Fluctuations near Quantum Critical Points of
Valence Transition and Superconductivity
Shinji WATANABE
Department of Applied Physics, University of Tokyo
Hongo 7-3-1, Bunkyo-ku Tokyo 113-0033
1
Introduction
The superconductivity emerging on the verge
of the magnetically ordered phase has been observed in various materials such as d- and felectron systems and organic compounds. The
relation between the superconductivity and
the magnetism has been intensively studied
and roles of spin fluctuations in terms of mechanisms of the superconductivity which appears
near the magnetic-instability point at zero
temperature T = 0, i.e., the quantum critical
point (QCP) have been argued.
Recently, the superconductivity whose
mechanism seems to be different from these
types has been observed in Ce compounds and
attracts much attention: In CeCu2 Ge2 [1], the
superconducting transition temperature TSC
has a maximum in the pressure regime far
from the antiferromagnetic (AF) QCP, where
the coefficient A in the resistivity ρ = ρ0 +AT n
drops rapidly with n ∼ 2. Since A is proportional to the effective mass [2] (m∗ /m)2
which is related to the f-electron number as
m∗ /m = (1 − nf /2)/(1 − nf ) [3], this implies
that TSC has a maximum around the sharp
valence change of Ce. Similar behavior has
been observed in the isostructural compound
CeCu2 Si2 [4] and in CeCu2 (Si1−x Gex )2 [5].
Recently, an NQR measurement has revealed
that TSC increases under hydrostatic pressure
in CeIrIn5 although AF spin fluctuation
is suppressed [6], suggesting the different
mechanism for the superconductivity from
ordinary AF spin fluctuation.
As known as the γ-α transition [7], Ce
metal shows the first-order valence transition
(FOVT) in its temperature-pressure phase diagram. The valence change of Ce is due
to the 4f level located near the Fermi level,
which can hybridize easily with the conduction band by applying pressure. At the critical point, the valence susceptibility diverges
as diverging density fluctuation in the liquidgas transition. When the critical temperature is suppressed by controlling the chemical substitution and applying pressure, and
enters the Fermi-degeneracy regime, diverging
valence fluctuation is considered to be coupled with the Fermi-surface instability. This
multiple instability in the quantum-degeneracy
regime seems to be a key mechanism for understanding the instabilities observed in the
above Ce compounds. However, theoretical
understanding of the mechanism has not been
fully achieved although a pioneering work on
the valence-fluctuation-mediated superconductivity [8] and later discussions [9] have been
made.
Here, we report our recent studies on the nature of the QCP of the FOVT as well as the
effects of fluctuations near the QCP. We show
that the superconducting correlation becomes
developed in the Kondo regime near the sharp
valence crossover on the basis of the densitymatrix renormalization group (DMRG) [10]
calculation for the one-dimensional periodic
Anderson model which will be introduced in
Sec.2. In Sec.3, the FOVT as well as the emergence of the QCP will be demonstrated and
the ground-state phase diagram for the valence
transition will be presented in Sec.4. In Sec.5
we will discuss the effects of valence fluctuations and will show that the superconducting
correlations develop near the QCP. The detailed analysis of the Tomonaga-Luttinger parameter will be performed in two ways in Sec.6
and Sec.7, respectively. Finally, we summarize
this report in Sec.8.
2
specially mentioned. Here, we consider infinite U for simplicity and basic results are unchanged if we set a finite value of U as far as U
is dominant among the parameters in eq. (1).
The number of states up to 1500 taken in our
DMRG calculation allows well convergence for
the open-boundary condition up to N = 80.
Model
We consider the periodic Anderson model with
Coulomb repulsion between f and conduction
electrons:
H = −t
†
(ciσ ci+1σ + c†i+1σ ciσ ) + εf
iσ
+V
fiσ ciσ + c†iσ fiσ + U
+Ufc
3
nfiσ
iσ
†
iσ
N
N
nfi↑ nfi↓
Figure 1 shows the system size dependence of
the f-electron number for Ufc = 7.0. We see
that nf ∼ 1 is realized for εf = −6.0. As
εf increases, nf at each system size decreases
and we see that the system-size dependence
of nf drastically changes between εf = −5.641
(solid triangle) and -5.639 (open diamond).
For εf = −5.641, by taking the linear extrapolation using the data of N = 56, 64, 72 and
80, nf is estimated at 0.930 in the N → ∞
limit. For εf = −5.639, by using the data of
N = 24, 32, 40, 48, 56 and 64, the linear extrapolation to the N → ∞ limit gives nf = 0.486.
This sudden jump in nf indicates the firstorder transition between the Kondo state and
the mixed-valence (MV) state.
Since the linear dependence on 1/N holds
quite well as seen in Fig. 1, we obtain nf extrapolated linearly to the N → ∞ limit, which
is shown in Fig. 2(a). We see that nf jumps
between εf = −5.641 and -5.639 as shown in
Fig. 1, indicating the FOVT.
The origin of the first-order transition is
understood as follows: When εf is deep, the
Kondo state with nf = 1 is realized to get the
energy gain of the f-level. When εf is larger,
the f electron is moved into the conduction
band to avoid the energy loss by the shallow
εf . As Ufc becomes large, the first-order transition takes place between the Kondo and the
mixed valence (MV) states, since the large Ufc
forces the electrons to pour exclusively into either the f level or the conduction band. An
important point here is that in the MV state
the f electrons are moved into the conduction
band to avoid the energy loss by Ufc as minimum as possible, so that nf = 1/4 is realized
for large Ufc . This is in striking contrast to
i=1
nfi nci ,
(1)
i=1
where ciσ (c†iσ ) and fiσ (fi†σ ) are the annihilation
(creation) operators of the conduction and f
electrons, respectively, at the ith site with spin
σ in the N -lattice sites. Here, t is the transfer integral for the conduction electrons and εf
is the one-body f -level energy. The third term
describes the hybridization between f and conduction electrons. The term with U denotes
the Coulomb repulsion for f electrons and the
term with Ufc the Coulomb repulsion between
f and conduction electrons. Here, the number operator is defined by naiσ = a†iσ aiσ and
nai = σ naiσ for a = f and c.
The filling n is defined by n = (nf + nc )/2
with
na =
First-order valence transition
N
1 na N i=1 i
for a = f and c. We note that the half filling
is realized when n = 1 in this definition.
For U = Ufc = 0, the f orbital and the conduction band are hybridized and the chemical potential is located at the lower hybridized
band for n < 1. Then, the Fermi volume
contains the f -electron number and the large
Fermi surface is realized.
Throughout this report, the transfer integral
for conduction electrons t is taken as the energy unit. To discuss typical heavy-electron
states, we consider the filling near half, n =
7/8 with U = 100 and V = 0.1 to simulate
the infinite on-site Coulomb repulsion, unless
2
f
1
nf
0.8
0.6
0.4
0.2
0
0.05
0.1
-6.000
-5.800
-5.750
-5.700
-5.650
-5.641
-5.640
-5.639
-5.630
-5.600
-5.500
-5.300
nf
1/N
phase
-5
-4.5
-4
nf
the MF theory which predicts the transition
between nf = 1 and nf = 0 for large Ufc [12].
As Ufc is set to be smaller, the jump in nf
becomes smaller. Figure 2(b) shows nf extrapolated to the N → ∞ limit for Ufc = 6.0, which
still shows a jump in nf . On the other hand,
we see that no jump appears in nf in Fig. 2(c)
for Ufc = 5.0. These results indicate that the
QCP of the valence transition, at which the
valence susceptibility χf ≡ −∂nf /∂εf diverges
is located between Ufc = 5.0 and 6.0. The
accurate determination of the QCP concludes
, UfcQCP ) =
the location of the QCP as (εQCP
f
(−4.520, 5.846) [11]. We see that nf shows the
crossover from the Kondo to the MV states for
Ufc = 4.0 in Fig. 2(d) and the gradual crossover
for Ufc = 0.0 as seen in Fig. 2(e).
Ground-state
gram
-5.5
nf
Figure 1: System-size dependence of f-electron
number for t = 1.0, V = 0.1, U = 100 and
Ufc = 7.0 at n = 7/8.
4
1
0.8
0.6
0.4 (a)U =7.0
fc
0.2
0
-6
1
0.8
0.6
0.4 (b) U =6.0
fc
0.2
0
-5
1
0.8
0.6
0.4 (c) U =5.0
fc
0.2
0
-4
1
0.8
0.6
0.4 (d) U =4.0
fc
0.2
0
-3.5
-3
1
0.8
0.6
0.4 (e) U =0.0
fc
0.2
0
-1
-0.5
-3
nf
-3.5
nf
-2.5
εf
0
diaFigure 2: The f-electron number in the bulk
limit for t = 1, V = 0.1, U = 100 at n =
7/8 for (a)Ufc = 7.0, (b)6.0, (c)5.0, (d)4.0 and
(e)0.0. In (b), symbols are the same as those
in (a).
The ground-state phase diagram in the plane
of εf and Ufc is summarized in Fig. 3. The
FOVT (the solid line with the filled diamonds)
with the QCP (open circle) is determined by
the DMRG method. The dashed line with
open squares represent the crossover point between the Kondo and MV states, at which χf
has a maximum as a function of εf for Ufc
in the DMRG calculation. The FOVT (the
solid line with the filled triangles) with the
QCP
QCP (εQCP
f MF , Ufc MF ) = (0.189, 0.978) (open
3
diamond) determined by the slave-boson MF
theory is also shown in Fig. 3.
In contrast to the MF result (not shown in
Fig. 3), we have found no evidence of the phase
separation signaled by the diverging compressibility and the inhomogeneous distribution of
the charge density by the DMRG calculation [12]. Furthermore, a remarkable result is
that the QCP identified by the DMRG (open
circle) is shifted from the QCP by the MF theory (open diamond) to a substantially larger
|εf | and Ufc . This indicates that quantum fluctuations and correlation effects beyond the MF
theory destabilizes the FOVT and terminates
the phase separation. As a result, the crossover
between the Kondo and MV states is realized
in a wide region of the phase diagram.
As shown in Fig. 2, the sharp valence
crossover from the Kondo to MV states appears for Ufc = 4.0 and 5.0. In connection
to the Ce compounds such as CeCu2 (Ge/Si)2 ,
this sharp crossover regime with strong valence
fluctuation is most interesting, since the superconductivity is observed in this regime. This
point will be discussed in the next section.
5
Enhancement
correlation
of
7
Mixed Valence
6
Ufc
5
QCP
Kondo
slave boson
MF
4
3
2
1
0
-6
-5
-4
-3
εf
-2
-1
0
pairing
Figure 3: Ground-state phase diagram in the
plane of Ufc and εf for t = 1, V = 0.1, U = 100
at n = 7/8. First-order valence transition line
(solid line with solid diamonds) with quantum critical point (open circle) is determined
by the DMRG. Dashed line with open squares
represents crossover determined by the point
where χf shows the maximum as a function of
εf for several Ufc . First-order valence transition line (solid line with solid triangles) with
QCP (open diamond) determined by the slaveboson mean-field theory, is also illustrated for
comparison.
We have calculated the superconducting correab (i)P cd † (i + x) with
lation functions Pm±
m±
1
ab
(i) = √ (ai↑ bi+m↓ ± ai↓ bi+m↑ )
Pm±
2
(2)
for the singlet(-) and triplet(+) pairing. Here,
a, b, c and d denote the f and/or conduction
electron operators. We have calculated the
nearest-neighbor (NN) pairing (m = 1) and
also the onsite pairing (m = 0). Figure 4 shows
the superconducting correlations for Ufc = 5.0
where the sharp valence crossover occurs (see
Fig. 2(c)) for (a)εf = −3.71, (b)−3.9 and
(c)−4.2. To avoid the effect of the open boundary, we here fix the position of the operator at
the central site, i.e., i = N/2 for the onsite
pairing and i = N/2 − 1 for the NN pairing.
For εf = −3.71 at which the sharp valence
crossover takes place, we see that the pairing correlations decay rapidly as ∼ x−5 in
4
|<Pabm-(i) Pcdm-(i+x)+>|
|<Pabm-(i) Pcdm-(i+x)+>|
|<Pabm-(i) Pcdm-(i+x)+>|
10-2
10-3
10-4
10-5
cccc
ffff
10-6
ffcf
-7
10
cfcf
ffcc
10-8
cfcc
-9
10
0
10-210
10-3
10-4
10-5
10-6
10-7
-2
10-8
x
-9
10
0
10-210
10-3
10-4
10-5
10-6
10-7
10-8 (c)
10-9 0
10
Fig. 4(a). A remarkable result in Fig. 4(b)
is that the power of the decay of the correlations for εf = −3.9 is slightly smaller than
2 as clearly seen, for example, in the data of
cf P cc † | and |P cf P cf † | with dashed lines
|P0−
1−
0− 0−
indicating x−2 . We have confirmed that all
ab (i)P cd † (i + x)| show
the correlations of |Pm−
m−
nearly the same power which is smaller than
2. On the other hand, for εf = −4.2, i.e., in
the Kondo regime, it is found that the pairing correlations decay faster than the dashed
lines with ∼ x−2 in Fig. 4(c). We confirmed
that these results are not changed in the calculations for all the other system sizes we performed. These results indicate that the superconducting correlation is enhanced near the
sharp valence increase.
We also note here the general remarks on
the pairing correlations: The NN singletpairing correlation for the conduction electrons
cc P cc † | has the largest amplitudes and the
|P1−
1−
second largest is the correlation between onsite
f-c singlet pair and the NN c-c singlet paircf P cc † | as seen in Fig. 4. When the
ing |P0−
1−
sharp valence-crossover point is approached,
cf P cf † | correlation is eni.e., εf → −3.71, |P0−
0−
hanced prominently as seen in Fig. 4(a). This
indicates that valence fluctuations affects this
pairing correlation remarkably.
We have confirmed that the valence transition takes place by keeping the same Fermi
volume in the MF theory [12]. In the DMRG
calculation, the large Fermi surface is signaled
by the peak structure of the Friedel oscillation in the both sides of the FOVT [12]. We
have also confirmed that the charge gap and
the spin gap close in the Kondo, the MV and
the crossover regimes (see Fig. 6(a)). These
results suggest that the system is described
by the single-band Tomonaga-Luttinger (TL)
liquid, which is adiabatically connected from
the U = Ufc = 0 state where the f level and
the conduction band are hybridized and the
chemical potential is located at the lower hybridized band. In the vicinity of the QCP, it
is nontrivial whether this assumption holds, or
not. To clarify this point, we have tried to
obtain the central charge c by calculating the
system-size dependence of the ground-state energy. However, it is difficult to estimate the
-5
x
(a)
101
(b)
101
-2
x
x
101
Figure 4: Singlet pairing correlation funcab (i)P cd † (i + x)| with P ab (i) =
tions |Pm−
m−
m−
√
(ai↑ bi+m↓ − ai↓ bi+m↑ )/ 2 (see text) for (a)εf =
−3.71, (b)−3.90 and (c)−4.20 at Ufc = 5.0
with t = 1, V = 0.1, U = 100 and n =
cc P cc † | (open circle), |P ff P ff † |
7/8: |P1−
1−
1− 1−
cf P cc † | (shaded diamond),
(solid circle), |P0−
1−
†
ff P cc † | (shaded triangle), |P cf P cf |
|P1−
1−
0− 0−
†
ff P cf | (shaded
(shaded triangle) and |P1−
0−
square). Here, i = N/2 is set for m = 0 and
i = N/2 − 1 for m = 1. The dashed line ∼ x−2
is drawn for comparison.
5
value of c with sufficient accuracy under the
open-boundary condition. Hence, we here perform the analysis assuming that the system is
described by the single-component TL liquid.
Actually, under this assumption the consistent
values of Kρ by the independent evaluations
of the spin, charge and superconducting correlation functions are obtained at least in the
regime not very close to the QCP, which will
be shown below.
For the single component TL liquid without the spin and charge gaps, the singlet superconducting correlation decays as
x−1−1/Kρ [13]. Then, each paring correlaab (i)P cd † (j)| should decay with the
tion |Pm−
m−
same power with −1 − 1/Kρ in the distance
large enough, which is actually seen in Fig.4.
Since the charge and spin correlations decay
as x−1−Kρ [13], the above results shown in
Fig. 4(b) indicate Kρ > 1, namely, that the superconducting correlation becomes dominant
in the Kondo regime just near the sharp valence crossover.
6
the maximum (minimum) value.
The Kρ estimated in this manner is summarized in Fig. 5. We see that the error bars obtained from independent correlation functions
are rather small. This seems to imply that the
correct power-law decay of the TL liquid may
be captured. Here, it should be noted that at
Ufc = 0.0 for εf ≥ −1.0, Kρ is plotted by using the result from the SS correlation, since
the SS, CDW and SDW correlations do not
give the consistent Kρ in the TL form, which
has been also reported in the 2-leg ladder system [14]. For εf ≥ −3.75 at Ufc = 5.0 and for
εf ≥ −5.60 at Ufc = 7.0, Kρ is estimated from
4kF CDW and SS correlations.
For Ufc = 5.0, we see that Kρ exceeds 1
around εf = −3.9 in Fig. 5(b), indicating that
the superconducting correlation is dominant,
which is consistent with the result presented in
Fig. 4(b). For Ufc = 7.0, Kρ jumps at the firstorder transition point as in Fig. 5(a). For Ufc =
0, Kρ remains smaller than 1, indicating that
the paring correlation does not develop. These
results indicate that the superconducting correlation is enhanced in the Kondo regime near
sharp valence crossover close to the QCP of
the valence transition. This behavior seems to
be consistent with CeCu2 Ge2 [1], CeCu2 Si2 [4]
and CeCu2 (Si1−x Gex )2 [5].
Evaluation of Kρ by correlation functions
In order to estimate the TL parameter Kρ systematically, we have made a least-square fit for
the charge, spin and superconducting correlations defined by Ōi Ōj† for Ōi ≡ Oi − Oi and
cc (i), respectively, since
Oi = nci , Sicz and Pm−
the conduction electrons have a large amplitude in the pairing correlation as seen in Fig. 4,
which seem to give a reliable estimate. By assuming the asymptotic forms of the correlation functions of the single-component TL liquid [13], we have performed the least-square
†
with fixing i at
fit to the data for Ōi Ōi+x
the central site and the i + x sites, whose values are common in several system sizes. In
this way, we eliminate the effect from the open
boundary and obtain reliable TL parameters.
We have obtained three Kρ ’s from the singletsuperconducting (SS), CDW and SDW correlation functions. Then we employ the middle
point between their maximum and minimum
values as a mean value, with the error defined
by the difference between the middle point and
7
Evaluation of Kρ by charge
compressibility and charge
velocity
To get further insight into the non-monotonic
behavior of Kρ around εf = −3.9 for Ufc = 5.0
in Fig. 5(b), we have calculated the charge
compressibility κ ≡ ∂(2n)/∂µ, since the TL
parameter is expressed by the charge velocity
vc and κ as Kρ = πvc κ/2. The charge compressibility is expressed by κ = 2/(N∆c ) with
the charge gap
∆c ≡ [E(2nN + 2, 0)
+E(2nN − 2, 0) − 2E(2nN, 0)] /2,
(3)
where E(Ne , S) is the ground-state energy of
the subspace specified by the electron number Ne and total spin S. Figure 6 shows the
system-size dependence of (a)∆c and (b)κ for
6
0.4
∆c
0.3
0.2
0.1
εf
-4.20
-4.00
-3.90
-3.85
-3.80
-3.71
-3.65
0
0
1
1
(a)
0.1
0.2
π/(N+1)
1.5
Kρ
nf
0.5
(a)
0
-6.5
-6
-5.5
-5
1
κ
0.5
0
1
(b)
0.5
0
1
0.02
0.04
1/N
0.06
Kρ
nf
0.5
0
0.5
(b)
-4.5
-4
Figure 6: System-size dependence of (a) charge
gap ∆c and (b) charge compressibility for t =
1, V = 0.1, U = 100 at n = 7/8.
0
-3
-3.5
1
nf
Kρ
1
0.5
0
0.5
(c)
-1
-0.5
εf
0
0
Figure 5: nf vs. εf (open square) and Kρ vs.
εf (solid diamond) for t = 1, V = 0.1 and
U = 100 at n = 7/8 for (a)Ufc = 7.0, (b)5.0
and (c)0.0.
various εf at Ufc = 5.0. We see that the
metallic phase is realized in the sharp valence
crossover regime in Fig. 6(a). We extrapolate
κ to the N → ∞ limit assuming the linear
dependence on 1/N . The resultant κ in the
bulk limit is shown in Fig. 7. We see that
κ increases as εf increases, especially around
εf = −3.71. This implies that the total charge
fluctuation becomes large when the f electron
is moved into the conduction band and this
can be understood if we recall that the compressibility is expressed by the total charge correlations in the form of the thermal average:
κ = i,j (nfi + nci )(nfj + ncj )/(NT ).
A remarkable result here is that the compressibility stays constant around εf = −3.9
where Kρ estimated from the least-square fit
of the correlation functions has the maximum.
This indicates that the enhancement of Kρ is
caused by the enhancement of the charge velocity, but not by the compressibility. In order
to get further insight into this, we have calcu-
7
1.0
K* ρ
κ
1.5
κ , v*c
nf , Kρ , Κ∗ρ
2.0
v*c
nf
Kρ
1.0
0.5
0.5
0
-4.2
-4
-3.8
εf
-3.6
0
-3.4
Figure 7: Compressibility (solid triangle) obtained by κ = limN →∞ 2/(N∆c ) and charge
velocity (solid circle) obtained by vc∗ =
limN →∞ ∆c /∆q for t = 1, V = 0.1, U = 100
and Ufc = 5.0 at n = 7/8. Kρ (shaded diamond) obtained by the least-square fit of correlation functions, Kρ∗ = πvc∗ κ/2 (open diamond)
and nf (open square) are shown.
lated the charge velocity defined by
vc∗ ≡ lim
N →∞
∆c
∆q
(4)
with ∆q = π/(N + 1) instead of vc , since
the numerical accuracy is not enough for the
second-lowest eigenvalue in the Lanczos diagonalization [17]. We see that vc∗ (filled circles)
takes maximum around εf = −3.9 and this
suggests that the enhancement of vc is the origin of the enhancement of Kρ . We also note
that Kρ∗ = πκvc∗ /2 (open diamonds) is consistent with Kρ (shaded diamonds) obtained by
the least-square fit to the correlation functions
for εf ≤ −3.85 within the error bars. Since Kρ∗
is obtained by vc∗ and κ after the extrapolation
to N → ∞, this suggests that the estimate
of Kρ made by the finite sizes taken in this
work gives reliable values for the thermodynamic limit. A more accurate determination
of Kρ is left for future study.
The increase in vc∗ toward εf − 3.9 in the
Kondo regime arises from the decrease of the
density of states at the Fermi level, i.e., the
broadening of the Kondo peak. As εf further
increases, vc∗ is suppressed, since Ufc interrupts
the coherent motion of the electrons in the MV
state. This is the reason why vc∗ has a max-
imum near the sharp valence increase. This
suppression of vc∗ and coherence with increasing εf in the MV regime is not captured in
slave-boson MF theory [12]. The enhanced vc∗
implies the enlargement of the effective band
width. Therefore the coherent motion of electrons together with enhanced valence fluctuation is interpreted as the origin of the enhanced
pairing correlation.
The enhancement of Kρ around εf = −3.9
agrees with the following naive expectation:
In the deep Kondo regime, the SDW correlation is dominant by the RKKY interaction
and the sharp valence crossover point where
CDW correlation becomes dominant with enhanced χf . Between the two regimes, the superconducting correlation can be dominant,
since the regime midway between Kondo and
sharp valence crossover is unfavorable for both
the SDW and CDW correlations.
Here it should be stressed that the development of the pairing correlation is caused by
the enhanced charge velocity, but not by the
compressibility. This fact is in striking contrast to the ordinary models which have shown
Kρ > 1 such as the 1D t-J model [15] and 1D
t-V model at n = 1/4 [16]. In these models the Kρ > 1 regime appears just next to
the phase-separated regime in the ground-state
phase diagram. Then, enhanced Kρ is caused
by the enhanced κ reflecting the instability
toward the phase separation. In contrast, in
the present system, the enhancement of Kρ is
caused by the enhanced charge velocity, i.e.,
the increase of the effective bandwidth of electrons without diverging κ.
8
Summary
We have reported our recent studies on the nature of the QCP of the valence transition. The
DMRG calculations for the ground state show
that the first-order valence transition emerges
with the quantum critical point with diverging valence susceptibility. Instead of the phase
separation in the mean-field result, quantum
fluctuations generate a wide region of crossover
between the Kondo and mixed valence states.
It is found that the superconducting correlation is developed in the Kondo regime near
8
[15] M. Ogata, et al.: Phys. Rev. Lett. 66
(1991) 2388.
the quantum critical point of the valence transition. The origin is ascribed to the enhanced
coherent motion of electrons with valence fluctuation.
This report is based on the recent works
done with M. Imada and K. Miyake.
[16] K. Penc and F. Mila: Phys. Rev. B 49
(1994) 9670.
[17] When the charge velocity vc is defined
by vc ≡ lim∆q→0 ∆∗c /∆q for ∆∗c ≡
E1 (2nN, 0)−E(2nN, 0) with E1 being the
first-excited state, Kρ = πvc κ/2 holds. If
the symmetric condition εf + U/2 = 0 and
Ufc = 0 are satisfied in eq. (1), ∆c = ∆∗c ;
i.e., vc = vc∗ holds by charge SU(2) symmetry. It is expected that vc ∼ vc∗ holds
for eq. (1) in the Kondo regime.
[D class; 20000K (A), 16000K (B)]
References
[1] D. Jaccard, et al.: Physica B 259-261
(1999) 1.
[2] K. Miyake, et al.: Solid State Commun.
71 (1989) 1149.
[3] T. M. Rice and K. Ueda: Phys. Rev. B 34
(1986) 6420.
[4] A. T. Holmes, et al.: Phys. Rev. B 69
(2004) 024508.
[5] H. Q. Yuan, et al.: Science 302 (2003)
2104.
[6] S. Kawasaki, et al.: Phys. Rev. Lett. 94
(2005) 037007.
[7] D. C. Koskenmaki and K. A. Gschneidner,
“Handbook on the Physics and Chemistry
of Rare Earths”, edited by K. A. Gschneidner and L. Eyring (North-Holland, Amsterdam, 1978) Vol. 1, Chap. 4.
[8] Y. Onishi and K. Miyake: J. Phys. Soc.
Jpn. 69 (2000) 3955.
[9] P. Monthoux and G. G. Lonzarich: Phys.
Rev. B 69 (2004) 064517.
[10] S. R. White: Phys. Rev. Lett. 69 (1992)
2863, S. R. White: Phys. Rev. B 48,
(1993) 10345.
[11] S. Watanabe, M. Imada and K. Miyake:
J. Phys. Soc. Jpn. 75 (2006) 043710.
[12] S. Watanabe, M. Imada and K. Miyake:
in preparation.
[13] H. Schulz: Int. J. Mod. Phys. B 5 (1991)
57.
[14] R. M. Noack, et al.: cond-mat/9601047.
9