Ubiquity of Unconventional Phenomena Associated with Critical Valence Fluctuations in Heavy Fermion Metals

TL;DR

This paper discusses the widespread unconventional phenomena in heavy fermion metals linked to critical valence fluctuations, proposing that sharp valence crossover phenomena occur in several compounds under pressure and magnetic field.

Contribution

It introduces the idea that critical valence fluctuations are a unifying explanation for unconventional behaviors in heavy fermion metals, supported by experimental and theoretical analysis.

Findings

Unconventional phenomena are common in heavy fermion metals.

Sharp valence crossover phenomena are predicted in specific compounds.

Pressure and magnetic field tuning induce valence changes.

Abstract

Ubiquity of unconventional phenomena observed in a series of heavy fermion metals is discussed on the basis of an idea of critical valence fluctuations. After surveying experimental aspects of these unconventional behaviors in prototypical compounds, CeCu_2(Si,Ge)_2, under pressure, we propose that sharp valence crossover phenomena are realized in CeCu_6, CeRhIn_5, and Ce(Ir,Rh)Si_3 by tuning the pressure and the magnetic field simultaneously, on the basis of previous results for an extended Anderson lattice model with the Coulomb repulsion U_fc between localized f-electron and itinerant conduction electrons.

Tables

Table 1. Table 1: Theoretical results for a series of physical quantities by critical valence fluctuations (CVF) giving the exponent ζ 𝜁 \zeta as 0.5 ​ < ∼ ζ ​ < ∼ 0.7 0.5 fragments fragments similar-to 𝜁 fragments fragments similar-to 0.7 0.5\mathop{\vtop{\halign{#\cr$\hfil\displaystyle{<}\hfil$\crcr\kern 1.0pt\nointerlineskip\cr$\,\,\sim$ \crcr\kern 1.0pt\cr}}}\limits\zeta\mathop{\vtop{\halign{#\cr$\hfil\displaystyle{<}\hfil$\crcr\kern 1.0pt\nointerlineskip\cr$\,\,\sim$ \crcr\kern 1.0pt\cr}}}\limits 0.7 depending on the region temperature T 𝑇 T higher than T 0 subscript 𝑇 0 T_{0} , the extremely small temperature scale (see §6 and §7), and unconventional criticality observed in a series of materials. T 1.5 → T → superscript 𝑇 1.5 𝑇 T^{1.5}\to T in the column ρ ​ ( T ) 𝜌 𝑇 \rho(T) means that T 𝑇 T dependence of ρ ​ ( T ) 𝜌 𝑇 \rho(T) crosses over from T 1.5 superscript 𝑇 1.5 T^{1.5} at T < T 0 𝑇 subscript 𝑇 0 T<T_{0} to T 𝑇 T at T > T 0 𝑇 subscript 𝑇 0 T>T_{0} around T ∼ T 0 similar-to 𝑇 subscript 𝑇 0 T\sim T_{0} . The symbol * indicates that there is no available experiment. In the last low, theoretical prediction on the conventional criticality associated with antiferromagnetic (AF) quantum critical point (QCP) are shown, in which − T 1 / 2 superscript 𝑇 1 2 -T^{1/2} and − T 1 / 4 superscript 𝑇 1 4 -T^{1/4} indicate decrement from some constant values, respectively.

Theories & Materials	$\rho \left(T\right)$	$C\left(T\right)/T$	$\chi \left(T\right)$	$1/T_{1}T$	Refs.
CVF	$T^{1.5}\to T$	$-\log T$	$T^{-\zeta }$	$T^{-\zeta }$	[11, 12, 13]
YbCu5-xAl${}_x{}^{}(x\simeq 1.5)$	$T^{1.5}\to T$	$-\log T$	$T^{-0.66}$	*	[1, 2]
YbRh2Si2	$T$	$-\log T$	$T^{-0.6}$	$T^{-0.5}$	[3, 4]
$\beta$-YbAlB4	$T^{1.5}\to T$	$-\log T$	$T^{-0.5}$	*	[7]
$\alpha$-YbAl1-xFexB4 ($x\simeq 0.014$)	$T^{1.5}\to T$	$-\log T$	$T^{-0.5}$	*	[8]
Yb15Al34Au51	$T$	$-\log T$	$T^{-0.51}$	$T^{-0.51}$	[9]
Yb14Al35Au51 at $P\simeq 1.8$ GPa	$T$	*	$T^{-0.51}$	*	[19]
AF QCP	$T^{3/2}$	$-T^{1/2}$	$-T^{1/4}$	$T^{-3/4}$	[14, 15, 16]

Equations

$$\frac{m^{\ast }}{m_{\mathrm{band}}}=\frac{1-n_{\mathrm{f}}/2}{1-n_{\mathrm{f}}},$$

$$H_{\mathrm{EPAM}}$$

$${\epsilon }_{\mathrm{f}}+n_{\mathrm{c}}{U}_{\mathrm{fc}}=\mu ,$$

$$H=H_{\mathrm{EPAM}}-h\sum _i(S_{\mathrm{f}i}^z+S_{\mathrm{c}i}^z),$$

$${ϵ}_k=-2t(\cos k_{x}a+\cos k_{y}a),$$

Full text

1 Introduction

In the past decade or so, it gradually turned out that the critical-valence-transition or sharp-valence-crossover phenomena in heavy fermion metals is rather ubiquitous than first thought in the beginning of the present century. Indeed, unconventional quantum critical phenomena, which cannot be understood on the basis of the quantum criticality associated with magnetic transitions, have been observed in a series of heavy fermion metals, YbCu5-xAlx (x=3.5) [1, 2], YbRh2Si2 [3, 4], YbAuCu4 [5, 6], $\beta$-YbAlB4 [7], $\alpha$-YbAl1-xFexB4 (x=0.014 ) [8], and quasi-crystal compound Yb15Al34Au51 [9] and quasi-crystal-approximant Yb14Al35Au51 under pressure $P\simeq 1.8$GPa as well [10]. The non-Fermi liquid behaviors observed in these compounds can be explained in a coherent way by a scenario based on the critical valence fluctuations (CVF) using the mode-mode coupling approximation for CVF [11, 12], as summarized in Table 1. A recent highlight is that the so-called $T/B$ scaling behavior of the magnetization, observed in $\beta$-YbAlB4 [7, 17, 18], and quasi-crystal related compounds Yb15Al34Au51 [19] and Yb14Al35Au51 [10], has been theoretically derived by taking into account the effect of the magnetic field in the mode-mode coupling approximation scheme [20, 21].

The unconventional phenomena associated with sharp valence crossover have also been observed in a series of Ce-based heavy fermion metals, while the complete systematic behaviors shown in Table 1 were not observed because such Ce-based compounds are not considered to be just at the valence criticality but in the sharp valence crossover region. CeCu2Ge2 is the first example in which anomalous properties characteristic to the sharp valence crossover of Ce ion under pressure around $P=P_{\rm v}$ was reported [22, 23]. In particular, the superconducting transition temperature exhibits a drastic increase by about triple of that at around the magnetic critical pressure. The $T$-linear dependence in the resistivity $\rho(T)-\rho_{0}$, with $\rho_{0}$ being the residual resistivity, was also ovserved at around $P=P_{\rm v}$, together with a huge enhancement of $\rho_{0}$. After that, similar behaviors have been reported in CeCu2Si2 [24], CeCu2Si1.8Ge0.2 [25], and CeRhIn5 [26, 27], which can be comprehensively understood on the basis of a valence crossover scenario [28, 29]. It was also predicted [30, 31] that the position of the critical point of valence transition is well controlled by the magnetic field, which opens a possibility of realizing the critical point by tuning both pressure and magnetic field simultaneously. Recently, a symptom of such a phenomenon was reported in CeCu6 [32] which is considered to be located in the crossover region of valence transition [28, 33]. This suggests that the puzzling non-Fermi liquid properties observed in CeCu6-xAux (x$\simeq$0.1) [34] should be revisited from the viewpoint of this CVF scenario.

The essence of these intriguing phenomena cannot be captured within the so-called the Doniach paradigm [35] that is essentially based on the Kondo lattice picture in which the valence of Ce and Yb ion is fixed as C3+ and Yb3+. In other words, we had to develop a conceptually new physics. The purpose of the present paper is to sketch the history and the present status of unconventional phenomena associated with sharp valence crossover or enhanced valence fluctuations in Ce-based heavy fermion metals, and to discuss how these phenomena are ubiquitous than thought previously. The organization of the paper is as follows:

In Sect. 2, we review how the existence of critical point of valence transition in heavy fermion metals was recognized by showing the case of CeCu2(Si,Ge)2.

In Sect. 3, we present some fundamental properties of the extended Anderson lattice model (including the Coulomb repulsion between f and conduction electrons) which is the minimal model for describing essential aspects of valence transition and valence fluctuations observed in Ce- or Yb-based heavy fermion metals.

In Sect. 4, we show that the magnetic field gives appreciable influence on the valence transition of Ce ion discussing the case of CeCu6.

In Sect. 5, we discuss how the change of the Fermi surface observed in CeRhIn5 under pressure is understood as a sharp valence crossover phenomenon in a unified fashion without relying on an idea of the destruction of c-f hybridization.

In Sect. 6, summary is given.

2 Sharp crossover in valence of Ce in CeCu2(Si,Ge)2

In this section, we discuss a series of experimental evidence suggesting that the valence of Ce exhibits a sharp crossover under pressure in CeCu2Si2 [24], resulting in a series of anomalous properties associated with the sharp valence crossover, while such a trend was first reported [22] and discussed theoretically [23] for CeCu2Ge2.

A drastic decrement in the $A$ coefficient of the $T^{2}$ term in the resistivity, $\rho(T)\simeq\rho_{0}+AT^{2}$, by about two orders of magnitude around the pressure $P=P_{\rm v}\simeq 4.5$ GPa, as shown in Fig.  1(c), suggests a sharp valence crossover occurs at around $P=P_{\rm v}$. Note that $T_{1}^{\rm max}$ is an increasing function of pressure $P$ and simulates the variation of $P$. The residual resistivity $\rho_{0}$ exhibits a sharp and pronounced peak there, as shown in Fig. 1(b). This implies that the effective mass $m^{*}$ of the quasiparticles also decreases sharply there, since $A$ is scaled by $(m^{*})^{2}$ [36]. This decrement of $m^{*}$ implies in turn a sharp change in the valence of Ce ion, deviating from Ce3+, considering the fact that the following approximate (but canonical) formula holds in the strongly correlated limit [37, 38]:

[TABLE]

where $m_{\rm band}$ is the band mass without electron correlations, and $n_{\rm f}$ is the f-electron number per Ce ion.

Such a sharp crossover in the valence of Ce ion gives rise to a sharp crossover of the so-called Kadowaki-Woods (KW) ratio [36], $A/\gamma^{2}$, where $\gamma$ is the Sommerfeld coefficient of the electronic specific heat, from that of a strongly correlated class to a weakly correlated one. Note that $\gamma^{-1}$ is related to the temperature $T_{1}^{\rm max}$, where the resistivity $\rho(T)$ exhibits maximum, as shown in the inset of Fig. 1(c). This indicates, as discussed in Ref. [39], that the mass enhancement due to the dynamical electron correlation is quickly lost at around $P=P_{\rm v}$ [39].

This physical picture based on the sharp valence crossover of Ce has been reinforced by 63Cu-NQR measurements in CeCu2Si2 at temperature down to $T=3.1\,$K and under pressures up to $P=5.5\,$GPa covering $P_{\rm v}\simeq 4.5\,$GPa [40, 41, 42]. Indeed, the NQR frequency ${}^{63}\nu_{\rm Q}$ rather sharply deviates at above 4 GPa from the linear $P$-dependence in the low pressure range ($P\leq 3.5\,$GPa). The $P$-dependence of the deviation from the linear dependence in ${}^{63}\nu_{\rm Q}$ is shown in Fig. 2b). This deviation was estimated to correspond to the change of the valence $\Delta n_{\rm f}={0.04}$ by the first principles calculations [42]. This degree of change is consistent with the decrease of mass enhancement by $\sim 3.9$ ($\sim 15.3$ in the $A$ coefficient of the resistivity, as shown in Fig. 1(c)), if the change in $n_{\rm f}$ would be from $n_{\rm f}=0.99$ to $n_{\rm f}={0.95}$.

In Ref. [42], the change of valence in Ce was estimated from that in NQR frequency $\nu_{\rm Q}$ as follows. On the basis of the LDA calculations, the pressure ($P$) dependence of $\nu_{\rm Q}$ for “LaCu2Si2” and “CeCu2Si2” was estimated with the use of the observed $P$ dependence of the lattice parameter, where f electrons are assumed to be completely localized in the case of “LaCu2Si2”, i.e., essentially in $(4{\rm f})^{0}$ configuration, and itinerant but not strongly correlated in the case of “CeCu2Si2”. Both show a linear $P$ dependence with similar slopes: d$\nu_{\rm Q}/{\rm d}P\simeq 0.103$MHz for “LaCu2Si2” and d$\nu_{\rm Q}/{\rm d}P\simeq 0.089$MHz for “CeCu2Si2”. However, observed NQR frequency $\nu_{\rm Q}$ of a real CeCu2Si2 is located in between because 4f electrons in CeCu2Si2 are not fully itinerant but have a localized character reflecting strong correlations among 4f electrons. Indeed, $\nu_{\rm Q}(P=0)\simeq 3.73$MHz for “LaCu2Si2” and $\nu_{\rm Q}(P=0)\simeq 3.02$MHz for “CeCu2Si2”, while the observed one for the real CeCu2Si2 is $\nu_{\rm Q}(P=0)\simeq 3.43$MHz. The deviation in $\nu_{\rm Q}(P)$ from a $P$ linear dependence shown in Fig. 2(b) reflects a change of electronic contribution to $\nu_{\rm Q}$, while the linear $P$ dependence reflects the lattice contribution. The amount of deviation $\Delta\nu_{\rm Q}$ from $P=3.9$ to $P=4.5$ is about $\Delta\nu_{\rm Q}\simeq-0.03$MHz, which is $3.9$% of the difference $[\nu_{\rm Q}($”CeCu2Si2”$)-\nu_{\rm Q}($”LaCu2Si2”$)]\simeq-0.77$MHz at around $P=3.9$-$4.5\,$GPa. The negative value of $\Delta\nu_{\rm Q}$ implies an increase in the itinerant character of 4f electrons, and the ratio of 4f electrons of acquiring itinerant character is estimated as $0.03/0.77\simeq 0.04$ of the localized component, suggesting $\Delta n_{\rm f}\simeq 0.04$.

The huge peak of $\rho_{0}$ at $P\sim P_{\rm v}$, as shown in Fig. 1(b), can be explained by the effect that the disturbance in the ratio of numbers of f and conduction electrons around impurity extends to the correlation length $\xi_{\rm v}$ which grows appreciably toward $P=P_{\rm v}$ giving rise to strong scattering of quasiparticles. The microscopic justification has been discussed in Ref. [43]. This is in contrast to the effect of AF critical fluctuations on $\rho_{0}$ which is rather moderate, as discussed in Ref. [44]. Thus, the critical pressure $P_{\rm v}$ can be clearly manifested by the maximum of $\rho_{0}$.

Subtle but systematic tendencies shown in Fig. 1 near $P=P_{\rm v}$ are that the peak of the $T_{\rm c}$ and the Sommerfeld coefficient $\gamma(T=2~{}{\rm K})$ appears at slightly lower pressure than $P_{\rm v}$. These behaviors can be understood on the basis of explicit theoretical calculations in which almost local valence fluctuations of Ce are shown to develop around the pressure where the sharp valence crossover occurs [24, 45, 28], as supported by the density-matrix-renormalization-group (DMRG) calculation [46].

Another salient property associated with the sharp crossover of the valence is that the temperatures $T_{i}^{\rm max}$ ($i=1,2$), corresponding to two maxima in the the rsistivity $\rho(T)$, merge at $P\simeq P_{\rm v}$ as shown in Fig. 3 [24]. Although the existence of the two peaks in $\rho(T)$ and the mergence of them under pressure can be shown theoretically as an effect of smearing crystalline-electric-field (CEF) splitting by the increase of the c-f hybridization under pressure in general [47], it is non-trivial that the mergence occurs at around $P=P_{\rm v}$. This behavior was also observed in CeCu2Ge2 [22], CeAu2Si2 [48], and CeAl2 [49], suggesting a generic property associated with the sharp valence crossover of Ce ion as argued below [24].

A fundamental wisdom is that the so-called Kondo temperature $T_{\rm K}$, related to $T_{i}^{\rm max}$ ($i=1,2$), depends crucially on the degeneracy $(2\ell+1)$ of local f-state: $T_{\rm K}\sim D\exp[-1/(2\ell+1)N_{\rm F}|J|]$, where $D$ is half the bandwidth of conduction electrons, $N_{\rm F}$ the density of states of conduction electrons at the Fermi level, and $J$ the c-f exchange coupling constant [50]. Furthermore, even though the sixfold degeneracy of local 4f-state is lifted by the CEF effect, leaving the Kramers doublet ground state with the excited CEF levels with excitation energy $\Delta_{\rm CEF}$, the Kondo temperature $T_{\rm K}$ is still enhanced considerably by the effect of excited CEF levels [51].

It is also crucial to take into account the fact that the practical degeneracy of CEF levels, relevant to the Kondo effect, is affected by the broadening $\Delta E$ of the lowest CEF level. If $\Delta E\ll\Delta_{\rm CEF}$, the degeneracy relevant to $T_{\rm K}$ is twofold. On the other hand, if $\Delta E>\Delta_{\rm CEF}$, it increases to fourfold or sixfold. The level broadening is given by $\Delta E\simeq z\pi N_{\rm F}|V|^{2}$ where $|V|$ is the strength of c-f hybridization, and $z$ is the renormalization factor which gives the inverse of mass enhancement in the case of lattice systems. Then, it is crucial that $\Delta E$ is very sensitive to the valence of Ce ion because $z^{-1}$ is essentially given by eq. (1). In particular, the factor $z$ increases from the tiny value in the Kondo regime, $z\sim(1-n_{\rm f})\ll 1$, and approaches to 1 in the so-called valence fluctuation regime.

Since the factor $\pi N_{\rm F}|V|^{2}\gg\Delta_{\rm CEF}$ in general for Ce-based heavy fermion metals, the ratio $\Delta E/\Delta_{\rm CEF}$, which is much smaller than 1 in the Kondo regime, greatly exceeds 1 across the valence transformation around $P\sim P_{\rm v}$, leading to the increase of the practical degeneracy of f-state, irrespective of a sharpness of the valence transformation. Therefore, $T_{1}^{\rm max}$ should merge $T_{2}^{\rm max}$, which corresponds to fourfold or sixfold degeneracy of 4 f-state due to the effect of broadening of the CEF ground level. This explains why $T_{1}^{\rm max}$ increases and approaches $T_{2}^{\rm max}$ at around $P=P_{\rm v}$.

On the other hand, there have been trials of explaining this mergence of $T_{i}^{\rm max}$ ($i=1,2$) at around $P=P_{\rm v}$ as a phenomenon caused by a meta-orbital transition among CEF levels [52] or an interchange of CEF level scheme itself [53]. The latter theoretical prediction contradicts with experimental measurements both of the inelastic neutron scattering [54] and the non-resonant X-ray scattering (NRXS) [55, 56] which shows that the ground state of the CEF levels in CeCu2Si2 is $\Gamma_{7}^{1}\simeq-0.88|\pm 5/2\rangle+0.47|\mp 3/2\rangle$. The former prediction [52] does not immediately contradict with the result of NRXS measurement [56], while it is not so evident whether the condition for the meta-orbital transition to occur is satisfied in the CEF states of CeCu2Si2. Reference [52] assumes much larger c-f hybridization at the first-excited CEF level than that at the ground CEF level. However, the CEF excited state with the c-f hybridization, larger than that of the ground CEF state, seems to correspond to the highest CEF level, $\Gamma_{7}^{2}\simeq 0.47|\pm 5/2\rangle+0.88|\mp 3/2\rangle$, with excitation energy about 360 K [54].

3 Model for describing valence transition and fluctuations

A canonical model for describing the valence transition due to electronic origin is an extended periodic Anderson model (EPAM) that takes into account the Coulomb repulsion $U_{\rm fc}$ between f and conduction electrons. Explicit form of the EPAM is given as follows:

[TABLE]

where $U_{\rm fc}$ is the f-c Coulomb repulsion, which turns out a main origin of valence transition or fluctuations. The label $\sigma$ in Eq. (2) stands the degrees of freedoms of the Kramers doublet state of the ground CEF level. This model has been discussed in a variety of context [57, 58, 59, 60, 61, 62].

The model without hybridization $V$ is called the Falicov-Kimball model (FKM) which has been a canonical model for discussing the valence state of rare earth ions [63, 64]. The condition for the valence transition in the FKM is simply given by

[TABLE]

where $\mu$ is the chemical potential or the Fermi level in the Kondo limit where f electrons are essentially singly occupied, and $n_{\rm c}$ is the number of conduction electrons at f-site. This relation expresses the competition between the energy level of f-electron modified by the mean field arising from $U_{\rm fc}$, the f-c Coulomb repulsion, and the Fermi energy of conduction electrons.

Although the existence of the hybridization $V$ makes it difficult to solve the problem, the condition [Eq. (3)] remains to be valid in the mean-field level of approximation. Figure 4 shows the ground state phase diagrams of the EPAM in the $\varepsilon_{\rm f}$-$U_{\rm fc}$ plane that are obtained by the mean-field approximation using the slave boson technique by taking into account the strong correlation effect ($U_{\rm ff}=\infty$) [45], and by the DMRG method for the one-dimensional version of the Hamiltonian Eq. (2) with the tight-binding dispersion for the conduction electrons, i.e., $\epsilon_{k}=-2t\cos ka$ with $a$ bing the lattice constant [46]. The electron filling ${\bar{n}}\equiv(n_{\rm c}+n_{\rm f})/2$ is fixed as ${\bar{n}}=7/8$ in both calculations. A bunch of calculations on the Gutzwiller variational ansatz have also been performed [65, 66, 67, 68, 69]. The first order valence transition (FOVT) line is given essentially by the condition (3) in the mean-field approximation. The quantum critical end point (QCEP), i.e., the quantum critical point (QCP) of the FOVT shown by closed circles, shifts from the position given by the mean-field approximation to that given by asymptotically exact DMRG calculation owing to the strong quantum fluctuation effect. In this approach based on EPAM [Eq. (2)], a series of electronic properties associated with the critical valence fluctuations can be directly calculated within a required accuracy as summarized in Table 1.

Although the DMRG calculation in one dimension is asymptotically exact, it cannot give a definite conclusion about ordered states. Nevertheless, we can obtain a useful information on the ordered state in higher dimensions by analyzing the dominant long-range correlation function. As shown in Fig. 4, the superconducting tendency with spin-singlet and inter-site paring dominates over both the SDW and CDW tendencies in the Kondo regime, shaded region, near the crossover line. This suggests that the d-wave superconductivity is stabilized in the Kondo regime near the crossover line, which is consistent with the result obtained by the theory on the basis of slave-boson mean-field solutions supplemented by Gaussian fluctuations around them in the case of three dimensional free dispersion for the conduction electrons, i.e., $\epsilon_{k}=k^{2}/2m$, and with the same electron filling ${\bar{n}}=7/8$ [45].

4 Effect of magnetic field on valence transition and crossover: Case of CeCu6

In this section, the effect of the magnetic field on the critical valence transition or sharp crossover of valence in Ce-based heavy fermion metals is discussed. Since the valence transition or crossover is a phenomenon associated with a charge transfer process or sometimes it is referred to charge fluctuations phenomenon, it appears not to be affected considerably by the magnetic field. However, this is not the case. Indeed, the valence-transition temperature $T_{\rm v}\simeq 46$ K of $\alpha$-$\gamma$ transition at ambient pressure in Ce0.8La0.1Th0.1 (without magnetic field) is suppressed to $T_{\rm v}=0$ K by the magnetic field of about 50 T [70]. This fact suggests that the position of the QCEP of the valence transition is also greatly influenced by the magnetic field, or the QCEP is easily induced by the magnetic field if the system is located near the QCEP. Indeed, this is the case as discussed below.

The effect of the magnetic filed $h$ applied along the $z$-direction is taken into account by introducing the Zeeman term to the EPAM Hamiltonian [Eq. (2)] as

[TABLE]

where $S_{{\rm f}i}^{z}$ and $S_{{\rm c}i}^{z}$ are $z$-component of spin of f and conduction electrons, respectively, and the dispersion of conduction electrons in $H_{\rm EPAM}$ [Eq. (2)] is set as $\epsilon_{k}=k^{2}/2m-D$, which extends from $-D$ to $D$. Figure 5 shows how the QCEP of valence transition in the ground state moves by applying magnetic field $h$ on the basis of slave boson mean-field approximation which properly takes into account the correlation effect due to the strong on-site Coulomb interaction $U_{\rm ff}$ in Eq. (4) in the ground state [30, 31]. The unit of energy and magnetic field is taken as $D$, and the electron fillling is fixed as ${\bar{n}}=7/8$. The physical meaning of this sensitivity of the position of the QCEP is that the f-electron level of down spin, i.e., $S_{{\rm f}i}^{z}<0$, increases toward the Fermi level by the magnetic field, which promotes the valence transition, as discussed in Refs. [28, 30, 31].

This implies that the magnetic field is a good tuning parameter for searching the QCEP of valence transition. In particular, the QCEP can be precisely hit by simultaneously changing the magnetic field $h$ and the pressure $P$ if the system is located near the QCEP in the $\varepsilon_{\rm f}$-$U_{\rm fc}$ plane at the ambient state, as shown in Fig. 6(a). Indeed, since both $U_{\rm fc}$ and $\varepsilon_{\rm f}$ increase as $P$ increases in the Ce-based compounds, the point indicating the location of the system shifts toward right and upper direction as shown in Fig. 6(a). Therefore, the locus of the QCEP due to applying $h$ intersect s with that due to applying $P$.

Recently, a symptom of this phenomenon has been observed in CeCu6 [32]. Figure 6(b) shows the magnetic field dependence of the $A$ coefficient of the $T^{2}$ term in the resistivity under a series of pressures $P$s. At ambient pressure, $A$ is an almost monotonically decreasing function of $H$. By increasing pressure, it begins to show a clear maximum at $H_{\rm m}$ where the metamagnetic sharp increase in the magnetization is observed. Note that $A$ is shown in a logarithmic scale. Variations of the $A$ coefficient is rather prominent if it is plotted in a linear scale, as shown in Fig. 6(c) where $A$ is scaled by $A(0)$ at ambient pressure and $H$ is scaled by the metamagnetic field $H_{\rm m}$. The peak structure becomes sharper and sharper as $P$ increases, which strongly suggests that $A$ diverges at the QCEP because the $T$ dependence of the resistivity should exhibit the $T$-linear dependence there. Experiments at higher pressure and magnetic field are highly desired.

Indeed, it has already been reported that CeCu6 is located near the QCEP of valence transition and a sharp valence crossover occurs at $P\simeq 5$ GPa, although the decrement in the $A$ coefficient against the pressure is slightly moderate compared to that in CeCu2Si2 and CeCu2Ge2 [33, 28]. A characteristic temperature $T_{\rm F}^{*}$ (or the effective Fermi energy $\epsilon_{\rm F}^{*}$) of CeCu6 is very low of the order of 10 K reflecting the heaviness of its effective mass. This suggests that the QCEP is recovered by the magnetic field of the order of $T_{\rm F}^{*}$, i.e., H\mathop{\vtop{\halign{#\cr\hfil\displaystyle{>}\hfil$\crcr\kern 1.0pt\nointerlineskip\cr$,\sim \crcr\kern 1.0pt\cr}}}\limits 10 T and under a certain pressure $P_{\rm c}>2$ GPa, which is in consistent with the experiment reported in Ref. [32]. This physical picture is also consistent with the disappearance of magnetic correlations in the region H\mathop{\vtop{\halign{#\cr\hfil\displaystyle{>}\hfil$\crcr\kern 1.0pt\nointerlineskip\cr$,\sim \crcr\kern 1.0pt\cr}}}\limits 2.5 T, which was observed by the inelastic neutron scattering experiment [71].

The heavy fermion compound CeRu2Si2, exhibiting metamagnetic behavior, is also expected to exhibit a valence transition at $P\sim 4$ GPa, offering us another candidate for investigating the effect of magnetic field on valence transition [72].

5 Signature of sharp valence crossover in CeRhIn5 under pressure

CeRhIn5 is one of prototypical heavy fermion systems well studied experimentally. Its phase diagram is shown in Fig. 7: (a) in the $P$-$H$ plane at $T=0\,$K, and (b) in the $P$-$T$ plane at $H=0$ [26]. Other than the coexistence of superconductivity and antiferromagnetic or the proximity of them across the discontinuous phase boundary under pressure [26, 27], the de Haas-van Alphen (dHvA) measurement performed along the line $H\simeq 15\,$Tesla (indicated by an arrow) in Fig. 7(a) revealed the following remarkable facts [74]:

1. a)

The Fermi surfaces change at $P=P_{\rm c}$ from those expected for localized f electrons (giving no contribution to the Fermi volume as in LaRhIn5) to those for itinerant f electrons contributing to the Fermi volume. 2. b)

The cyclotron mass exhibits a sharp peak at around $P=P_{\rm c}$ even though the magnetic transition is the first order and is not associated with AF critical fluctuations.

These aspects are explained naturally as a phenomenon associated with the valence crossover of Ce ion on the basis of the EPAM Hamiltonian defined by Eq. (2) [75, 76]. Characteristic of our theory is that the first order magnetic transition is not accompanied by the localization of f electrons in the magnetic phase but remains itinerant with mass enhancement of quasiparticles, which is consistent with experiments [74]. This is in marked contrast to the local criticality theory on the so-called Kondo breakdown idea [79, 77, 78] in which the c-f hybridization in the AF phase is completely vanishing.

The relation between AF order and valence transition or sharp crossover can be understood on the EPAM [Eq. (2)] by treating it in the mean-field approximations both for the AF order and the slave boson which is introduced to take into account the strong local correlation effect between on-site f electrons [75, 76]. In order to simulate the above two facts observed in dHvA experiment, the dispersion of conduction electrons is set as

[TABLE]

which simulates the two dimensional $\beta_{2}$-branch observed by the dHvA measurements in Ref. [74].

Figure 8(a) shows the phase diagram in the ground state of EPAM [Eq. (2)], with the conduction electron dispersion given by Eq. (5), in the $\varepsilon_{\rm f}$-$U_{\rm fc}$ plane for the parameter set $t=1$, $V=0.2$, $U=\infty$ and the electron filling ${\bar{n}}=0.9$ [75]. A remarkable aspect is that the line of first-order valence-transition (shown by solid line with triangles) and that of valence crossover (shown by dashed line with open circles) almost coincides with the boundary between AF and paramagnetic phase (shown by solid line with squares). A solid circle represents the QCEP of valence transition. Figure 8(b) shows the valence susceptibility $\chi_{\rm v}\equiv-(\partial n_{\rm f}/\partial\varepsilon_{\rm f})$ which exhibits clear maxima on the line of valence crossover. Namely, the AF order is cut by the valence transition or valence crossover [80]. Figure 8(c) shows the $\varepsilon_{\rm f}$ dependence of the AF order parameter $m_{\rm s}$ exhibiting the first order transition at $\varepsilon_{\rm f}=0.283$.

Figure 9(a) shows the change of the Fermi wave number $k_{\rm F}$ along the line of $k_{x}=k_{y}$ for the parameter set $t=1$, $V=0.2$, $U=\infty$, $U_{\rm fc}=0.5$, and the electron filling ${\bar{n}}\equiv(n_{\rm c}+n_{\rm f})/2=0.9$ under the magnetic field $h=0.005$ with the Hamiltonian [Eq. (4)][75]. This shows that, associated with the onset of AF state by the first order transition, the Fermi surface changes discontinuously from the smaller size to the larger one as if the f-electrons were localized in the AF state or the transition to AF state were accompanied by the localization of f electrons. This is because the $k_{\rm F}$ in the AF state almost coincides with $k_{\rm F}^{\rm c}$ of the conduction electrons with the filling $\bar{n}_{\rm c}\equiv(n_{\rm c}/2)=0.4$ since $\bar{n}_{\rm c}=n-(n_{\rm f}/2)=0.9-(n_{\rm f}/2)$ is reduced to $\bar{n}_{\rm c}=0.4$ if the number of f electron per Ce is $n_{\rm f}=1.0$ corresponding to the “localization” of f-electrons at each Ce site. However, it is not the case. Indeed, the quasiparticles consist both of f- and conduction electrons there, and their effective mass is still enhanced in consistent with specific heat measurement in CeRhIn5 [81, 82], showing that the Sommerfeld constant in the AF state $\gamma=50$ mJ/moll$\cdot$K2 is about 10 times larger than $\gamma=5.7$ mJ/mol$\cdot$K2 in LaRhIn5. The origin of small Fermi surface is the effect of the band folding associated with the onset of the AF ordering [75], but not that of the localization of f electrons [77, 78, 79].

The enhancement in observed cyclotron mass, near the phase boundary of AF and paramagnetic states, is reproduced theoretically, as shown in Fig. 9(b). The origin of this enhancement is the band effect of folding or unfolding of the Fermi surface associated with the AF transition, which is in consistent with the above picture of the change of the Fermi surface at the transition [75].

To summarize the above discussions, the characteristic aspects a) and b) of CeRhIn5 obtained by the dHvA experiments [74], listed in P. 11, can be understood in a unified way as a phenomenon associated with the valence crossover of Ce ion under the pressure and the magnetic field.

Other circumstantial evidence for the sharp valence crossover to be realized in CeRhIn5 at $P=P_{\rm c}$ is summarized as follows:

The resistivity at $T=2.25\,$K just above $T_{\rm c}$ exhibits huge peak at around $P=P_{\rm c}$ [26, 83], which is a reminiscent of the case of CeCu2(Si,Ge)2. This indicates that the valence fluctuations are enhanced around $P=P_{\rm c}$, as discussed in Ref. [43]. 2. 2)

The exponent of $\alpha$, in $[\rho(T)-\rho_{0}]\propto T^{\alpha}$, approaches 1 near $P=P_{\rm c}$, as demonstrated in Ref. [27]. This gives the signature of existence of critical valence fluctuations [24]. 3. 3)

The so-called Kadowaki-Wood scaling, $\sqrt{A}/m^{*}={\rm const}.$, holds at P\mathop{\vtop{\halign{#\cr\hfil\displaystyle{<}\hfil$\crcr\kern 1.0pt\nointerlineskip\cr$,,\sim \crcr\kern 1.0pt\cr}}}\limits P_{\rm c}, while both $A$ and $m^{*}$ grow steeply as $P$ approach $P_{\rm c}$ from the lower pressure side. This implies that the divergent behaviors in $A$ and $m^{*}$ are not due to the AF critical fluctuations which would make $A/(m^{*})^{2}$ diverge there.

Concluding this section, we here discuss a symptom suggesting the existence of the QCEP of the valence transition on the phase boundary between AF and paramagnetic state (at $P\simeq P_{\rm c}$) at higher magnetic field outside the region explored experimentally so far. A crucial point is that the pressure dependence of the SC transition temperature $T_{\rm c}$ and the upper critical field $H_{{\rm c}2}$ are quite different (see Fig. 7). Namely, the former is almost flat at P\mathop{\vtop{\halign{#\cr\hfil\displaystyle{>}\hfil$\crcr\kern 1.0pt\nointerlineskip\cr$,\sim \crcr\kern 1.0pt\cr}}}\limits P_{\rm c} [Fig. 7(a)] while the latter prominently increases as the pressure approaches $P_{\rm c}$ [Fig. 7(b)]. This suggests that the SC pairing interaction is promoted by the magnetic field $H$. One of such possibilities is that the QCEP of valence transition is located at the magnetic field $H=H^{*}$ ($>10\,$Tesla) on the phase boundary at $P=P_{\rm c}$ between the AF and the paramagnetic state in the phase diagram Fig. 7(a). This is reasonable, considering the fact that its phase boundary coincides with the valence crossover lines as shown in Fig. 8 [75], and the SC state is stabilized in the region where a sharp crossover of valence occurs [45, 46, 24, 28].

It is also interesting to note that a similar trend can be seen in CeIrSi3 [84] and CeRhSi3 [85]. Indeed, Fig. 10(a) shows the $P-T$ phase diagram of CeIrSi3 [84], in which the pressure ${\bar{P}}_{\rm c}$, where the smooth extrapolation of the Néel temperature $T_{\rm N}$ in the SC phase vanishes, and the pressure $P_{\rm c}^{*}$, where the SC transition temperature $T_{\rm sc}$ takes a broad maximum, is markedly different. Furthermore, the uppercritical field $H_{{\rm c}2}$ exhibits a sharp and huge enhancement around $P=P_{\rm c}^{*}$, as shown in Fig. 10(b). This implies that the pairing interaction is increased sharply as the magnetic field $H$ is increased, suggesting that the QCEP of valence transition is induced by the magnetic field at around $P=P_{\rm c}^{*}$ as in the case of CeRhIn5 at $P=P_{\rm c}$ shown in Fig. 7, although the definition of $P=P_{\rm c}$ and $P=P_{\rm c}^{*}$ are different in these two references [84] and [26]. We also note that a similar enhancement of $H_{{\rm c}2}$ was observed in UGe2 at P\mathop{\vtop{\halign{#\cr\hfil\displaystyle{>}\hfil$\crcr\kern 1.0pt\nointerlineskip\cr$,\sim \crcr\kern 1.0pt\cr}}}\limits P_{x} [86], which was explained nicely by the effect that the paring interaction is enhanced by the effect of field induced magnetic transition between two types of ferromagnetic states in UGe2 [87].

6 Summary

We have presented a story how the idea of the critical valence transition or sharp valence crossover of heavy fermion metals has been developed, and how these phenomena are ubiquitous than first thought in the beginning of the present century. Although we have discussed relatively well established cases, there would be other potential systems which have not been well recognized so far.

Acknowledgements

We acknowledge J. Flouquet, A. T. Holmes, D. Jaccard, H. Maebashi, O. Narikiyo, Y. Onishi, and A. Tsuruta for long collaborations on physics of critical valence fluctuations on which a considerable part of the present article is based. Discussions and conversations with K. Deguchi, K. Fujiwara, H. Kobayashi, T. C. Kobayashi, K. Kuga, Y. Matsumoto, S. Nakatsuji, and N. K. Sato on experimental data are also acknowledged. This work is supported by the Grant-in-Aid for Scientific Research (No. 24540378, No. 25400369, No. 15K05177, and No. 16H01077) from the Japan Society for the Promotion of Science. One of us (S.W.) was supported by JASRI (Proposal No. 0046 in 2012B, 2013A, 2013B, 2014A, 2014B, and 2015A).