Progress of Theoretical Physics Supplement No. 106, 1991
95
Heavy Electrons in the Mott-Transition Region
Fusayoshi J.
0HKAW A
Department of Physics, Hokkaido University, Sapporo 060
(Received August 29, 1991)
Recent work on the Hubbard model is reviewed. What occur in the Mott-transition
region are the formation of the Gutzwiller heavy-electron band within the Mott-Hubbard gap,
and magnetic or superconducting instability of heavy-electron liquids by exchange interactions. Application to high-temperature superconductivity in the cuprate oxides is presented..
§ 1.
Introduction
The Hubbard model is one of the simplest and the most useful effective
Hamiltonians in condensed-matter physics. It simulates many interesting phenomena, for example, the Mott transition. In the Hubbard approximation/> the band is
split to the lower Hubbard band (LHB) and the upper Hubbard band (UHB) for U/D
)> 1, U and D being on-site interaction and a half of bare bandwidth, respectively.
Because the chemical potential is in the Mott-Hubbard gap in the half-filled case n=1,
n being the number of electrons per unit cell; the system is an insulator and the ground
state is antiferromagnetic. However the system presumably becomes metallic for n
=Fl. The Gutzwiller variational approximation2 > implies the formation of a heavyelectron band (HEB) for n~l. The mass-enhancement factor is given by rPm
~1/11-nl.
Because electron correlation is crudely treated, the Hubbard approximation is no
more valid for low-energy phenomena. Because it is concerned· with the ground
state, on the other hand, the Gutzwiller approximation can only tell low-energy
phenomena. Therefore it is speculated that in the Mott-transition region the density
of states has three peaks such as the LHB, the UHB and the HEB as shown in Fig. 1.
The purpose of this paper is to review recent work which has demonstrated the
formation of heavy-electron liquids and their instability by exchange interactions in
the Mott-transition region. Applications to high-temperature superconductivity
(high- Tc SC) and other topics are presented.
§ 2.
2.1.
Heavy-electron liquids
Confinement and localization of auxiliary particles
Because auxiliary particles are just auxiliary, various auxiliary-particle models
are possible. Here the method of Barnes3 > is used with a 'modification to treat the
mapping rigorously. The Hubbard model in a simple cubic lattice in d dimension is
96
F. J. Ohkawa
mapped to
3£ =~(.sa- CJfLBH)slCJSiCJ + fL~ el ei
i
' iCf
+(2.sa- JL+ U)~d/ di
i
where Ea is the band center, fLBH the
Zeeman energy, fL the chemical potential
of electrons, <ij) nearest-neighbor sites,
g JCf = eiS JCf + CJSi-Cfd/ and Qi = ~CJS JCf SiCJ
+el ei+d/ di. Here slCJ, el and d/ are
creation operators of auxiliary particles,
which are tentatively called s, e and d
particles, respectively. Two types of
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
Fig. 1. There are two effects of strong correlation.
and d particles are fermions. Because
The REB is formed at the top of the LHB for
electrons are mapped to fermionic pair
n < 1 and n ~ 1. The chemical potential is
excitations created by gi~, such pair
renormalized about a half of the bare bandwidth. Unoccupied conduction bands drawn
excitations are called fermionic excitons
by dashed lines and occupied valence bands are
or simply electrons.
shown in both cases of U/D=O and U/D-:J>l.
There exists a local gauge symmetry, [3£, Qi]=O for any site i. Therefore subspaces with different configurations {Qi} are disjoint to each other. · The
mapping is rigorous, only if the Hilbert space is restricted to {Qi=I at any sites}.
Because no single-particle excitations are possible within the restricted subspace, auxiliary 'particles are confined. In order to argue explicitly single-particle excitations, the
Hilbert space should be extended to {Qi=f=I at some sites}. The chemical potential ,.t
of auxiliary particles is introduced to satisfy ~i<Qi>=N, where <Qi> is the thermal
average over disjoint subspaces and N is the number of lattice sites; and U=/1 Ui ~ +oo
is introduced to exclude unphysical subspaces. Because [3£, QJ=O, single-particle
Green functions of auxiliary particles are site diagonal. Auxiliary particles are not
only confined but also localized.
Green functions of fermionic excitons are given by GCJ(i.s, k)=l/[KCJ- 1(i.s)-.s(k)],
where .s(k)=- t./2/d Y(k) with Y(k)=(l/N)~ <iJ) cos[ik·(Ri- Rj)] is the bare dispersion relation; the bare bandwidth 2D is O(ltl) for any d in the Hubbard model
(2·1); and KCJ(i.s, k)= ~zKCJ< 0 (i.s, k), where K(j< 0 (i.s, k) is the irreducible function. Its
definition is schematically shown in Figs. 2 and 3.
Heavy Electrons in the Matt- Transition Region
2.2.
+
+
------ -=====-=::::-:=-~=-===
Fig. 2. Double solid-dashed lines stand for fer·
mionic excitons, solid lines for s particles,
dashed lines for e or d particles, and dotted
lines for tJZ/d or c(k).
Fig. 3. Irreducible functions of fermionic excitons.
-cr
~
\
"'--_::,..-
I
/
d
Fig. 4. The Hubbard approximation.
/
97
Single-site approximation
The effects of t/2/d and Uoo can be
included perturbatively. In any SSA,
K(J(ic, k) does not depend on k. The
simplest SSA shown in Fig. 4, where
internal lines are bare, reproduces the
Hubbard approximation. 4 > When bare
lines are substituted by renormalized
lines in Fig. 4, it is another SSA beyond
the Hubbard approximation. Such a
SSA has been developed for U/D~ +oo
by setting Uoo=0. 5 > A HEB is formed at
the top of the LHB for n < 1 as shown in
Fig. 5. Th.e mass enhancement is as
large as c/Jm-::::::,1/(1-n). This SSA is
consistent with both the Hubbard
approximation and the Gutzwiller
approximation. However the Fermiliquid relations are not satisfied. Such a
drawback is improved in the best SSA. 6 >
Write down each skeleton diagram.
Internal auxiliary-particle lines can be
classified into two types such as purely
single-particle lines and a part of internal fermionic-exciton lines as shown in
Fig. 6. In any SSA, the single-particle
lines are restricted to a single site in a
diagram. The fermionic-exciton lines
can have different site indexes. Bare
lines are substituted by renormalized
lines. Summing up all the single-site
diagrams are reduced to solving the
auxiliary-particle Anderson model with
the same statistics, fl., 11, Ea, U, H ~nd Uoo
as those in the "Hubbard" model (2 ·1),
respectively. In the "Anderson" model,
Ea is the level of localized electrons. By
comparing all the diagrams, it is easy to
see that K(J(ic) in the best SSA is equal
to that of the "Anderson" model, if
renormalized fermionic-exciton lines
multiplied by transfer matrixes in the
"Hubbard" model are the same as renormalized conduction-electron lines multi-
98
F.]. Ohkawa
0
;;
0.4
.......
n=0.8
+
'0
b
a:
E
1--1
0.2
i
-2
0
Energy
2
e/0
Fig. 5. Density of states of electrons for n=0.8
(Ref. 5)). The chemical potential is at c=O.
Here R(f(ic) is the site-diagonal Green function.
<11
·----~----1
··---~----_:
j
' ~
// I
------~----
-/
(b)
Fig. 6. In the SSA, i= j is imposed in diagram (b),
i and j being site indexes. Wavy lines stand
for Uoo.
plied by hybridization matrixes in the "Anderson" model:
1-Kcr(ic)Lcr(ic)
(2· 2)
with Mcr(ic)=(l/N)~kc2 (k)Gcr(ic, k), and
( . )-}dX-Llcr(x)
L crZE-
zc-x
(2·3)
with L1cr(c)=(7C/N)~k8(c+ f.l-cccr(k)) Vi(k), where Vcr(k) is the hybridization matrix,
and cccr(k) is the dispersion relation of conduction electrons, in the Anderson model.
Not only Kcr(ic) but also Llcr(c) should be solved self-consistently in the "Anderson"
model to satisfy Eq. (2 · 2).
The self-consistent condition (2 · 2) is equivalent with
(2·4)
which. is nothing but the self-consistent condition of the best SSA in the real-particle
model. In any diagram of the self-energy, ~cr(ic)=....:.Kcr- 1 (ic)+ic+f.l-Ea+CJf.l.BH,
interaction lines U are restricted to a single site in a diagram. Consequently internal
lines are site diagonal. Equation (2·4) requires that site-diagonal lines of renormalized electrons in the Hubbard model are equal to lines of renormalized local,ized
electrons in the Anderson model. Not only ~cr(ic) but also Llcr(c) should be solved
self-consistently in the Anderson model to satisfy Eq. (2·4).
The best SSA can also be developed in other models, for example, in the periodic
Anderson model. Applications to magnetism in actinide- and lanthanide-based
heavy-electron compounds are presented elsewhere.7)
The ground state in the best SSA is a normal Fermi liquid, because the ground
99
Heavy Electrons in the Matt- Transition Region
state of the Anderson model is a normal Fermi
expanded as 10 l
liquid.s)~Io>.
The self-energy can be
2:'c1(c+ i0)=2:'o+(1- ¢m)c+(1- c/Js)6f-J.BH -(1- c/Jc)Llca + ···
(2·5)
with ¢m~(¢s+ c/Jc)/2 for li.sl~ TK, l,uBHI~ TK and ILJ.sal~ TK, where Ll.sa is a variation of
Ea and TK~iCLl/4¢m is the Kondo temperature to show the energy scale of local spin
fluctuations. Here Llc1(0) is simply denoted by LJ, and iCLl~2D from Eq. (2·4). The
Fermi-liquid relations can be proved within the SSA by following Luttinger.ll) For
example, the Fermi surface sum rule is given by
nc1=(- ; )
1~Im{ln[ -1/GO'( + iO, k)]},
(2·6)
the spin susceptibility at T=OK by
(2· 7)
and so on. As long as charge fluctuations are so depressed as c/Jc~l, the Wilson ratio
is almost as large as two: Ws=¢s/¢m~2.
For n < 1 and n ~ 1, f-1. is at the top of the LHB, that is, ,u- Ea ~ 2:'o ~ D; and ¢m ~ 1/ (1
- n). 5 l On the other hand, ,u- ca=l:'o= U/2 and ¢m~exp( U/iCLl) for n=1 in the
symmetric Anderson model. 10l As long as ¢m');> 1, the HEB is formed at f-1., and its
bandwidth is about 2D* =4 TK. The self-consistent condition (2 · 4) shows that the
density of states in the best SSA is the same as that in the Anderson model. Two
broad bands are also formed around ca and ca + U. They are nothing but the LHB
and the UHB, respectively.
The dispersion relation of heavy electrons is approximately given by .s~(k)=[ca
+2:'o+.s(k)-,u]/¢m. In the auxiliary-particle model, perturbations are not applied to
fermionic excitons but to auxiliary particles. Therefore vertex functions should be
evaluated in order to argue low-energy phenomena in terms of fermionic excitons.
Vertex functions are obtained by cutting an internal auxiliary-particle line in
fermionic-exciton lines. Therefore vertex functions to exchange interactions are
given by c/Js. Dynamical spin susceptibility is approximately given by6 l
(2·8)
with /(.s)=l/[exp(.s/T)+1]. In the limit w=O and lql~o, the approximate result
(2 · 8) is only Ws times as large as the rigorous result (2 · 7) within the best SSA.
§ 3.
Intersite exchange interactions
Weak-coupling regime
Intersite interactions can be included perturbatively. 6 l The superexchange interaction12l is mediated by virtual double-exchange of e and d particles. Its matrix
~.1.
100
F.]. Ohkawa
element is calculated as
(3·1)
with Is~ -8t 2 /dU; and 67]aP with r;=x, y and z the (a,B)-th component of the Pauli
matrix. In Eq. (3 ·1), operators in the double bracket have been shown to show that
the interaction works between s particles. Virtual single-exchange of e or d particle
gives correlated-hopping interaction. It works between s particles and fermionic
excitons. The matrix element is given by
(3·2)
with le~-8t 2 /d(fl.-ca) and ]d~-8t 2 /dU. Site j and j' in <i,jj') are nearestneighbors of site i with j-=1= j'. The sum of the terms proportional to ls and ]d
corresponds to the ]-term of the t-1 modeJ.l 3 > Although they are evaluated for Uoo=O,
the results are consistent with the t-] model.
Electrons have already been renormalized in the SSA. They are further
renormalized by including intersite spin fluctuations. For example, the dynamical
susceptibility is calculated in the random phase approximation (RP A) as6 >
XsCim, q)
Xs (O)( im, q)
(3·3)
with ls(q)= ls Y(q), where the correlated-hopping interaction is ignored. Because
xs<o>~1/TK, the weak-coupling regime can be approximately defined by ls(q)<4 TK for
any q. It is expected that various treatments work well such as the mean-field
approximation (MFA), the RP A, and the self-consistent renormalization approximation (SCR) 14> based on heavy-electron liquids, as long as the second renormalization by
intersite spin fluctuations is relatively small.
3.2.
1/d expansion from the best SSA
The momentum conservation is not essential in the d = + oo dimension. 15 )-Is)
Almost all the leading-order terms with respect to 1/d are included in the best SSA.
The other leading-order effect is a MFA type of magnetism. 6 > The perturbation with
respect to the transfer matrix is nothing but the 1/d expansion. The superexchange
interaction ls(q) is 0(1/d0 ) at the zone-boundary wave vector Q. Therefore the
ground state is antiferromagnetic in the strong-coupling regime approximately
defined by ls( Q) >4 TK. However the critical fluctuations are higher-order effects,
because ls(q) is 0(1/ /d) for almost all q. Therefore the MFA based on the best
SSA is rigorous for d= +oo. 6 >
In the weak-coupling regime, the ground state is paramagnetic. However it is
not any riormal Fermi liquid because of higher-order effects with respect to 1/d. The
most probable instability is anisotropic SC. In the d= +oo dimension, the ordered
ground state is degenerate with the normal Fermi liquid in the best SSA. 6 > The
situations are the same even when any other intersite interactions such as the
Heavy Electrons in the Matt- Transition Region
101
correlated-hopping interaction are included, although the weak- and strong-coupling
conditions are changed a little.
The 1/d expansion from the heavy-electron liquid in the best SSA is nothing but
the Fermi-liquid theory of Landau. Therefore it is straightforward to apply the
perturbative scheme to examining lower-temperature phases. Application to high- Tc
SC in the cuprate oxides is presented in § 5.
§ 4.
Mean-field approximation of auxiliary-particle models
Another type of the MFA of auxiliary-particle models and its extensions have
·been developed by many groups. 19 >- 26 > 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 LHB and
the UHB. It can be naively argued that results are only applicable to high-energy
phenomena in a crude approximation of non-variational theories, and that results
become applicable 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<slc1Sjc1>1 2, l<el ej>l 2 and l<d/ dj>l 2 are finite for i=l=- j.
Therefore auxiliary particles are itinerant in the MFA. Itinerant s and e particles
are nothing but 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 prove4 > 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
best SSA.
§ 5.
5.1.
Application to high-temperature superconductivity
Model of the cuprate oxides
It is widely accepted that high- Tc SC is realized in Cu02 layers. The simplest
effective Hamiltonian is the Hubbard model in the simple square lattice. 28 > Therefore
c(k)= -2t[cos(kxa)+cos(kya)], a bei~g the lattice constant, is assumed throughout
this section; and D=4ltl. Because the weak-coupling condition is nothing but the
stability condition against antiferromagnetism, paramagnetic oxides are in the
weak-coupling regime. The weak-coupling condition also corresponds to lti/U < 1
- n, which is presumably satisfied in superconducting oxides.
Preferential doping of holes into oxygen ions simply means that the charge
susceptibility is much larger at oxygen ions than at Cu ions. They are different
problems; which orbits holes are introduced in, and which orbits heavy electrons are
composed of. It is implicitly assumed in the single-band model that heavy electrons
102
F.]. Ohkawa
are composed of 3d electrons on Cu ions and 2p electrons on oxygen ions.
5.2.
High-energy anomalies of normal states
In the absence of U, th~ chemical potential fJ. is at the center of the bare band for
n~l. In the presence of U, the center of the LHB is not renormalized, but fJ. is at the
top of the LHB for n < 1. Therefore occupied bands including the LHB become
deeper by about a half of the LHB as shown in Fig. 1. Observed shift29 > about L/fJ.
~ 1""' 2 eV is consistent with an estimated shift from band calculations.
In any heavy-electron liquid, single-particle states for given k are distributed in
a wide energy range, for example, in the LHB, the HEB and the UHB. Excitations
among such states give optical conductivity. The intensity of excitations from the
LHB to the UHB is proportional to t 2 /U. On the other hand, the intensities of the
Drude contribution within the HEB, excitations from the LHB to the HEB, and
excitations from the HEB to the UHB are proportional to the spectral weight of the
HEB, which is approximately given by II- nl. They can explain observed anomalous
optical conductivity, cr(m). 30 > Optical scattering is approximately given by I(m)
cx::cva(m)/[1-exp( -cv/T)] (Ref. 31)). Therefore the excitations from the LHB to 'the
HEB can explain observed Raman spectra at mid-infrared frequencies. 32 >
A gap of SDW can be open on the Fermi surface of heavy electrons, when ls is
large enough. Its magnitude is O(llsl) or smaller than that, which is much smaller
than the Matt-Hubbard gap. Because excitations across the SDW gap are possible,
optical and Raman spectra are almost the same between insulating and metallic
phases except at low frequencies.
Another high-energy anomaly is the tunneling conductance proportional to Vo
+ cl VBI in a wide range of the bias voltage VB such as IVBI < 1 volt. Because the
anomaly is also observed for (BiO)z layers in BizSrzCaCuzOsH (Ref. 33)), which are
almost independent of CuOz layers, it is plausible that the anomaly is not related with
electronic properties of superconducting CuOz layers.
5.3.
Low-energy anomalies of normal states
In the best SSA, the Fermi surface sum rule is satisfied and the mass enhancement
is independent of momenta. Therefore the Fermi surface is almost the same as the
bare one in the weak-coupling regime. The feature is consistent with photoemission
measurements of the Fermi surface. 34>· 35 >
Because of the nesting of the Fermi surface and the superexchange interaction,
antiferromagnetic spin fluctuations with q~ Q can also be developed, with Q=( + 7r/a,
+ 7r/a) in the square lattice. They explain the gap-like behavior of neutron spectra,
anomalous nuclear magnetic relaxation rate of Cu ions, temperature-linear resistivity
in a wide temperature range. 36 > There exists another low-energy scale co of antiferromagnetic spin fluctuations. Experimentally, TK~lOOOK and co~IOOK to explain
the low-energy anomalies.
Once the temperature-linear resistivity is explained, the Wiedemann-Franz law
gives the thermal conductivity almost independent of temperature. 37 > The
temperature-linear resistivity is consistent with the life-time broadening linear in
energy. 35 >
Heavy Electrons in the Mott- Transition Region
103
5.4. Anisotropic Cooper pairs between heavy electrons
Cooper pairs can be formed by the superexchange interaction. 28 )' 38 ) It was
argued39 ) that in the t-1 model isotropic Cooper-pairing states, flux states/0 ) and spin
chiral states41 ) are less favorable than d -symmetry Cooper-pairing states with order
parameter .d(k)=(1/4)cG(T)[cos(kxa)-cos(kya)]. Superconducting properties of dsymmetry Cooper-pairing states were examined in the weak-coupling treatment of
SC. 42 ) The observed Tc::::::100K can be reproduced by llsl Ws2::::::1000K, which is much
smaller than experimental llsl::::::: 1000K. The life-time broadening of heavy electrons
is estimated to be as large as y:=:::: T from the resistivity R::::::150 f-lQcm at T=300 K
and the phase relaxation time n/r<P::::::l.3 kBT in the magnetoresistivity. 43 ) If such
pair-breaking is assumed, the reduction of Tc is large and experimental llsl :::::::1000 K
gives Tc::::::100 K. 44 ) On the other hand, the pair-breaking is reduced at low temperatures because of the growing SC gaps. Therefore it is speculated that B=2cG(O)/Tc
is much larger than a weak-coupling value 4.3 for d-symmetry Cooper pairs. 42 )
The penetration depth is given by )..2(0)=c 2m*/4ne 2Ncun for T=OK, 44 ) with m*
the effective mass of heavy electrons, and Ncu the concentration of Cu02 units.
Therefore )..(0):::::::1200 A for Ncu=2.4 X 1022 cm- 3, n=0.85 and m*=10me, me being the
free electron mass. It is consistent with observed )..(0)::::::: 1400 A. 45 )- 47 )
Figure 7 shows calculated )..( T) by assuming cG( T) =(1/2)BTc tanh[2j Tc/T -1]
for no pair-breaking r=O, while Fig. 8 shows;..( T) for r=0.3 nkB Tc. 44 ) Although they
are difficult to be realized in the Hubbard model, Figs. 7 and 8 also show)..( T) of ssymmetry pairs by assuming .d(k)=(1/2)cG( T). Here Tc/D*=0.05 is assumed with
D* = D/¢m; and D*::::::: 2 X 103 K experimentally. The linear- T dependence is expected
for r=O, while the T 2-dependence is expected for r>O, at low temperatures. The
T 2-dependence of Biz(Sr, Ca)3Cuz0y (Ref. 48)) can be explained by d-symmetry SC
with pair-breaking.
Observed )..( T) of YBazCu307 obeys an empirical law )..( T)/t1(0)=[1-(T/Tc) 4]112
(Ref. 47) ), which is shown by dashed lines in Figs. 7 and 8. The empirical law above
0.35 Tc can be fitted by isotropic SC with B::::::5 or d-symmetry SC with B::::::8. The
2.5
u
fLO
~
r==O
r=0.37tksTc
u
1LO
2.0
0
<""<!
" 1.5
C0
7.0 (d)
8.0 (d)
02
5.0 ( s)
6.0 ( s)
7.0 (d)
8.0 (d)
5.0 ( s)
6.0 ( s)
1-
0.0
0.5
1.0
T/Tc
Fig. 7. A( T) /A(0.35 Tc) versus T/Tc for various
2cG(0)/Tc and r=O. Here d stands for dsymmetry, and s for isotropic s-symmetry.
0.0
0.5
T/Tc
1.0
Fig. 8. A( T)/A(0.35 Tc) versus T/Tc for various
2cG(0)/Tc and r=0.37rks Tc.
104
F.]. Ohkawa
number B;:::::; 8 is more likely, because the reduction of Tc is large as argued above.
Various experiments also give B;:::::; 8 such as tunneling, Raman spectra, and optical
absorption. Anisotropic SC can explain ..1( T) above 35 K, but it can never explain
t1(T) below 35K.
Optical conductivity o(w) or NMR 1/Tl is given by the following integral
_l_~A(k )[ 1 + c~(k)c~(k+ q)- r;Ll(k)Ll(k+ q)
N k
'q
E(k)E(k+q)
J
(5·1)
with r;= -1 (type-II), E 2 (k)=[c~(k))2+ Ll 2 (k) and A(k, q) a certain factor in each
case. Antiferromagnetic spin fluctuations with q;:::::; Q mainly contribute to 1/Tl of Cu
ions in CuOz planes. Because Ll(k+ Q)= -Ll(k) for d-symmetry Cooper pairs, 1/Tl
should be similar to the behavior , of the type-I ( r; = + 1). No coherence peak is
expected in 1/TI. On the other hand, only the homogeneous part (q=O) contributes
to o(w). Therefore the large coherence peak of the type-II is expected in o(w).
Actually large peaks in o(w) have recently been observed. 49 >'50 > The different appearance of the coherence peaks in o(w) and 1/Tl implies anisotropic SC.
5.5.
Second superconducting transition driven by CD W
The two-band Hubbard model should be used for YBazCu307 with two CuOz
layers in unit cells. Every Bloch state can be classified according to "parity" with
respect to the symmetry operation of exchanging the two layers. Bonding electrons
between the two layers are of even "parity", while antibonding electrons are of odd
"parity". Singlet Cooper pairs can also be classified according to the "parity". The
Bloch parts of intraband pairs are of even "parity", while those of interband pairs are
of odd "parity". Either Cooper pair appears at Tc.
The possibility of the second SC transition in YBazCu307 was pointed out that
off-diagonal-long-range-order (ODLRO) of another SC and diagonal-long-range-order
(DLRO) of charge density wave (CDW) appear at the same temperature Tcz;=:::;35 K. 51 >
The order parameter of DLRO in Ref. 51) should be regarded as a combined one of
CDW and lattice distortion, both of which are of odd "parity". Because the two
layers are not equivalent in the presence of the odd- "parity" CDW, the even-"parity"
SC and the odd- "parity" SCare inevitably coupled. Therefore the instability of the
CDW below Tc means nothing but the second SC transition. When interlayer repulsion, which favors the odd- "parity" CDW or the disproportion of carriers in the two
layers, is strong enough, the second SC transition can be driven by the CDW.
It was pointed out51 > that the softening of odd- "parity" phonons occurs at Tcz,
which vibrate the two CuOz layers out of phase. The predicted softening is consistent
with small anomalies around 35K observed by many groups. 52 >- 58 > In particular, it is
consistent with small but sharp anomalies of the transversal NQR 1/Tz in CuOzlayers,
no or much smaller anomalies in CuO chains, and no anomalies in the longitudinal
NQR 1/Tr. 54 >- 58 > The softening of odd-"parity" phonons means that the odd-"parity"
CDW is also realized. If the softening is real, therefore, the second SC transition
must also be real.
According to the phenomenological theory, 51 > the specific-heat jump at Tcz is small
because of weak inter layer coupling. No specific-heat jump around 35 K has been
Heavy Electrons in the Matt- Transition Region
105
reported so far.
Although the driving of the second SC transition by the CDW was not considered,
it was pointed out59 > that SC gaps can be open everywhere on the Fermi surface below
Tc2 in orthorhombic lattices. Because the reduction of Tc from about 90 K to about
55 K implies the reduction of Tc2 from about 35 K to about 0 K, it is speculated that SC
gaps have nodes even at T = 0 K in oxygen-depleted YBa2Cu301-s with Tc <55 K. In
such low- Tc compounds, actually, there are significant intensities of inelastic neutron
scattering at low energies, 60 >' 61 > and the T 2 -term of ll( T) at low temperatures. 62 > It is
unusual that when Tc is decreased, the neutron gap becomes small more rapidly than
Tc. 60 >' 61 > It is likely that there exist two types of SC, and that the neutron gap scales
with Tc2.
If the second transition is real, it is easy to explain not only anisotropic-like
behaviors above Tc2 but also isotropic-like behaviors below Tc2. In order to elucidate
the machanism of high- Tc SC, therefore, it should be clarified if the second transition
is real or not in YBa2Cu307.
§ 6.
Application to ferromagnetism
When the number of electrons is N -1, N being the number of unit cells, the
ground state for U/D~ +oo is a fully polarized ferromagnetic state called the
Nagaoka state. 63 > The ferromagnetism in the one-hole case has raised a longstanding
issue whether the single-band Hubbard model simulates ferromagnetism in almost
half-filled cases. The superexchange interaction and the correlated-hopping interaction due to d particles vanish in the limit U/D~ +oo. The second-order perturbation
of the correlated-hopping interaction due to e particles gives an RKKY-like ferromagnetic exchange interaction. The most probable instability is triplet SC rather
than ferromagnetism in the d = + oo dimension. 6>
The possibility of the first-order transition is another thing to be clarified.
Kinetic energy is only relevant in the limit U/D~ +oo. Because the bandwidth of the
LHB is about D, the energy gain per holes is about D/2 in the Hubbard approximation. On the other hand, the energy gain is Din the Nagaoka state. Therefore the
ground state in the Hubbard approximation is unstable against ferromagnetism.
According to a variational Monte-Carlo study (VMC), 64 > the Nagaoka state has lower
energy than paramagnetic states for n >0.61 in the square lattice. Because the VMC
gives an upper limit of the energy of paramagnetic states, the critical value of nc=0.6I
should be regarded as being a lower limit. Further work is necessary to settle the
possibility of the first-order transition.
Acknowledgements
The present paper is dedicated to Professor Akio Yoshimori.
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