Progress of Theoretical Physics Supplement No. 108, 1992
209
Magnetism in Heavy.Electron Liquids
Fusayoshi J.
0HKAW A
Department of Physics, Hokkaido University, Sapporo 060
(Received August 30, 1991)
One of the most characteristic features of heavy-electron liquids is interplay between
local and intersite spin fluctuations. Recent developments in including both spin fluctuations
are reviewed. Local spin fluctuations, which are responsible for the formation of heavy
electrons, are completely included in the single-site approximation (SSA). The SSA is
reduced to solving the Anderson model selfconsistently. Intersite spin fluctuations are
perturbatively included. Such a perturbative scheme is nothing but the 1/d expansion from
the d= +oo dimension. The mean-field approximation based on the SSA is rigorous for d
= +oo. Applications of the 1/d expansion to paramagnons, metamagnetism, tiny-moment
magnetism, and other topics are presented.
§ 1.
Introduction
Certain I electron compounds are called heavy-electron systems, because their
low-energy properties can be understood in terms of heavy quasiparticles.l)- 3 >
Because they are also characterized by near localization of I electrons, the single-site
Kondo effect is a good starting point. 4 > Local spin fluctuations are responsible for the
formation of heavy electrons. Various treatments have been applied to include local
spin fluctuations such as extensions of the single-site Kondo effect, 5 >- 7 > and the
Gutzwiller approximation. 8 >· 9 >
On the other hand, they are also characterized by competition-between the Kondo
effect and the Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange interaction, or
interplay between local and intersite spin fluctuations. Actually neutron scattering
experiments have demonstrated coexistence of both spin fluctuations. 10>,n> A variety
of low-temperature phases is easily understood, if the competition is in a critical
situation. When the RKKY exchange interaction is weaker than the Kondo effect, a
compound is paramagnetic, that is, a normal Fermi liquid or superconducting.
Otherwise it is magnetic at enough low temperatures.
The purpose of this paper is to review recent developments in including both spin
fluctuations. 12> First, heavy-electron liquids are constructed in the single-site approximation (SSA). Secondly, the RKKY exchange interaction is perturbatively included. Such a perturbative scheme is applied to examining paramagnons,
metamagnetism, tiny-moment magnetism, and other topics.
§ 2.
Heavy electrons in the single-site approximation
Let us consider the periodic Anderson model:
210
F.]. Ohkawa
(2·1)
with Chko- and f/o- creation operators of conduction and f electrons, respectively; and
f.LBH the Zeeman energy. Two or more than two conduction bands are assumed to
make the system metallic without fail as shown in Fig. 1.
Let us examine the selfenergy of I electrons in the "best" SSA. All the skeleton
diagrams are included, but interaction lines U are restricted to a single site in a
diagram. Consequently internal lines off electrons are site diagonal. Bare lines are
substituted by renormalized ones Ro-(is)=(1/N)~kGffo-Uc, k) with
(2· 2)
with Ko-(is)=[is+ fl.- cf+ rJfl.BH- .So-(is)]- 1 and Cno-(is, k)=[is+ fl.- sn(k)+ rJfl.BH]- 1.
The selfenergy .So-( is) is independent of kin any SSA. The SSA is reduced to solving
selfconsistently the Anderson model with the same cf, H and U as those in Eq. (2 ·1),
respectively; and the chemical potential f.1. is also set the same. By comparing all the
diagrams, it is easy to see that the selfenergies are the same between the two models,
if
Ro-(is) = [Ko- - 1 (is) __
l (1dx-4...:;_o-(,__x-<-)]rc)
zs-x
1
(2· 3)
is satisfied, where the right-hand side of
Eq. (2 · 3) is the rigorous Green function
in the impurity model; and Llo-(c)
=(rc/N)~k8(s+ f.L- cco-(k)) Vi(k) with
cco-(k) and Vo-(k) the dispersion relation
of conduction electrons and the hybridization matrix, respectively, in the imk
purity model. Not only l'o-(is) but also
Lla(s) ·should be solved selfconsistently in
(a)
(b)
the impurity model.
Fig. 1. Hybridization gaps. The chemical potenBecause the ground state of the
tial is denoted by fl..
Anderson model is a normal Fermi liquid/3)-15) the ground state in the SSA is also a normal Fermi liquid. The selfenergy
can be expanded as
.Eo-(is)=.Eo(H)+ [1- ¢m(H)]is+ rJf.l.B[ H
-1H<Ps(x)dx J+[1- <Pc(H)]Llfl.+ ···
(2·4)
with 2¢m(H)~ <Ps(H)+ ¢c(H) for finite H, lei~ TK, and iLlf.Li~ TK. Here Llf.L is variation of f.L, and TK=rcL1/4¢m is local Kondo temperature to show the energy scale of
211
Magnetism in Heavy-Electron Liquids
local spin fluctuations. Here and in the following part, LltY(O) is simply denoted by Ll.
The Fermi-liquid relations can be proved within the SSA by following
Luttinger. 16> The Fermi surface sum rule is given by
ntY=1~(- ;)Im{ln[-1/GffiY(+iO, k)]+~ln[-1/CntY(+iO, k)]},
(2··5)
and the derivative spin susceptibility at T=O K by
1
xs< 0>(iw=O, jqj-? 0)= N :2J(-l)Im[ ¢s(H)GHIY( + iO, k)+ :2JGnntY( + iO, k)] (2 ·6)
ktY
1C
with GnntY(ic, k)= CM(ic, k)+ C ~tY(ic, k) Vn 2 (k)GffiY(ic, k).
<. TK is given by Croc= rroc T + 0( T 3 ) with
n
The specific heat at T
As long as charge fluctuations are so depressed as ¢c(H)<.1, the Wilson ratio is
almost as large as two for any H:
In the symmetrical case nf=1, with nf=(1/N):2JitY<!/ofi(J>, we obtain So= U/2, and
15
f..t= U/2; and ¢m(0)~(1/2)exp(U/7rLl) for U/7rLl>l. l In asymmetric cases, Fermi
liquid parameters are evaluated in the Appendix. For example, ¢m(0)~(7r 2 /8)/(1
- nf ). As long as ¢m')> 1, a heavy-electron band (HEB) is formed at f-l. The selfconsistent condition (2 · 3) shows that the density of states in the SSA is the same as that
in the Anderson model. Therefore two broad bands are also formed around cf and
cf+ U. They are the lower band (LB) of singly occupied states, and the upper band
(UB) of doubly occupied states, respectively.
§ 3.
3.1.
Auxiliary-particle model
Confinement and localization
Because auxiliary particles are just auxiliary, the periodic Anderson model (2 ·1)
can be mapped to various auxiliary-particle models. Here the method of Barnes 17l is
used with a modification to treat the mapping rigorously:
212
F.]. Ohkawa
w.
creation operators of auxiliary particles.
They are tentatively called s, e and d
=
particles, respectively. Two types of
e
statistics are possible. In one method, s
particles are fermions, while e and d
particles are bosons. In the other
method, s particles are bosons, while e
and d particles are fermions. Because
f electrons correspond to fermionic pair
excitations created by gla, they are
called fermionic excitons or simply f
CJ
electrons.
K<4>
There exists the local gauge symmeCJ try,
[j[ aux, Qi] =0 for any site i. Theree
fore subspaces with different configuraFig. 2. Irreducible functions of fermionic excitons.
tions {Qi} are disjoint to each other.
Solid lines stand for s particles, and dashed
The mapping is rigorous, only if the
lines for e or d particles.
Hilbert space is restricted to {Qi=l at
any site}. Auxiliary particles are confined, because no single-particle excitations are
possible within the restricted subspace. In order to argue explicitly single-particle
excitations, the Hilbert space should be extended to {Qi=t=l at some sites}. The
chemical potential t1 is introduced to satisfy -:Ei<Qi>=N, where <Qi> is the thermal
average over disjoint subspaces, and Uoo/l U! ___, +oo is introduced to exclude unphysical subspaces. Because of [j[aux, Qi]=O, single-particle Green functions of
auxiliary particles are site diagonal. Therefore auxiliary particles are not only
confined but also localized.
Itinerant excitations are fermionic excitons. By defining four irreducible functions Ka~l)(is) shown in Fig. 2, Green functions of fermionic excitons are given by
Eq. (2·2) with Ka(ic, k)=I;zKa< 0 (ic, k).
CJ
W
3.2. Single-site approximation
Effects of Vn(k) and Uoo are examined in the diagrammatic expansion. In any
SSA, Ka(is, k) is independent of k. The simplest SSA shown in Fig. 3, where any
internal line is bare, :reproduces the LB and the UB:
(3· 2)
n
\
"'-~ /
e
and
-(J
I
~
\
"'~-/
is+ f-1- cf+ 6f-1sH- U
(3· 3)
I
/
d
Fig. 3. The simplest single-site approximation.
with
ne=(l/N)~i<ei t ei),
X ~i<di t
nd=(l/N)
di), and na=(l/N)~i<slasia>;
and Ka< >(ic)=O, and Ka< 4 >(is)=O.
3
Magnetism in Heavy-Electron Liquids
213
The "best" SSA can also be formulated in the auxiliary-particle model. 12> The
selfconsistent condition is exactly equivalent with Eq. (2·3). Not only Krt(ic) but also
Llrt(is) should be solved selfconsistently in the impurity "Anderson" model.
3.3.
Dynamical susceptibility
Green functions of fermionic excitons with Isl ~ TK are approximately given by
Gffrt,a(is, k)=(l/Zm)/[ic-c~rt(k)] with Zm=¢m+'JJnYn 2 (k)~¢m with Yn(k)= Vn(k)
Cncr( + iO, k). Here c~rt(k) is the dispersion relation of heavy electrons defined by the
ath pole of Gffrt(is, k). Because perturbations are applied to auxiliary particles,
vertex functions should be evaluated in order to argue low-energy phenomena in
terms of fermionic excitons. They are obtained by cutting an internal auxiliaryparticle line in fermionic-exciton lines. Therefore vertex functions are given by cPs in
the limit of small transferred frequencies.
Dynamical spin susceptibility off electrons is approximately calculated as
x/0 >(iw, q)= -1a~rt T~¢s 2 Gffrt,a( ie-+ iw, k+ ~
q)cffrt,a(is, k- ~ q)
= W 2II(z"w, q)
(3·4)
with
(3·5)
·with /(s)=l/[exp(s/T)+ 1]. Not only interband contributions but also contributions
of conduction electrons are ignored, and two vertex functions are included in Eq.
(3·4). Because approximate Green functions are used, the summation over frequencies is restricted to Isl < TK. Therefore the over-counting of diagrams is partly
avoided. Actually, in the limit of w=O and lql~o, the real part of Eq. (3·4) is w~2
times as large as the first term of Eq. (2 •6), which is rigorous within the SSA. This
scheme gives the correct magnitude of the w-linear imaginary part, because Shiba's
argumenti 8 > for the impurity model is also applied to the periodic model. Therefore
we assume
Xf<o>(w+iO, q)= WRe[ll(w+iO, q)]+iW2 Im[l1(w+iO, q)].
3.4.
(3·6)
Mean-field approximation of auxiliary-particle models
The mean-field approximation (MFA) and its extensions have been developed by
many groups 19>-z6 > in the auxiliary-particle periodic Anderson model, or in the
auxiliary-particle Hubbard model. Low-lying spin fluctuations in the MFA are
similar to those in the Gutzwiller approximation and the SSA. However charge
fluctuations are totally different between them. In the MFA, low-lying charge
fluctuations are accumulated, while high-energy charge fluctuations are absent. The
MFA cannot describe high-energy phenomena such as the formation of the LB and the
UB. It is naively argued that results are only applicable to high-energy phenomena
in a crude approximation of non-variational methods, and that results become appli-
214
F. J. Ohkawa
cable even to low-energy phenomena in a higher approximation. It is curious if the
MFA could only describe low-energy phenomena.
Because the broken local gauge symmetry_ is assumed in the MFA, unphysical
expectation values such as l<slo-sjo->1 2 , l<ei t ej>l 2 and l<di t dj>l 2 are finite for i=l= j.
Auxiliary particles are itinerant in the MFA. Itinerant s and e particles are called
spinons and holons, respectively, in the MFA of the resonating valence bond (RVB)
theory. 27) When the mapping is rigorously considered, on the other hand, auxiliary
particles are confined and localized. Therefore it is easy to prove28 ) that auxiliary
fermions have no Fermi surfaces, and that auxiliary bosons never show the Bose
condensation. Because they are developed in totally different Hilbert spaces, there is
no correspondence between low-lying spin fluctuations of the MFA and the SSA.
§ 4.
4.L
Exchange interactions
One-Parameter scaling
Intersite terms can be included perturbatively.12) Virtual exchange of e or d
particle gives c-f exchange interaction. Its matrix element is given by
1
' nm (k , P )(z(JTJ
e i(k-p)·R·J
ap)( 1 rs);;
z(JTJ
\\
t
t
\\SiaSipCnkrCmp8/l
(4 ·1)
with
1
]nm(k, p)=4 Vn(k) Vm(P)( U -fl. + cf
1
)
+ f.l.-cf
(4 ·2)
and aTJaP with r;=x, y and z the (aP)-component of the Pauli matrix. Operators in the
double bracket have been shown in Eq. (4 ·1) to show outgoing and incoming particles.
The second-order perturbation of the c-f exchange interaction gives the RKKY
exchange interaction between s particles:
](iw, q)=]intra(iw, q)+ ]inter(iw, q)- ]o(iw),
(4 ·3)
]Intra ( iw, q)
(4 ·4)
where
with M(k, q)=I;nmfnm(k+(1/2)q, k-(1/2)q) Yn(k+(1/2)q) Ym(k-(1/2)q), is its
intra band contribution, while ]Inter(iw, q) is its interband contribution. No vertex
functions appear for conduction electrons, or in Eq. (4·4). Because on-site parts have
already been included in the SSA to construct heavy-electron liquids,
]o(iw) =
_1~[]Intra( iw, q) + ]Inter(iw, q)]
. (4·5)
is subtracted in Eq. (4·3). Therefore ~q](iw, q)=O, and the RKKY exchange interaction can never be positive in the whole Brillouin zone. If the nesting of the Fermi
215
Magnetism in Heavy-Electron Liquids
surface is significant, there appears a large peak of !(0, q) for finite q, and !(0, lql ~ 0)
can be negative. On the other hand, !(0, lqi~O)>O in two- or three-dimension, if a
free-electron like dispersion relation is assumed for ~~(f(k). Therefore it is likely that
!(0, lqi~O)>O in many compounds.
Because heavy electrons are admixtures of f and conduction electrons,
conduction-electron components are also renormalized, and /Intra(O, lqi~O)oc1/¢m.
However !Interoc(1/¢m)0 approximately, because excitations of large energies lml > TK
contribute to ]tnter. If ]Inter can be ignored, the RKKY exchange interaction !(0, q) is
approximately proportional to TK. The so called one-parameter scaling29 >is approximately satisfied in low-energy phenomena, in particular, accurately for q=O.
4.2.
Weak-coupling regime
Electrons have already been renormalized into heavy electrons in the SSA.
Heavy electrons are further renormalized by !(0, q). In the random phase
approximation (RP A), for example, dynamical susceptibility is calculated as
Xs(m+ iO, q)
1-
(4 ·6)
~ J(m+ iO, q)x/ >(w+ iO, q) .
0
Because x/ 0 >(o, q) ~ 1/TK, a stability condition against magnetism is roughly given by
!(0, q)<4 TK for any q, which defines the weak-coupling regime. Although a more
accurate stability condition depends on band structure or the nesting of the Fermi
surface, well developed magnetic moments can never be realized in the weak-coupling
regime. Because !(0, q) is renormalized to satisfy the one-parameter scaling, normal
heavy-electron liquids are more stable than predicted by previous theories.s>,ao> It is
expected that various treatments of !(0, q) work well such as the MFA, the R.PA, and
the selfconsistent renormalization (SCR) approximation, 31 > as long as the second
renormalization by intersite spin fluctuations is relatively small.
Static homogeneous susceptibility is given by xs(O, 0)=2 Wp*(O)/v with p*(c)
=(1/N)~ka(~- ca *(k)) and v=1-!(0, 0) Wp*(0)/2.
For the sake of simplicity, a single Fermi surface in the ath branch is considered,
and its dispersion relation is assumed to be given by ~a *(k)=n 2(k 2 - kl)/2m* with kF
·the Fermi wave vector and m* the effective mass of heavy electrons. The specificheat coefficient due to local spin fluctuations is given by Yioc= Jr 2 ks/2 TF* with ks TF*
=n2 kF2 /2m* being renormalized Fermi energy. Because y=Jr 2 ks/6TK in the Kondo
effect, TK= TF* /3 approximately. Equation (3·8) is expanded as
x/ 0 >(w+io, q)=2Wp*(o){l-r[ ~ ( 2%FY
+( n:q YJ+iJrW n:q +.. ·}
(4·7)
with VF=IikF/m*, Because the Fermi liquid relation (2·6) tells nothing about the
expansion coefficients of the real part for finite m and finite q, a numerical· factor r
of the order of unity is assumed in Eq. (4 · 7). Similarly !(m + iO, q) is expanded as
2
J(m+iO, q)=J(O, O)j{~-r[l(_!i_) +(-m-) ]+iJrW-m-+ .. ·}
j
3 2kF
2
ftVFq
ftVFq
(4·8)
216
F.]. Ohkawa
with j being a numerical constant of the order of unity, which includes a part of the
effects of ]Inter. The imaginary parts of Eqs. (4 · 7) and (4 · 8) are only finite for
(1)
2fikFVF
<
q
2kF -
( q
2kF
)2 .
(4 ·9)
1/d expansion from the SSA
The momentum conservation is not essential in the d = + oo dimension. 32>- 35 >
Almost all the leading-order terms with respect to 1/d are included in the SSA. The
other leading-order effect is a MFA type of magnetism. 36 >
In the periodic Anderson model in d dimension, both of transfer integrals of
conduction electrons and hybridization matrixes are proportional to 1/dz 12 for l-th
nearest-neighbor pairs in order to give non-trivial solutions. Because any intersite
term is given by the perturbation with respect to 1/dz12 , any intersite exchange
interaction is 0(1/ /d) in almost the entire Brillouin zone. The ground state is
paramagnetic in the weak-coupling regime. Although intersite terms can induce a
phase transition such as superconductivity, any ordering is a higher-order effect with
respect to 1/d. Accordingly, the ordered ground state is degenerate with the normal
heavy-electron liquid in the SSA in the d = + oo dimension.
However effects of exchange interactions can be 0(1/d 0 ), when the mean fields
from nearest neighbors contribute coherently. Another leading-order effect is a
MFA type of antiferromagnetism in the strong-coupling regime, which is approximately defined by /(0, Q) >4 TK, Q being its ordering vector. However the critical
fluctuations are higher-order effects, because /(0, q) is 0(1/ /d) for almost all q.
The MFA based on the SSA is rigorous for d = + oo.
The 1/d expansion from the ·heavy-electron liquids in the SSA is nothing but the
Fermi liquid theory of Landau. Therefore it is straightforward to apply the perturbative scheme of exchange interactions to examining lower-temperature phases.
Various applications of the 1/d expansion are presented in the following sections.
4.3.
Paramagnons
§ 5.
5.1.
Specific heat
Corrections to the free energy by the RKKY exchange interaction are calculated
in the RP A as 12 >
LJQ(T)=
~1:oodwcoth( 2 k~T) 2 ;N~rm{ln[1- !J(w+iO, q)x/0 >(w+i0, q)J},
(5 ·1a)
7r
XT
an
w
(1)
Zrtw;(i
-1
r
A+
]
r :q r
~ ( 2kF + ( fi
(5·1b)
Magnetism in Heavy-Electron Liquids
217
with A=vj/(1+j)r. When the system is close to ferromagnetic instability, A<t:l.
Factor 3/2 in Eq. (5·1) comes from the contributions of longitudinal spin fluctuations
as well as transverse spin fluctuations.
Because Re[f(w+ iO, q)] can never be positive in the en~ire region of the Brillouin
zone, the cutoff qc is naturally introduced in Eq. (5 ·1a). This is distinct from the
cutoff qc-:::::-2kF in the original paramagnon theory.37),ss> Because expansion forms are
used, however, a phenomenological qc. is introduced in Eq. (5·1b). For example, Yc
=qc/2kF-:::::-0.5, if Re[J(w+ iO, q)] is positive in about a half of the Brillouin zone.
Because it depends on band structure, qc oryc is left as a parameter to be determined
experimentally.
The upper limit of the w integration and the lower limit of the q integration in
Eq. (5·1b) come from the restriction that the imaginary parts of Eqs. (4~7) and (4·8)
are n:on-zero in the region of Eq. (4 · 9). In Eq. (5 ·1b), w/nkFVF<t:1 is assumed to set the
lower limit of the q integration.
Specific heat due to paramagnons is given by LJC=- T[8 2LIQ/8T 2 ]. Its
asymptotic form at low temperatures is calculated as
LJC=ypT-8T 3ln
.
T +···
(5· 2)
Tsr
with
(5·3)
and
a=~[(
160
w)s+
w
12AJ~
r
r Jr
Tp
2
3
•
(5·4)
Here Tp=ATF* and Tsr=8A 312 TF* with 8 a constant of the order of unity. The
specific heat due to paramagnOJ:?.S is large for Yc 2 /3A~ 1, that is, when the enhancement factor is large and paramagnons are widely extended in the Brillouin zone.
C\1
1<0
""
a.
>-..
I
I-
-2
""u
<J
Fig. 4. (LJC- yp)/8T 2 as a function of ln( T/A 112 Tp)
for Yc=0.4 and various A. Dotted lines
express asymptotic expressions.
Fig. 5. (LJC_:_ yp)/8T 2 as a function of ln( T/A 112 Tp)
for A=0.3 and various Yc.
218
F. J. Ohkawa
1.0
IC.
).....
"u 0.5
<l
0.0
0
10
5
CT/0.2A
112
TP)
2
0
200
100
T
Fig. 6. LJC!rPT as a function of (T/0.2A 112 Tp) 2 for
A=0.3 and various Yc.
~200
0
E
.......
--,
..s 100
,
1--
u
0
2
Fig. 8. C/T of UAlz as a function of T • A solid
line is an experimental result of Ref. 40), and a
dashed line is assumed to be rlac+ pyz with !3
=0.20 mJ/mol·K4 from Ref. 40).
300
400
( K2 )
Fig. 7. C/T of UPts as a function of T 2 • ·A solid
line is an experimental result of Ref. 42), and a
dashed line is assumed to be r1oc+ (3T 2 with (3
=0.79 mJ/mol·K4 from Ref. 44).
The second term proportional to ( W/r)
x (AjTp 3 ) in Eq. (5·4) was not obtained
in the original paramagnon theory. 37 >.ss>
It can be ignored, unless A is as large
as unity .
At elevated temperatures, LJC is
numerically calculated. Figure 4 shows
( L1 CIT - y P) /8 T 2 as a function of
ln(T/A 112 Tp) for Yc=0.4, W/r=2, and
various A, while Fig. 5 shows the same
quantity for A=0.3, W/r=2, and various
Yc. Constant 8 or Tsr depends on A, Yc,
and W/r. However, once Tsr is given,
the asymptotic expression (5·2) can be
approximately used at temperatures T < 0.5 Tsr, and accurately used at T < 0.3 Tsr for
any set of parameters. This feature is consistent with numerical results by a
phenomenological theory. 39 > It can be seen from Figs. 4 and 5 that 8 ~ 0.2 or Tsr~ 0.2
XA 112 Tp for A~0.4 and Yc~0.3. Figure 6 shows LJCjypT as a function of (T/0.2
XA 112 Tp) 2 • The specific heat due to paramagnons becomes small above Tsr for A
=0.3 and Yc< 0.5. On the other hand, it is still large above Tsr for smaller A and large
Yc.
5.2.
Application
By taking the numbers of TF* and TP in units of Kelvin, "/loc=4.10 X 104 /TF* m]
/mol· K 2 , yp=6.15 X 104 (Yc2 jTp)( W/r) m] /mol· K2 or yp=l.23 X l0 5 (y/ /Tp) m] /mol· K 2 ,
and 8=1.50Xl05 (l/Tp)3(W/r) 3 mJ/mol·K4 or o=l.20Xl06 (l/Tp) 3 m]/mol·K\ where
W/r=2, (r/W) 2A<{l, and Yc 2 /3A<{l are assumed.
Experimental data can be fitted by C- /3T 3 = yT- oT 3ln( Tsr/T), where the
phonon contribution /3T 3 should be subtracted from observed data. Experimental y,
219
Magnetism in Heavy-Electron Liquids
Table I. Observed y, 8, and Tst of UAb, CeSh.s5 and UPta.
r(exp) mo·
~JK2
8(exp)
m:R!K4
Tsiexp)K
UAb
(Ref. 40))
150
1.6
14
CeSh,ss
(Ref. 41))
240
0.7
15
UPta
(Ref. 42))
420
1.4
15
Table II. Estimated parameters according to the present theory. Theoretical Tst in this
table should be compared to experimental Tst in Table I.
YJac
mJ
mol·K 2
YP
mJ
mol·K 2
n*K
Tp K
A
Yc
Tst K
TK K
~so
~100
820
91
0.11
0.27
6
270
CeSh.ss
~100
~140
410
120
0.29
0.37
13
140
UPta
~150
~270
270
95
0.35
0.46
15
90
UAb
o and
Tst of Refs. 40) ""42) are listed in Table I. Then let us divide observed r into
r1oc and yp. For example, Fig. 7 shows experimental C/T as a function of T 2 of
DPts. 42 > Because impurities suppress the logarithmic term, 43 > the phonon. part /3T 2
with /3=0.8 m] /mol· K 4 is assumed from data of alloyed compounds DPts-xPdx. 44 >
By taking into account that LJC/T due to paramagnons at Tst is about a half of that
at T=O K as shown in Fig. 6, a dashed line can be drawn in Fig. 7, which is assumed
to be the sum of Ytoc and /3T 2 • Then we obtain Ytoc~150mJ/mol·K 2 , and yp~270
mJ/mol·K 2 , which give TF*~270 K or TK~go K and Tpjy/~460 K. On the other
hand, experimental o about 1.4 mJ/mol·K 4 gives Tp~95 K. Therefore the parameters in the paramagnon theory are estimated as A~0.35, Yc~0.46, and Tsf~15 K by
assuming 8=0.2. Estimated Tst~15 K listed in Table II should be compared with
experimental Tsf~15 K listed in Table I. The same analysis can be made for DAb
of Ref. 40) and CeSh.s5 of Ref. 41). Figure 8 shows experimental C/T as a function
of T 2 of DAlz. 40 > Estimated numbers are also shown in Table II.
Although many features are ignored such as band structure effects, specific heat
due to antiferromagnetic spin fluctuations, T 3 -term due to local spin fluctuations, and
so on, the analyses give an agreement between experimental Tst and theoretical Tsf,
in particular, for CeSh.s5 and DPts. Therefore it is likely that paramagnons are
responsible for the observed T 3 ln T term of CeSh.s5 and DPts, and that they are at
leastresponsible for a part of the upturn of the specific heat of DAb. Not only local
spin· fluctuations but also paramagnons contribute to the low-temperature specific
heat.
§ 6.
6.1.
Metamagnetism
Mean-field approximation
In the MFA, molecular fields are given by
220
F.]. Ohkawa
(6·1)
with m=(1/N)~krJ6</lrJ!krJ> in the presence of homogeneous fields. On the other
hand, the selfenergy can be expanded as in Eq. (2·4), where His replaced by Hefr, and
L/f.1.=0. Because the polarization of conduction electrons can be ignored, it is
assumed that the same field Heff is applied to both electrons. From the Fermi surface
sum rule, polarization at T=O K is given by
(6· 2)
with
(6· 3)
The main part of the polarization comes from I electrons in the LB. If an unrenormalized density of states
(6·4)
is defined, Eq. (6·2) is equivalent with
(6·5)
with
(6·6)
Here it is assumed that the density of state is symmetrical around the chemical
potential in the absence of fields. The selfconsistent equations to be solved are
Eqs. (6·1) and (6·5).
Derivative susceptibility is calculated as
(6· 7)
When the denominator of Eq. (6·7) vanishes, a metamagnetic transition occurs.
Kondo volume collapse
·It is well known 45 >' 46 > that TK depends on volumes as TK(x)= TK(O)e-x with x
= F(LI V/V), where L1 V/V is the relative change of volume. The volume dependence
of TK can play an important role in metamagnetism of Kondo lattices, 47 > because of
large Gruneisen constant r. For example, it is as large as T=180 in CeRu2Siz. 48 > In
order to include magnetostriction, another SSA is taken for the free energy in a
similar scheme to that in the Appendix. By treating the RKKY exchange interaction
in the MFA, effective free energy at T=O K per unit cell is given by
6.2.
Magnetism in Heavy-Electron Liquids
221
0.3
- / TK 2 + (JLBHeff) 2 + E1at . ·
(6· 8)
0.1
0 •0 ,___,__,___,___.__,__._..___.__,__--L...--.>__,
2
0
1
e/ /j
Fig. 9. Unrenormalized density of states, PtJ(c),
multiplied by L1 for 7J = 1, various c1, and cz
=1.3.
Lattice energy can be expanded as E1at
= TK(O)[- x + (1/2K)x 2 + ···] around the
equilibrium volume, x=O, at T=O K for
H =0; the linear term - TK(O)x is added
to make x=O an equilibrium point. A
stability condition in the absence of
fields is given by O< K< 1 for compressibility.
Equilibrium conditions in the presence of fields are given by m= -(1/N)
X (oFmean/OJLBHeff) and oFmean/ox =0.
They give Eq. (6·5) with
1
Ll
p(f(c)= Pimp(c)=Ji cz+ Llz
(6·9)
and
x=K[1
e-zx
/ e- 2 x+ h~ff
J
(6·10)
with heff = JLBHefd TK( 0), respectively. In deriving Eq. (6 · 10), the one-parameter sealing has been assumed for any x.
6.3.
Application to metamagnetism of CeRu2Si2
In order to include effects of magnetostriction, Eq. (6·10) is added to a set of the
selfconsistent equations. Therefore Eqs. (6·1), (6·5) and (6·10) should be solved
selfconsistently to determine m and x as a function of H.
In order to reproduce the hybridization gap, at ·least three bands of heavyelectrons should be included as shown in Fig. 1. Here a phenomenological model of
the density of states is used:
(6 ·11)
with
O~IYI<I
I<lyl.
(6 ·12)
Parameter c1 stands for the depth of the pseudogap, and cz stands for the magnitude
of the pseudogap. However r; is not so essential. We set cz=l.3 and r;=l. Figure
9 shows p(f(c) for various c1. The magnitude of /(0, lqi~O; H=O) is chosen to satisfy
222
F.]. Ohkawa
0.6
0.2
>
""-.
>
0.4
<I
L
E
0.2
o.o~==~~~~~--~~~~~
o.o~~~~~~~~~~~~
0.0
0.1
II
X
0.0
0.4
0.2
0.2
h
Fig. 10. Polarization m as a function of h
= {J.BH/TK(O). Experimental data of Ref. 49)
are shown by a dotted line, where saturated
magnetization 2.1 f.i.B is assumed.
0.4
h
Fig. 11. Volume expansion x=TllV/V as a function of h=fJ.BH/TK(O). Experimental data of
Ref. 50) are shown by a dotted line, where r
= 180 is assumed.
[am/ahh . . o= 1 with h= fJ.sH/TK(O), because it is approximately satisfied in
CeRuzSiz. 29 >' 48 >' 49 > If its interband contribution is ignored, the RKKY exchange interaction scales with TK. Therefore the· one-parameter scaling is assumed for any x as
observed in CeRuzSiz.29 >
Calculated properties are consistent with observations of CeRuzSiz as shown in
Figs~ 10""' 13, if we take c1 =0.425 and K=0.5. In this analysis, K is chosen to obtain an
agreement of magnetostriction between theory and experiment. Figure 10 shows
calculated magnetizations as a function of h as well as observed data, where saturated magnetization 2.1 fl.s is. assumed to, plot observed data. 49 > Figure 11 shows
magnetostrictions as a function of h, where F=180 is assumed to plot observed
data. 50 > Figure 12 shows an enhancement factor defined by
(6·13)
as a function of h. If magnetostriction is ignored, Eq. (6·13) should be unity at
metamagnetic points. However it is at most about 0.5, which means that the second
term of
(_
0
+J
u
co
4-
0
0.4
+J
c
QJ
E
QJ
u
c
co
..c
c
QJ
0.2
0 0 L--.L-.-1.---'---'---'----'-'---'---'---'---'
0.4
0.2
0.0
0
h
Fig. 12. Enhancement factor as a function of h
= fJ.BH/TK(O).
0.5~~~~~~~~~~~~
0.0
0.4
0.2
0.6
h
Fig. 13. Specific-heat coefficient
of h=fJ.BH/TK(O).
Y1oc
as a function
223
Magnetism in Heavy-Electron Liquids
( om ) dV
dm ( om )
dH= oH v+ oV HdH'
(6·14)
is large in the process of magnetization. The divergence of the derivative susceptibility comes from dV/dH.
Figure 13 shows the specific-heat coefficient due to local spin fluctuations given by
(6 ·15)
Because of the hybridization gap, r1oc is enhanced at metamagnetic points. The mass
enhancement due to paramagnons is also expected around metamagnetic points.
Three parameters are relevant in this theory. One is ferromagnetic exchange
interaction. Another is magnetostriction. The other is a pseudogap due to the
hybridization between I and conduction bands. The collapse of antiferromagnetic
spin fluctuations at metamagnetic points has been observed in neutron scattering, 51 >
but no development of antiferromagnetic spin fluctuations has been found. Therefore
it is unlikely that antiferromagnetic spin fluctuations play an role.
§ 7.
Tiny-moment magnetism
Mean-field approximation
7.1.
By assuming a longitudinal spin density wave (SDW) state with wave vector Q,
anomalous Green functions of I electrons are given by
-1
- O'r( Q)
- ar*( Q)
.ic-
co-* (k- ~ Q)
(7·1)
with basis (lk+QJ2,o-, lk-QJ2,o-).
¢mr( Q)= T~ 1~
~ /(0,
In the MFA, the anomalous average is determined by
2
Q)¢s 0'G}}2J( ic, p+
~ Q, p- ~ Q),
where G}YJ is the (z])-th component of GffrJ' defined by Eq. (7 ·1).
given by
(7 ·2)
The gap-equation is
(7 ·3)
with
E±(k;
Q)= ~ [co-*( k-+Q )+co-*( k++Q)
+/(co-*( k- ~ Q )-co-*( k+ ~ Q )Y +4lr(Q)I
2
].
(7·4)
Let us assume that the nesting of the Fermi surface is the main driving force of
224
F.]. Ohkawa
SDW. The dispersion relation of nested parts can be approximated by onedimensional dispersion relation as
ctJ*(k)=
I
c( k + ~ Q):
c( ~ Q) :
11
k;;-
where
k;;
k11 < 0,
k;;
>0 .
is the component of k parallel to Q.
(7 ·5)
Then Eq. (7·3) becomes simple as
1 wz](O Q) 1 ~' 1
-z- T=:f
' N 7: 2E(k) tanh( E(k))
2T
1- 1
(7 ·6)
with E(k)=J c 2 (kz+(1/2)Q) 2 +Ir(Q)I 2 for k;/50, and
(7 ·7)
The sumn;J-ation over kin Eq. (7·6) is restricted to the nested region, while in Eq. (7·7)
the summation is over the other whole region.
Because Eq. (7·6) is nothing but the Bardeen-Cooper-Schrieffer (BCS) gap equation, the Neel temperature ·TN is given by TN-::::. TKexp( -1/t1), where ,.1=(1/2) W 2 ]
X (0, Q)pN*(0)/(1- ~) is the dimensionless coupling constant, with
PN*(O)= hi?'B(Eo(k))
(7 ·8)
with Eo(p)=c(k;;+(1/2)Q) for k;;SO; the summation in Eq. (7·8) is also restricted to
the nested region. The order parameter is approximately calculated as 2lr(Q)I=3.5
2
X TN[1-( T/TN) )1 12 • Once <Sz( Q)> is given, ¢mr( Q)= -(1/2)](0, Q)¢s<Sz( Q)> in the
MFA, where cPs on the right-hand side is the vertex correction to mean fields.
Therefore moments of the SDW are given by
(7·9)
Equation (7 ·9) gives a trivial result that moments are small for TN<.](O, Q).
Because ](0, Q)<4 TK in the weak-coupling regime, TN<. TK should also be satisfied.
It means that only a part of heavy electrons contribute to SDW, and that tiny-moment
SDW occurs as one of the most typical states of itinerant magnetism. It should be
noted that the main part of tiny magnetization comes from the polarization in the LB.
7.2.
Discussion
The competition between the Kondo effect and the RKKY exchange interaction
was examined in a similar manner to the treatment in § 6.2. 52 ) The magnetization at
T=O K is given by
(7 ·10)
for (1/4)/(0, Q) > TK, and m(Q)=O otherwise. When moments are well developed,
fine structure of the dispersion relation can play no important roles. When
Magnetism in Heavy-Electron Liquids
(is+
1
- cJr( Q) )iL1sign(s)
1
1 .)
1
.
'
is+zLJsign(s)
-cJr*(Q)
Gffct ( zs, k+'[Q, k-'[Q = cPm ·
225
(7 ·11)
is used instead of Eq. (7 ·1), Eq. (7 ·10) is also reproduced in the scheme argued in this
paper.
Moments can be small in any SDW state for TN<t:.J(O, Q). For example, they are
small for !(0, Q) ~4 TK in Eq. (7 ·10). However, it is expected from the square-root
dependence of Eq. (7 ·10) that the magnitude of moments can be changed very much,
when either !(0, Q) or TK is changed, for example, by pressures or alloying.
If the Fermi surface has some fine structure or nesting, on the other hand,
tiny-moment SDW is possible even for !(0, Q)<4 TK, that is, in the weak-coupling
regime. The magnitude of moments depends on structures of the Fermi surface.
Poor nesting gives low TN and small moments.
Tiny-moment SDW has been observed by several groups. 53 )- 57 ) Because antiferromagnetic spin fluctuations are developed even at temperatures as high as 100 Kin
actual heavy-electron compounds, (1/4)!(0, Q) should be of the order of 100 K. On
the other hand, the Neel temperature TN are of the order of 1 ""'"'10 K. They are
consistent with observed moments of the order of 0.01 /LB"'-'0.1 /LB.
Because observed r is of the order of 0.1 J /K 2mol, TK is estimated as being of the
order of 100 K. Therefore (1/4)!(0, Q) and TK presumably compete with each other
in actual compounds. The competition is also implied by weak stability of heavyelectron compounds. For example, alloying or pressures can induce phase transitions between paramagnetic and antiferromagnetic states. When the competition is
in a crucial situation, poor nesting can give finite TN.
The temperature dependence of observed neutron scattering intensity of UPt3 is
similar to that in the MFA, 56 ) but the development of intensity is suppressed in
superconducting states. The suppression implies that the competition between tinymoment SDW and anisotropic superconductivity occurs in UPt3, and that a part of the
Fermi surface is responsible for superconductivity while the other part of the Fermi
surface is responsible for tiny-moment SDW. Such competition is plausible, if there
are many sheets of Fermi surfaces.
§ 8.
Other topics
Large magnetoresistance is observed at low temperatures in many typical Kondo
lattices. Large Grtineisen constants of TK imply that TK can be different along
dislocation lines or around impurities. Randomness of TK from site to site can give
large magnetoresistance at low temperatures T«:.. TK. According to a phase-shift
analysis based on the SSA, 58 ) magnetoresistance is positive for ttBH < TK, while it is
negative for ttBH > TK. It reaches its peak value, when fields are so strong that
magnetization might be about a half of its saturated value.
Anisotropic Cooper pairs can be formed by spin fiuctuations, 59 )' 60 ) or the dynamical polarization of the RKKY exchange interaction. 61 ) It is plausible that magnetism
and superconductivity compete with each other in heavy-electron compounds, because
the RKKY exchange interaction is the main driving force of both types of instability.
226
F.]. Ohkawa
Which is realized first presumably depends on -fine structures of the Fermi surface.
The SSA can also be developed in other models. For example, the SSA in the
Hubbard model is also reduced to solving the Anderson model selfconsistently. 36 > As
far as we are concerned with low-energy phenomena, the essential features are the
formation of heavy electrons and exchange interactions between heavy electrons.
Because they are included even in the Hubbard model, therefore, the existence of
conduction electrons is not necessarily relevant. It is argued that high-temperature
cuprate-oxide superconductors are also heavy-electron compounds described by the
Hubbard model. 62 >
Varma proposed that diamagnetic high-temperature superconductors BaxKI-x
BiOs can be described by a Hubbard-type model with attractive U, which is due to
nonlinear charge transfer screening. 63 > The SSA can also be developed in such a
model. It is argued that they are also diamagnetic heavy-electron compounds, where
local charge fluctuations are responsible for the formation of heavy electrons. 64 >
In the SSA, the HEB is formed between the LB and the UB. Optical conductivity
of heavy-electron liquids is composed of various contributions such as the Drude
contribution within the HEB and excitations among the LB, the HEB and the UB.
Intensities of their spectra are proportional to the spectral weight of the HEB except
the excitations from the LB to the UB. The optical conductivity of heavy-electron
liquids should be anomalous at TK< w~l Ul. The optical anomaly should be common
in heavy-electron systems such as lanthanide- and actinide-based I electron compounds, the cuprate oxides, and the non-cuprate oxides BaxKI-xBiOs.
Acknowledgements
The present paper is dedicated to Professor Tadao Kasuya.
Appendix
--Asymmetric Anderson Model-Because U)>j,u-cfl in actual heavy-electron compounds, an Anderson model
with U/JrLl ~ + oo is examined. Here L1 is the hybridization energy, whose energy
dependence is ignored. By following Yosida's treatment of the s-d model, 65 > the
ground state energy Egis given by
Eg=cf_f-l_
~ln~-/TK2 +(,uBH) 2 ,
(A ·1)
in the Kondo limit with TK=J DLl exp[Jr(.sr.:._ ,u)/2L1] the Kondo temperature and D the
width of conduction bands. Magnetization is given by
m=
(A·2)
and derivative spin susceptibility by
Xs
(A·3)
227
Magnetism in Heavy-Electron Liquids
The number of I electrons is given by
n
=
oEu
f
TK
= 1-__!!_
2
2Ll / TK 2 + (JJ.BH) 2
osf
(A·4)
'
and charge susceptibility by
-
onf - ( J[
Xc- - Ocf - 2Ll
)
2
TK [ TK + 2(JJ.BH) ]
[ TK 2 + (JJ.BH) 2 )3 12
2
2
2
(A·5)
From Eqs. (A· 2) and (A ·4), the number of I electrons with spin CJ is given by
n~(H) =
i[ +
1
6 ./
Ti~~"H)' J· .
(A·6)
The Kondo temperature is also written as TK=(2/Jr)Ll(l- nAO)).
On the other hand, the selfenergy of I electrons can be expanded in the same form
as Eq. (2·4). By following Langreth's66 > and Shiba's 18 > treatments, the Friedel sum
rule is expressed as
cf + ~o(H)- CJfJ.B
with Bo-(H) = Jrno-(H).
Xs(H)
1Hdxc/Js(x) + ; ~Jdx ~~~
1CL1
Jr[LBH
2/ TK 2 + (JJ.BH) 2
J
+~H)']"' sec'[ 2./ T ;,:~H)' J
with c/Js(O)=JrL1/2TK or c/Js(O)=(Jr 2 /4)/(1-nAO)).
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(A· 7)
(A·8)
Therefore c/Jc(H)~l, c/Jm(H)=c/Js(H)/2, and
,p,(H) = ,p,(0) [TK'
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
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~ Jr- Bo-(H) J
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r/>s(H) cosz[
and Xc(H)~l/JrLl.
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