve outstanding puzzles observed in Yb- and Ce-basedsystems. First, we show how our results explain theisostructural FOVT observed in YbInCu and the sharpcontrast between YbAgCu and YbCdCu in their mag-netic responses. Second, we focus on the peculiar mag-netic response in CeIrIn , where the first-order transitionline emerges in the temperature-magnetic-field phase dia-gram, giving rise to non-Fermi liquid behavior. Third, weargue that the first-order like disappearance of antiferro-magnetism (AF) and the change of de Haas-van Alphen(dHvA) signal observed in CeRhIn at P ∼ . h >
10 Tesla may be explained byour model. systems4.1.1 Isostructural FOVT in YbInCu YbInCu is known as a typical Yb compound that ex-hibits the isostructural FOVT at T = 42 K
10, 11, 61 be-tween the high-temperature phase with Yb +2 . and thelow-temperature phase with Yb +2 . . Namely, in thehole picture, ¯ n f jumps from 0.97 to 0.84 as temperaturedecreases. This can be understood qualitatively from theresult shown in Fig. 5(a): in the FOVT region, the largerhole-density phase is realized in the high- T phase (theKondo phase), because of the free-energy gain due tothe larger entropy. Since this high- T phase has a smallerf-electron number, the volume of the system is consideredto be small in comparison with that for the low- T phase.Hence, as temperature decreases, the first-order transi-tion to the smaller hole-density phase (the MV phase) isrealized with volume expansion. The collapse of T v undera magnetic field was found in macroscopic magnetizationmeasurement, explained in phenomenological approachand well confirmed by microscopic X-ray experiments. As mentioned below eq. (2), the band-structure calcu-lations as well as as photoemission measurements suggestthe importance of the V and U fc terms in eq. (2) at theFOVT in YbInCu . Here, we should also note the pos-sibility that band structures such as semimetallic struc-tures also play an important role in the FOVT as pointedout in refs. 48 and 65. Although the accurate estimationof the values of the model parameters of a model Hamil-tonian on the basis of first-principles calculations is animportant task in the future, we here discuss the basicproperties of YbXCu on the basis of eq. (2). and YbCdCu When X=In is used to replace the other elements,YbXCu does not show the FOVT, but shows merelythe valence crossover. For example, X=Ag andX=Cd
68, 69 for YbXCu show neither the FOVT nor themagnetic transition and they have the paramagnetic-metal ground state. The Kondo temperatures of bothmaterials estimated from the magnetic susceptibilitydata are nearly the same, T K ∼
200 K. A strikingpoint is that when the magnetic field is applied to thesesystems, only X=Ag shows a metamagnetic behavior inthe magnetization curve, while X=Cd merely shows thegradual increase in magnetization. Our results explain why such a sharp increase in mag-netization emerges only for X=Ag, but not for X=Cdin spite of the fact that both have nearly the same T K ’s. Figure 9 shows the schematic contour plot of thef-hole number per site, ¯ n f , which can also be regardedas the contour plot of the Kondo temperature T K inthe U fc - ε f plane, because T K is a function of ¯ n f , as in T K ∝ (1 − ¯ n f ) / (1 − ¯ n f / In the small- U fc and small- ε f region, ¯ n f approaches 1, so that T K becomes small. In thelarge- U fc and large- ε f region, ¯ n f is smaller than 1, givingrise to a large T K . As ( U fc , ε f ) approaches the QCP fromthe valence-crossover regime for U < U
QCPfc , contour linesof T K ’s get close, and at the FOVT line T K ’s show dis-continuous jumps. Since the compounds with X=Ag andX=Cd have nearly the same T K , both are considered to Full Paper
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CdAg T QCP(b) e f U fc e f U fc (a) CdAg
Fig. 9. (Color online) (a) Schematic contour plot of ¯ n f , i.e., theKondo temperature T K in the U fc - ε f plane for the Hamiltonian(1) at h = 0. The first-order valence-transition line (solid line)terminates at the QCP (filled circle). The arrow represents adistance of ∼
40 T between the QCP and YbAgCu . YbCdCu is located far away from the QCP. (b) Schematic T - ε f - U fc phasediagram. YbAgCu reaches the valence-crossover surface with adistance of about T = 40 K. YbCdCu is too far from the QCPand hence has too large a distance from the valence-crossoversurface. be located near the same contour area (see Fig. 9). How-ever, the compound with X=Ag appears located moreclosely to the QCP of the FOVT than X=Cd, with avalence fluctuation energy, i.e., equivalent to a Zeemanenergy of approximately 40 T. A sharp increase in mag-netization will appear in the case of X=Ag; by applying h ∼
40 T, the field-induced QCP of the valence transition(or sharp valence crossover) is reached.A sharp contrast between YbAgCu and YbCdCu was also observed in the T dependence of the magneticsusceptibility χ ( T ). Although both show nearly thesame χ (0)’s reflecting the fact that both have nearly thesame T K ’s, a broad maximum in χ ( T ) appears at ap-proximately T = 40 K only in YbAgCu , but a mono-tonic decrease in χ ( T ) appears in YbCdCu as T in-creases. This can be understood if YbAgCu reachesthe valence crossover surface from a distance of approxi-mately T = 40 K, while YbCdCu reaches it at too largea temperature T interval, as shown in Fig. 9(b). Since themagnetic susceptibility χ ( T ) has a peak at the valencecrossover temperature T ∗ v (see Figs. 6(a) and 8(a)), thepeak of χ ( T ) at T = 40 K in YbAgCu is explained nat-urally. Indeed the proximity of YbAgCu to the QCP isreflected in the thermal volume expansion directly linkedto the pressure dependence of the entropy: the volumeexpansion was observed below T = 40 K simultane-ously with the sharp valence crossover from Yb +2 . toYb +2 . in YbAgCu , in contrast to YbCdCu . Hence, the viewpoint of the closeness to the QCP of theFOVT expressed in Figs. 9(a) and 9(b) not only explainsthe metamagnetic behavior but also the peak of the uni-form susceptibility χ ( T ) consistently. Both phenomenaare coupled with the local origin of each phenomenon. Our results also explain the peculiar magnetic re-sponse in CeIrIn , which shows a jump in the m - h curveat 42 T. Capan et al . have observed that resid-ual resistivity increases, and the Sommerfeld constant in the specific heat shows a diverging increase toward themetamagnetic-transition field ∼
25 T. Furthermore, theyhave found that as h increases, the power of the resistiv-ity ρ ∼ T α at low temperatures decreases from α = 1 . α = 1 . ρ ∝ T . plot. Exper-imental effort should be made to properly confirm theexpectation that the T -linear resistivity and the peak ofthe residual resistivity will be observed around h ∼
25 T.Capan et al . have pointed out that these anomalous be-haviors may be related to the metamagnetic transitionthat forms a first-order-transition line in the T - h phasediagram and that this may be the origin of the non Fermi-liquid normal state observed at h = 0, although its mech-anism has not yet been clarified.Our results suggest that the mechanism is the valencefluctuation: This can be readily understood if CeIrIn islocated inside the enclosed area of the QCP line for h = 0in Fig. 7. Namely, at h = 0, the system is considered to belocated in the gradual valence-crossover regime (i.e., for U < U
QCPfc in Fig. 7), since no evidence of the first-ordertransition has been observed at any physical quantitiesas a function of T at h = 0. However, when h is applied,the QCP of the FOVT reaches and eventually goes acrossthe point of the system, causing metamagnetic transitionin the magnetization curve. Since it has been shown the-oretically that the residual resistivity is enhanced nearthe QCP and that the T -linear resistivity is expectedin a wide- T region, the observed non-Fermi-liquid be-havior is quite consistent. Furthermore, the first-ordertransition emerges in the T - h phase diagram in agree-ment with our predictions.We here remind the readers of the fact that CeIrIn and CeCoIn have nearly the same crystalline-electric-field (CEF) structures and that a change in CEF levelunder a magnetic field cannot explain the metamag-netic increase in magnetization in CeIrIn , as pointedout in ref. 72. Also note that almost the same Fermisurfaces in both systems have been obtained by thede Haas-van Alphen measurements as well as the first-principles band structure calculations.
77, 78
However, nei-ther the enhancement of residual resistivity nor themetamagnetic-transition line in the T - h phase diagramhas been observed in CeCoIn , in contrast to that inthe case of CeIrIn at zero pressure. These results reem-phasize that a distinct energy scale other than theKondo temperature is indispensable for understandingthe Ce115 systems.In order to directly verify the above scenario, detec-tion of the Ce valence change at T v ( h ) in the T - h phasediagram is highly desired by measurements such as theX-ray adsorption spectra and the NQR electric gradient.The notable result is that h scan can lead to a h reen-try into the valence critical domain. Our approach as-sumes a paramagnetic ground state; in some ( U fc , ε f )windows, long-range magnetism will appear as in thecase of YbInCu under pressure ( P > Si may be a singular spectacularcase where in the ( h, T ) phase diagram the magnetismand valence fully interact. Such interplay also occurs in . Phys. Soc. Jpn.
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Author Name 11
CeRhIn . At zero pressure, CeRhIn is a heavy-fermionantiferromagnet with the Neel temperature T N ≈ . P > P ≈ . Sharp valence crossover has also been suggested inCeRhIn under pressure near P = 2 . ρ ( T = 2 .
25 K ≈ T SC ), with T SC being thesuperconducting transition temperature showing a peak,as well as the T -linear resistivity in the wide- T rangeemerges. Hence, the sharp valence crossover origi-nating from the QCP of the valence transition may beinduced by applying pressure to CeRhIn . Namely, theviewpoint of the closeness to the QCP of the FOVT isimportant in elucidating the T - h - P (chemical doping)phase diagram of these compounds in a consistent way.The sharp peak of ρ ( T = 2 .
25 K) cannot be explainedby the many-body correction due to critical AF fluctua-tions, but can be understood by the enhanced valencefluctuations. Other lines of evidence for the crucial roles of FOVTin CeRhIn under the magnetic field are as follows:1) According to ref. 81, near P = 2 . m ∗ /m and the coefficient of the T term of theresistivity A exhibit a rather sharp enhancement. How-ever, m ∗ /m scales with A / , suggesting that the massenhancement is mainly driven by the “local correlationeffect” (but not due to critical antiferromagnetic fluctua-tions) just as in the case of the metamagnetic transitionof CeRu Si discussed in refs. 55 and 56. The enhance-ment of m ∗ /m can be interpreted as that of the quasi-particle density of states near the hybridization gap orpseudo gap, which can be approached at approximately P = P v ≃ .
81, 83, 84 and the change of the dHvA signalobserved in CeRhIn at P = 2 . can be naturally understoodas a FOVT induced by the magnetic field.2) According to ref. 81, the upper critical field H c2 ex-hibits a rather sharp peak at P = P v where the Fermisurface exhibits sharp change from “localized” to “itin-erant” under a magnetic field, while the superconduct-ing transition temperature T sc is essentially flat around P = P v . This fact can be interpreted as an effect of thegrowth of the paring interaction due to the effect of ap-proaching the magnetic-field-induced critical point of va-lence transition. Such behavior reminds us of the case ofUGe at P x = 1 . H c2 with a concaveshape.
86, 87
3) The P dependence of the low-temperature resis-tivity ρ ( T = 2 .
25 K) has a peak at P = 2 . T -linear dependence of ρ ( T ) in thevicinity of P = 2 . can be naturally explained by the present mechanism.
16, 20
We stress here that the “localized”-to-“itinerant”change in electron character reported in the dHvA mea-surement can be explained by the Ce-valence jump orsharp crossover at P = 2 .
23, 56 i.e., c-f hybridization is always switched onin sharp contrast to that in the Kondo breakdown sce-nario. Our mechanism is also consistent with the ex-perimental fact that the effective mass of electrons is en-hanced even at P = 0 with the Sommerfeld constant γ ≈
56 mJmol − K − , which is about 10 times en-hanced from the LaRhIn value,
78, 83 strongly indicatingthe AF state with the c-f hybridization. Furthermore, themass enhancement observed toward P = 2 .
81, 85 in-side the AF phase can also be explained by the presentmechanism. Hence, it should be stressed that the valenceQCP itself is the source of locality emerging in CeRhIn without invoking a collapse of Kondo temperature.As shown by the phase diagram of CeRh x Ir − x In , CeIrIn at ambient pressure is moderately far from theAF QCP with a distance of about x ∼ .
5. A slight in-crease in the nuclear spin-lattice relaxation rate 1 /T T indicates that moderate spin fluctuations may exist atleast at ambient pressure. It has been also reportedthat the magnetotransport measurements under pres-sures can be understood from the effects of AF spin fluc-tuations. However, the clear difference between CeIrIn and CeCoIn emerging in the T - h phase diagram men-tioned in § , whose transition tempera-ture increases even though AF spin fluctuation is sup-pressed under pressure. Since superconductivity will beenhanced near the valence QCP,
21, 23 the present view-point offers a new scenario that the proximity of QCP ofthe FOVT is the main origin of the superconductivity.The superconducting window reveals phenomenon otherthan spin fluctuation; the occurrence of superconductiv-ity is a unique opportunity for scanning through differentpairing channels. We have already pointed out in the in-troduction that in many heavy fermion compounds evenfor the magnetic QCP the interplay between spin and va-lence fluctuations is the main reason for collapse of thelong-range magnetism.
Detailed discussions for each material have been givenin from § § are clearly explained by our mechanism. Theeffects of the semimatallic band structure on the FOVTas well as the qualitative evaluation of U fc are issues tobe studied in the future for the complete understandingof the valence transition of YbInCu .2) The metamagnetism at h = 40 T and the peak Full Paper
Author Name structure in χ ( T ) at T = 40 K in YbAgCu but notin YbCdCu are naturally explained by our mechanism.The experimental fact that the valence crossover occursat T = 40 K for h = 0 in YbAgCu is consistent with ourtheory. The direct observation of the Yb-valence changeunder a magnetic field at approximately h = 40 T at lowtemperatures is highly desired to confirm our theoreticalproposal.3) We point out that the field-induced FOVT explainsthe T - h phase diagram as well as the non-Fermi-liquidcritical behavior observed in CeIrIn . It has been re-ported that magnetotransport measurement under pres-sure can be explained by AF spin fluctuations. Wethink that, in addition to the influence of the AF QCP,the viewpoint of the closeness to the QCP of the FOVTis necessary for the comprehensive understanding ofCeIrIn . To examine our theoretical proposal, it is highlydesired to experimentally determine whether the changein Ce valence occurs at the FOVT T v ( h ) in the h - T phasediagram.4) We point out that the anomalous behaviors at ap-proximately P ∼ . canbe naturally explained if the FOVT or sharp valencecrossover of Ce takes place at such a pressure. We thinkthat such behavior cannot be explained solely by the AFQCP scenario. Our picture gives a natural explanation ofthe origin of the locality as well as the non-Fermi liquidbehavior without relying on artificial assumptions suchas the Kondo breakdown. It is highly desired to exper-imentally determine whether the Ce valence changes atapproximately P ∼ . is neces-sary for elucidating the P - h - T phase diagram toward acomplete understanding of the Ce115 system.
5. Conclusions
We have clarified the mechanism of novel phenom-ena in heavy-fermion systems emerging under a magneticfield and have discussed the significance of the proxim-ity to the FOVT as a potential origin of the anoma-lous electronic properties of Ce- and Yb-based heavy-fermions. We have shown that FOVT temperature issuppressed by applying a magnetic field, which correctlyconnects the high-temperature result derived from theatomic picture of the valence-fluctuating ion to the zero-temperature limit consistently with the observations inCe metal and YbInCu . The important result is that evenin intermediate-valence materials, by applying a mag-netic field, the QCP of the FOVT is induced. The QCPshows a nonmonotonic field dependence in the ground-state phase diagram, giving rise to the emergence ofmetamagnetism with diverging magnetic susceptibility.The driving force of the field-induced QCP is clarifiedto be a cooperative mechanism of the Zeeman effect andthe Kondo effect, which creates a distinct energy scalefrom the Kondo temperature.The use of an extended periodic Anderson model ex-plains how quite similar valences may lead to quite differ-ent h responses. Our model clarifies why metamagneticbehavior appears in YbAgCu but not in YbCdCu , in spite of the fact that both have nearly the same Kondotemperatures. The closeness to the QCP of the FOVTgives the distinct energy scale, which is a key conceptto understanding the properties of YbXCu (X=In, Ag,and Cd) systematically. This viewpoint also explains pe-culiar magnetic response in CeIrIn where the first-ordertransition line in the T - h phase diagram appears withfield-induced critical phenomena. The viewpoint of thecloseness to the QCP of the FOVT is also indispensablefor understanding CeYIn (Y=In, Co, and Rh) system-atically.As shown in the present study, the QCP of the FOVTand its fluctuations exerts profound influences on Ce-and Yb-based materials as a potential origin of anoma-lous behavior. Most of such materials are considered tobe located in the intermediate valence regime, i.e., in theregion for U fc < U QCPfc in Fig. 7 due to the intersite ori-gin of U fc . However, by applying a magnetic field, thevalence-crossover surface as well as the critical point isinduced, which causes various anomalies described in thispaper. The ( P, h ) valence transition mechanism clarifiedin this paper can be a key origin of unresolved phenom-ena in the family of the materials.
Acknowledgments
S. W. and K. M. thank S. Wada and A. Yamamotofor showing us their experimental data prior to publi-cation, with enlightening discussions on their analyses.They also acknowledge H. Harima for helpful discus-sions about the band structures of Ce- and Yb-basedheavy-fermions as well as their model parameters. S. W.is grateful to T. Miyake for estimating the magnitudeof the Coulomb repulsions in the model for Ce metalbased on first-principles calculations. This work is sup-ported by a Grant-in-Aid for Scientific Research on Pri-ority Areas (No. 18740191) from the Ministry of Edu-cation, Culture, Sports, Science, and Technology, Japan,and is supported in part by a Grant-in-Aid for ScientificResearch (No. 19340099) by the Japan Society for thePromotion of Science (JSPS). J. F. is supported by theGlobal COE program (G10) of JSPS for supporting hisvisit of the Graduate School of Engineering Science atOsaka University where the final stage of this work wasperformed. Part of our computation has been performedat the supercomputer center at the Institute for SolidState Physics, the University of Tokyo.
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