Post on 30-Dec-2015
description
Heavy Fermion Superconductivity: Competition and Cooperation of SpinFluctuations and Valence Fluctuations
K. Miyake KISOKO, Osaka University
KISOKO = Graduate School of Engineering Science
% Fundamental concepts of superconductivity of heavy fermion metals (mainly Ce-based compounds): Brief introduction of experiments and theoretical wisdom
% Two kinds of SC mechanism for heavy fermion metals, spin fluctuations and valence fluctuations: Experiments and theoretical attempts.
% Outlook for the future & New universality class of QCP associated with critical end point of valence transition in heavy fermions & Connection to high-Tc cuprates
% Signatures of valence transition or crossover in Fermi surface change of CeRhIn5
Outline of the talk
% Effect of magnetic field on valence transition
Former collaborators Y. Onishi (NEC), O. Narikiyo (Kyushu Univ.), H. Maebashi (ISSP, Univ. Tokyo) , A. T. Holmes (Univ. Birmingham), D. Jaccard (Univ. Geneva), M. Imada (Univ. Tokyo) T. Sugibayashi (Ehime Univ.),
Current collaborators
S. Watanabe (Osaka Univ. ⇒ Kyushu Institute of Technology), A. Tsuruta (Osaka Univ. ), J. Flouquet (CEA / Grenoble, ESPCI)
Fundamental concepts of superconductivity of heavy fermion metals (mainly Ce-based compounds):
experiments and theoretical wisdom
1979: year of paradigm change of superconductivity
Report of superconductivity in CeCu2Si2 which is barely magnetic material Steglich et al: PRL 43 (1979) 1892
Long silence till1984 and “Strum und Drang” of research developments after then
ρ
T(K) T(K)
103 times larger than usual metals
C/T m*(N/V)∝ 1/3
heavy electron
zero resistance
Meissner effect
BCS superconductivity is very fragile against magnetic impurities
La1-xGdx
Tc
Ferromagnetic state
T(K)
Gd (%)
Matthais et al : PRL (1958)
(Osaka Univ. 2002)
General clue for pairing interaction in heavy fermions % Pairing interaction among quasi-particles
% Coupling constant for Cooper pairing
In order that Tc is high enough to be observable, Pairing and heaviness of quasi-particles should be the same origin: magnetic fluctuations, quadruple fluctuations, etc.
k
-kk’
-k’
(weight of quasi-particle in electron)
Electronic (spin-fluctuation or charge-fluctuation) mechanism
Nakajima: Prog. Theor. Phys. 50 (1973) 1101.Anderson & Brinkman: Phys. Rev. Lett. 30 (1973) 1108
p
-pp’
-p’
pairing interaction
cf. Kohn-Luttinger : PRL 15 (1965) 524
finite Tc “in principle” even for purely repulsive interaction
Theory for superfluid 3HePairing interaction in triplet channel
Para-magnon = ferromagnetic spin fluctuations
Spin triplet P-wave pairing
Anderson-Brinkman: Phys. Rev. Lett. 30 (1973) 1108.
Ferromagnetic spin-fluctuation mechanism was so successful for understanding the existence of 3He-A phase (ABM phase)
The same process is valid also for antiferromagnetic spin fluctuationsKM, Schmitt-Rink, Varma: PRB 34 (1986) 6554.
Scalapino, Loh, Hirsch: PRB 34 (1986) 8190.
spin triplet
spin singlet
Pairing interaction in singlet channel
Kuroda: Prog. Theor. Phys. 51 (1974) 1269 .
Analogy between heavy fermion SC and 3He was stressed till mid ’80‘, early stage of research of heavy fermion SC
General expression of pairing interaction in RPA
AF fluctuations promote “d-wave”
CeIn3
Mathur et al: Nature 394 (1998) 39
Since mid ‘90, SC’s appeared near pressure induced AF-QCP
Grosche et al: Physica B 223/224 (1995) 50CePd2Si2
Non-Fermi liquid behavior
moderate enhancement
AF spin fluctuations should play an important role : Recent Dogma
3d AF-QCP
Another SC mechanism for heavy fermions, enhanced valence fluctuations:
Experiments
Suggesting new SC mechanism of repulsive origin
Two-types of P-induced heavy fermion SC near AF quantum critical point (QCP)
CePd2Si2: Grosche et al, Physica B 223/224 (1996) 50
CeIn3: Mathur et al, Nature 394 (1998) 39
CeCu2Si2: Steglich et al, PRL 43 (1979) 1892
CeRhIn5: Heeger et al, PRL 84 (2000) 4986
CeCu2Ge2: Jaccard et al, IRAPT’97; Physica B 259/261 (1996) 1
CeCu2Si2: Thomas et al,
J. Phys. Condens. matter 8 (1996) L51
CeCu2Si2: Bellarbi et al, PRB 30 (1984)1182
Strong-coupling theory near magnetic QCP Moriya et al: JPSJ 52 (1900) 2905 Monthoux &Lonzarich PRB 63 (2001) 054529
QCP of Valence transition in
D. Jaccard et al, Physica B 259-261 (1999) 1
20)( ATT
TTC ~)( f
f*
1
2/1
n
n
m
m
cf.
Gutzwiller arguments
Rapid decrease of2*
2
m
mA Rapid change of nf
(valence of Ce)211 s6d5f4:Ce )(4fCe 13
M. Rice & K. Ueda,PRB 34 (1986) 6420
CT
K.M. et al, PhysicaB 259-261 (1999) 676
22GeCeCu
Ce
CuGe
Kadowaki & Woods: SSC 58 (1986) 507
T-linear resistivity (T>Tc) in CeCu2Ge2 near P~Pv
Jaccard et al: Physica B 230-232 (1997) 297
H. Q. Yuan et al, Science 302 (2003) 2104
CeCu2(Si1-xGex)2
x=0 (Thomas et al ’96)
x=1 (Jaccard et al ’97)
x=0.1 (Yuan et al ’03) Two distinct Tc domes !
x=0 (Holmes et al ’03)
Enhancement of 0 at P=4GPa
(T) - 0 T at P=4GPa ∝
Pressure scale is shifted such that Pc=0
SCES’02 Krakow
(T) - 0 T∝
Holmes, Jaccard, KM: PRB 69 (2004) 024508
1) Enhancement of Tc (SC) 2) Enhancement of 0
3) T-linear reesistivity (T) ~ T 4) Shoulder of =C/T (at T=Tc)
Signature of critical valence fluctuations observed in CeCu2(Ge,Si)2 around the critical pressure Pv
H. Q. Yuan et al, Science 302 (2003) 2104
Two separate domes of Tc and 2) & 3) Pressure
PV
Fujiwara, Kobayashi et al: JPSJ 77 (2008) 123711
NQR relaxation rates
(1/T1T) A ∝ ∝ γ2
Line nodes: T3 – law at all pressures
K. Fujiwara: SCES2011
6
suggesting sharpvalence crossover
Another SC mechanism for heavy fermions, enhanced valence fluctuations:
Theoretical attempts.
1f n 1f n
1f n
(Kondo regime)
1f n
0 1 2 3 4 5 6 7 8 9 10-1.0
-0.5
0.0
0.5
1.0
Pairing Interaction in 3d
(0) (r
) (
0)
kFr
r ~ a : lattice const. (if kF~/a)
Real-space picture of pairing potential (0)(q)Strong on-site repulsion
short-range attractiond-wave pairing
'
f'
ffc
1
ffff
iii
N
iii
N
iiiii
N
ii
ijijji
nnU
nnUfccfVncccctH
Extended periodic Anderson model (PAM) with f-c Coulomb repulsion Ufc
% Phase diagram at T=0 K in U-f
% Superconductivity with d-wave symmetry in the valence crossover region
1st-order valence transition
slave boson MF
QCP
paramagnetic metal
crossoverKondo
Mixed Valence
+n U ~f fcc
super-conductivity
Watanabe et al.: JPSJ 75 (2006) 043710
Onishi & KM: JPSJ 69 (2000) 3955
PAM
pressure
slave-boson mean-field & fluctuations
1d DMRG calculations
1st-order valence transition
Effect of impurity extends to over long-range region
Effect of impurity remains as short-ranged
0 : highly enhanced
0 : not enhanced
V diverging as P → PV
Charge distribution around impurity at far from P~PV
Charge distribution around impurity at around P~PV
f-electron
impurity
cond-electron
cond-electron
f-electron
impurity
V
(b)
(a)
Intuitive picture for enhancement of residual resistivity
Enhancement of 0 due to critical valence fluctuations as many-body effect on impurity potential
In the forward scattering limit (k ~ 0), Ward-Pitaevskii identity
KM, Maebashi: JPSJ 71 (2002) 1007
Renormalized impurity potential
Residual resistivity
divergent at criticality ( ) : higher order corrections do not change the result. (cf. Rutherford scattering)
p+k/2
p-k/2
cf. Betbeder-Matibet & Nozieres: Ann. Phys. 37 (1966) 17 for single component Fermi liquid
A. T. Holmes, D. Jaccard, KM: Phys. Rev B 69 (2004) 024508
CeCu2Si2
Self-energy due to critical valence fluctuations
Umklapp scattering 0<q<3kF/2
Peak structure of effective mass at P=Pv
1st version
Z. Fisk et al: J. Appl. Phys. 55 (1984) 1921
Ce: transition
critical point Ce
Location of critical point (Pvc,Tvc) in P-T planedepends on the details of materials parameters
Variety of valence transition
Typical example of 1st order valence transition
Assumption: there exist compounds such that Tvc~0 or Tvc << EF*
S. Watanabe, A. Tsuruta, KM, J. Flouquet: PRL 100 (2008) 236401 & JPSJ 78 (2009) 104706.
Effect of magnetic field on valence transition and fluctuations causing a metamagnetic behavior
Drymiotis et al: J. Phys.: Condens. Matter 17 (2005) L77
Drastic effect of magnetic field on valence transition
Expected Phase Diagram in P-T-B space
How about in the case Tc<0 (i.e., in the crossover regime) ?
B B
Tc
Kondo
Mixed valence
QCP
Field-induced VQCP
Metamagnetic jump appears in crossover regime
UVD ,5.0 ,1
8/7n
3 dimension42.1fc U
i
zi
zi SShHH cf
PAM Dm
kk
2)(
2
ccff
2
1 nnnnm
Ufc
f
Field induced VQCP Reduction of TK Metamagnetism
S. Watanabe et al, PRL 100 (2008) 236401
),( QCPfc
QCPf U
at h =0.01
at h =0.02
cf. A. J. Millis, A. J. Schofield, G. G. Lonzarich & S. A. Grigera, PRL 88 (2002) 217204
D
D
fk
)(k
E. C. Palm, et al: Physica B329-333(2003) 587
1st-order transition at h~42T
S. Kawasaki, et al: PRL 96(2006)147001
CeIrIn5
T. Takeuchi, et al:JPSJ 70(2001) 877
f (P)
Ufc
V-QCP
Crossover of valence
1st order V-T
In-NQR Q starts to change at P ~ 2.1GPa
Yashima & Kitaoka (2010)
Raymond & JaccardJ. Low Temp. Phys. 120 (2000) 107
Jaccard et al: Physica B 259-261 (1999) 1
Promising candidate for magnetic field induced QCP-VT
CeCu6
f (P)
Ufc
V-QCP
Crossover of valence
1st order V-T
Fine tuning possible by changing H and P
(P)
Y. Hirose et al (Onuki group): J. Phys. Soc. Jpn. Suppl. (2012) accepted for publication
Magneto resistivity of CeCu6 under pressure
J // b, H // c
●
●●
●●
● ●% Signature of H and P induced QCP-VT
% Measurements at higher H and P expected to exhibit much sharper structure
Signatures of valence transition or crossover in Fermi surface change of CeRhIn5
S. Watanabe & K.M. JPSJ 79 (2010) 033707
Knebel et al: JPSJ 77 (2008) 144704
NK SatoGroup (Nagoya)
dHvA result in CeRhIn5H. Shishido, R. Settai, H. Harima & Y. Onuki,J. Phys. Soc. Jpn. 74 (2005) 1103
“Small” Fermi surfaces similar to those of LaRhIn5 for P<Pc
“Large” Fermi surfaces similar to those of CeCoIn5 for P>Pc
drastic change at Pc=2.35 GPa
Pc
Knebel et al: JPSJ 77 (2008) 144704
cf. Park et al: Nature 440 (2006) 65
What is the natureof transition?
Transport anomalies in CeRhIn5
G. Knebel et al., JPSJ 77 (2008) 114704
Pc
T. Muramatsu et al., JPSJ 70 (2001) 3362
T-linear resistivity emerges most prominently near P=Pc
Tlnln /)( 0
P (GPa)Pc
0)K 25.2( T
ATT 0)(
T. Park et al., Nature 456 (2008) 366
“residual resistivity” has a peak at P=Pc
= 1
Signature of sharp valence crossover
scaling under pressure
Cyclotron mass of 2 branch by dHvA at H=12~17 T
20)( ATT at H=15 T
Am /*
G. Knebel et al., JPSJ 77 (2008) 114704
Shishido et al., JPSJ 74 (2005) 1103
m* scales with AMass enhancement near P=Pc not from AF QCP but from band effect
Pc
cf. K.M. et al., Solid State Commun. 71 (1989) 1149
Extended periodic Anderson model
'
c'
ffc
1
ffffPAM
i
ii
N
iii
N
iiiii
N
ii
ijijji
nnU
nnUfccfVncccctH
iii ff
Nn
1f
iii cc
Nn
1c
0.92/)( cf nnn
c
f
tV
U ,ffcU
:9.0n
fccf Un fccf Un
1~fn
Kondo Mixed Valence
1f n
c
f
c
ffirst-order transition
k
E(k)
yxk kkt coscos2 2D-like2 branch
square lattice
filling: (cf. half filling n = 1)
S. Watanabe, A. Tsuruta, K. Miyake & J. Flouquet, JPSJ 78 (2009) 104706
kQkQkQkk
kQkQkQkk
kQkQkQkk
kQkQkQkk
kQkQkQkk
kQkkQkk
pffffp
Z
N
V
pffffp
Z
N
V
dUffffd
Z
N
V
effffe
Z
N
V
ffffN
dpp
ffffN
pp
dppe
0'
0'
0 2'
0'
12
2
1
222
22
2222
Slave-boson mean field theoryG. Kotliar & A. E. Ruckenstein, PRL 57 (1996) 1362
dppe , , , : probability for empty, singly-, & doubly-occupied states
, ’ , : Lagrange multipliers
2222 11
pepd
dpepZ
: mas renormalization factor
Z/1
7 equations aresolved self-consistently
Q = (,): AF-ordered vector
At quantum critical end point (QCP) of first-order valence transition, valence fluctuation diverges:
Ground state phase diagram
-2 -1 0 1 20
0.5
1
1.5
f
Ufc
-1 0 1 2 3 40
0.2
0.4
0.6
0.8
1
f
v
Ufc=1.0
n = 0.9
0.5
0.0
f
nf
Ufc
0.51.5
0.0
1.0
1.177 (QCP)
-2 -1 0 1 20
0.5
1
1.5
Kondo
Mixed Valence
-2 -1 0 1 20
0.5
1
1.5
f
ms
Ufc=0.5
22s ppm
AF-paramagnetic boundary almost coincides with valence transition & valence crossover line
Suppression of AF order by valence fluctuations
in CeRhIn5vc PP
t = 1, V = 0.2, U =∞
f
fv
n
H=0
Drastic change of Fermi surfaceh=0.005t = 1, V = 0.2, U =∞, Ufc=0.5,n = 0.9,
“Small” Fermi surface changes to large Fermi surfaceat AF to paramagnetic transition discontinously
cFk :kF for conduction band k
at nc=0.8
kF
283.0cf
S. Watanabe et al., JPSJ 79 (2010) 033707
Mass enhancement
h=0.005t = 1, V = 0.2, U =∞, Ufc=0.5,n = 0.9,
Gap between original lower hybridized band & the folded bandincreasesf-dominant flat part of the folded band approaches Fermi level Mass enhancement by band effect
const./ * mAThis explains scaling
k
kEN
D )(1
)(
G. Knebel et al., JPSJ 77 (2008) 114704
cf
As f increases toward , Zincreasescf
EE
S. Watanabe et al., JPSJ 79 (2010) 033707
Comparison with dHvA measurement
For f=-0.4, D()=0.84 is about 10 times larger than Dc()=0.092 at nc=0.8
H. Shishido et al., JPSJ 74 (2005) 1103
At P = 0= 50 mJ/molK2 in CeRhIn5 is about 10 timeslarger than = 5.7 mJ/molK2 in LaRhIn5
missing 2 branch for P >Pc Larger Dfor f >
cf
h makes D small CeCoIn5
cf
Pc
(A) (B) (C)
(C’)
VCeCu2(Si,Ge)2 β-YbAlB4 etc. CeRhIn5 under H
Effect of hybridization strength on P-T phase diagram
S. Watanabe & KM: J. Phys. Condens. Matter, 23 (2011) 094217.
CeCoIn5, CeIrIn5
New universality class of QCP: Critical end point of valence transition in heavy fermions
S. Watanabe, KM: PRL 105 (2010) 186403
Unconventional criticality in-YbAlB4
T(K)Y. Matsumoto et al., arXiv:0908.1242
S. Nakatsuji et al., Nature Phys. 4 (2008) 603
2)(3
4
e2
B
22
W
g
kR B
Enhanced Wilson ratio
Uniform magnetic susceptibility is enhanced as ~ T -0.5 even though the system in not close to the FM phase
-logT
~ T -0.5 ~ T
Unconventional criticality
C/T 0 Q 1/T1T
YbRh2Si2 T -lnT T -0.6 T -0.5
-YbAlB4 T 1.5 T -lnT T -0.5 exp. desired
Fermi liquid T 2 constant
3D F T 5/3 -lnT T -4/3 C.W. T -4/3
2D AF T -lnT c -lnT/T C.W. -lnT/T2D F T 4/3 T -1/3 -1/(T lnT ) C.W. -1/(T lnT )3/2
T. Moriya, Spin Fluctuations in Itinerant Electron Magnetism (Springre-Verlag, Berlin, 1985)T. Moriya & K. Ueda, Rep. Prog. Phys. 66 (2003) 1299 J. A. Hertz, PRB 14 (1976) 1165 A. J. Millis, PRB 48 (1993) 7183
RG study:
Self Consistent Renormalization (SCR) theory for spin fluctuations:
3D AF T 3/2 c-T 1/2 c-T1/4 T -3/2 C.W. T -3/4
'
c'
ffc
1
ffffPAM
i
ii
N
iii
N
iiiii
N
ii
ijijji
nnU
nnUfccfVncccctH
c
f
tV
U ,ffcU
Quantum criticality of VQCP
Construction of mode-coupling theory for valence fluctuationsstarting from HPAM
cf. Hubbard model SCR theory for spin fluctuationsLocal correlation effect by U
K. M, J. Phys.:Condens. Matter 19 (2007) 125201S. Watanabe & K. M., Phys. Status Solidi B 247 (2010) 490
P
T
P
T
P
T
CeVQCP
1st ordervalence transition
valencecrossovercritical
end point
Ce, Yb compoundsMost of
large U small U
-YbAlB4
fc fc
Periodic Anderson model
Key Origin Strong locality of valence fluctuation mode arising from local correlations of f electrons
QCP due to Critical Valence Transition
v ~ (T) ~ T - 0.5 < < 0.7~ ~ 1/(T1T ) ~ (T) ~ T -
T-linear resistivityC/T ~ -lnT
Unconventional criticality is caused by quantum valence criticality
Uniform spin susceptibility diverges in paramagnetic metal even without proximity to FM phase
Large Wilson ratio RW>>2
Unified understanding expected for unconventional NFL in
-YbAlB4 , YbRh2Si2 , YbAuCu4 (J. L. Sarrao et al 1999)Ce0.9-xLaxTh0.1 (J. C. Lashley et al 2006)YbCu5-xAlx (E. Bauer et al 1997), etc.
Connection to high-Tc cuprates
Mukuda et al: JPSJ 77 (2008) 124706
True phase diagram in high-Tc cuprates free from disorder
cf. Varma et al: Solid State Commun. 62 (1987) 681 p-d charge transfer mechanism for cuprates due to Udp
S. Shimizu et al: JPSJ 80 (2011) 043706
5 layered high-Tc cuprate superconductor
& Fundamental concepts of superconductivity of heavy fermion metals (mainly Ce-based compounds)
& Two kinds of mechanism for heavy fermion metals, spin fluctuations and valence fluctuations: Experiments and theoretical attempts.
& Outlook % New universality class of QCP associated with critical valence transition in heavy fermions % Connection to high-Tc cuprates
& Critical valence transition or crossover seems to be crucial for understanding CeRhIn5, and also for other Ce115 and Pu115
Summary
& Magnetic field is a good tuning parameter on valence transition
cf. Kondo volume collapse mechanism using phonons (cf. Razafimandimby, Fulde, Keller, Z. Phys. B 54 (1984) 111)
Phonon mechanism seems to be irrelevant to heavy fermion superconductivity
Static effect is very small according to microscopic analysis based on periodic Anderson model. Jich et al: Phys. Rev B 35 (1987) 1692
Dynamical valence susceptibility
),(1
),(~),( fc
fc
fcff
n
nn iqU
iqiq
0
ffff )0,(),(e ),( qnqndiq nin
f f
c c
ff f
f
f f
f fc c
'
c'
ffc ii nnU
fcU
f
f
' c
' c
RPA:
At critical end point as well as QCP, 0)0,0(1 fcfc U
m pmnm
n
ipGiiqpGN
Tiq
),(),(
),(
cf
fc
KTh
0for increases /1~)0,0( f
fc
h
decreases
QCPfcU fcU
f
KTh
K
Kffc
for decreases~for /1 /1~)0,0(
ThTh
increases
QCPfcU fcU
f
20 )( V
f
)(k
YbCu5-xAlx
C/T ~ -logT
E. Bauer et al, PRB 56 (1997) 711
-2/3(T ) ~ T
T (K)
x =1.5:Y
b v
alen
ce
2.0
3.0
2.8
2.6
2.4
2.2
0 21.20.80.4 1.6x
x
x
T
VQCP0 K
300 K
10 K
x =1.5Yb valence crossover occurs near x ~ 1.5
cf. K. Yamamoto et al, JPSJ 76 (2007) 124705 H. Yamaoka et al, PRB 80 (2009) 035120
X-ray LIII absorption edge measurements
CeIrIn5
12T
15T
17T
C. Capan et al PRB 70 (2004) 180502R
Residual resistivity increases
T-linear resistivity
Valence crossover line in T-H phase diagram
Q-CEP of 1st-order valence transitionS. Watanabe et al, PRL 100 (2008) 236401
convex shape
FL region
CeIrIn5
C. Capan et al., PRB 80 (2009) 094518
Hc=28 T
0 has a peak at Hc
Fermi surface volume does not change at H=Hc
Consistent with field-induced V-QCP
Watanabe, Tsuruta, K.M. & Flouquet, PRL 100 (2008) 236401 ; JPSJ 78 (2009) 104706
Kondo
MV
f
fcU
Fermi surface is always largei.e., c-f hybridization is alwaysfinite (f electrons are always itinerant)
Adiabatic continuation holdswith Luttinger’s sum rulesatisfied
S. Watanabe et al., JPSJ 75 (2006) 043710
Fermi surface in AF phase
i
zi
zi SShHH cf
PAM dHvA: H=15T h=0.005
Contour plot of lower hybridized band ),( yx kkE
:Fermi surface ofconduction bandk at nc=0.8
0.92/)( cf nnn
Small Fermi surface
1 0.8
i.e., V =0 in HPAM
== =
Fermi surface in AF phase for V=0.2 is nearly the same as small Fermi surface for V=0
283.0cf
For P <Pc Fermi surface in CeRhIn5 is very similar to LaRhIn5
h=0.005t = 1, V = 0.2, U =∞, Ufc=0.5,n = 0.9,
EE
Knebel et al: JPSJ 77 (2008) 144704
f (P)
Ufc
V-QCP
Crossover of valence
1st order V-T
Expected QCP-VT
f (P)
Ufc
V-QCP
Crossover of valence
1st order V-T
H=0H
cf. Park et al: Nature 440 (2006) 65
AF
Watanabe (Next Talk)
1st-order valence transition surface
valence crossoversurface
0K
VQCPUfc
T
f
Quantum valence criticality
v ~ (T) ~ T - 0.5 < < 0.7~ ~ 1/(T1T ) ~ T -
T-linear resistivityC/T ~ -logT
Large Wilson ratio RW>>2 S. Watanabe & K.M., PRL 105 (2010) 186403
Mode-coupling theory for valence fluctuations gives:
YbRh2Si2 T -logT T -0.6 T -0.5
-YbAlB4 T 1.5 T -logT T -0.5 exp. desiredP. Gegenwart et al, Nature 4 (2008) 186
S. Nakatsuji et al, Nature Phys. 4 (2008) 603
C/T 1/(T1T )
S. Watanabe et al, JPSJ 78 (2009) 104706
Quantum criticality in YbRh2Si2
O. Trovarelli, C. Geibel, S. Mederle, C. Langhammer, F.M. Grosche, P. Gegenwart, M. Lang, G. Sparn, & F. Steglich, PRL 85 (2000) 626
Heavy electron metal YbRh2Si2 shows magnetic transition at TN = 65 mK
cB
aB //
~ T
-logT
~ T -
0.6
YbRh2Si2
K. Ishida et al., PRL 89 (2002) 107202
1/(T1T ) ~ T -1/2
'0 LLL
mkk kkkkkkkkmkkm
km mkkmkkkkmkmkkm
bbN
Nbfc
N
V
ffccL
' '''
s''
s
''
f'0
h.c.
im imm
imimimim nnN
Unn
UL
'
c'
ffcfcfc
2'
2fcU
kk
'
s
'fc
ff
'
1
2 kkkkkkN
U
s
1 1PAM 1
N
i
N
miiimimi bbffH
Lagrange multiplier i
, m = 1, ..., N
To construct mode-coupling theory for valence fluctuations by Ufc after taking account of local correlations by U, we employ slave-boson large-N expansion framework:
HPAM :
Lagrangianbi : slave-boson operator
:U
For with , we introduce identity
For , saddle point solution is obtainedby stationary condition S0=0:
)(000
LdS
)(''0
LdS
'0 SSS SbbffccDZ exp)(
“perturbed” “unperturbed” U Ufc
0exp S
0 , 0 00
b
SS
q=q , bq=bq
'exp S
imimimimimimim cffc
N
UidDS
0
fc2 2exp'exp
][exp SDZ
s
2
s
10
2 ˆˆTrln)(][ NbN
NVGqS
qmm
),( liqq Tll 2
'f0s
s*
10
/
/ˆkk
n
kn
iNVb
NVbiG
ff0
fc0
cf0
cc0
0ˆ
GG
GGG
Gaussian fixed point
m q qqq iimmmm qqqq
N
vqqS
321
3
1321
32
2 )()()()()(2
1][
4321
4
14321
4 )()()()(qqqq i
immmm qqqqqN
v
10v
cfcf0
ffcc0
fcfc2 ),( ,,
21,
llll iqiqiq
N
UUiq
),(),(, 00s
0 lncd
knn
abl
abcd iiqkGikGN
Tiq
Renormalization group analysis: d = 3 , z =3
J. A. Hertz, PRB 14 (1976) 1165A. J. Millis, PRB 48 (1993) 7183
3for 0)( lim jsv js
Higher order terms than Gaussian term are irrelevant
Gaussian fixed point
Tnn )12(
q’=sq, ’=sz
q
qc
qc/s
cc/sz
0
)( 2
)( fc qU
q mm
Mode coupling theory of valence fluctuations
m q qqq iimmmmmlq qqqq
N
vqqCAqS
321
3
1321
32 )()()()()( 2
1][
4321
4
14321
4 )()()()(qqqq i
immmm qqqqqN
v
mql
lmlmlq iqiqCAqS ),(),( 2
1][ 2
eff
Constructbest Gaussian taking account of mode couplings for up to j=4 terms
Feynman’s inequality: )(~
effeffeff FSSTFF
0)(
~
d
Fd
Self-consistent equation for
ql lqCAqT
N
v
||
1122
s
40
},max{ -1ilq
CCq li : mean free path by impurity scattering
K.M., O. Narikiyo & Y. Onishi, Physica B 259-261 (1999) 676
Variational principle
Divergence of uniform spin susceptibility
Critical valence fluctuations are qualitatively described by RPA framework with respect to Ufc
10v
cfcf0
ffcc0
fcfc2 ),( ,,
21,
llll iqiqiq
N
UUiq
ll iqiq ,, cfcf0
ffcc0
1
ffcc0
fc-1fc
0v ,
21),(
ll iq
N
UUiq
0
-fff )0()(e ),( qqi
l SSdiq lDynamical f-spin susceptibilityhas common structure to v(q,il )
f f
c c
f f =
At V-QCP, renormalized valence fluctuation v(0,0) divergesf (0,0) diverges
)0,0(2
3)( f2
f2B gTUniform spin susceptibility
diverges
DMRG
S. Watanabe & K.M., arXiv:0906.3986
result
f
c
Watanabe et al.: JPSJ 75 (2006) 043710
Almost flat dispersion of valence fluctuation mode
8/7cf nnn Dm
kk
c
2
2D=1, V=0.5, U=
f
E
-D
D
0,0, cfcf0
ffcc0 qq
ffc
c (q,
0)0
cfc
f (q,0
)0
{
Almost q-independent dispersionemerges in Kondo regime & also in valence-fluctuation regime
Local correlation effect U
10v
cfcf0
ffcc0
fcfc2 ),( ,, 1,
llll iqiqiq
N
UUiq
qCAq l ||2
0
A: extremely small !!
f=-1.0
f=-0.5
f= 0.0
0 , 0 00
b
SS
Unconventional criticality by valence fluctuationsIn clean system Cq=C/q in d=3, for AqB
2< ~
2B q
y
0T
Tt
C
qT
2
3B
0
BccB / / qqxqqx
AA
3/2ty ty when at V-QCP (y0=0)
v(0,0) = -1 3/2 t (0,0) )0,0(2
3)( v
f2f
2B gt
Now we consider low-T regime (T<<T*F)
T0 is extremely small due to small A
(V-QCP) t =T/T0 is enlarged even for low T
Least square fit of y(t) for 5<t<100551.0 ty
v ~ (t) ~ t - 0.5 < < 0.7~ ~
1/(T1T ) ~ (t) ~ t -
shown below
y1=1
T-linear resistivity
0
Rv
3
0),(Im]1)()[(
1)(
c
qqdqnndT
Tq
In y > 1 ( t > 5 ) regime, T-linear resistivity appears
~ ~
qiCAqq 2
Rv
1),(
Cq=C/q
If A is extremely small, dynamical exponent z may be regarded as z = in Cq=C/qz-2When z = , (T ) T for T 0 limit Locality of valence fluctuation
is origin of T-linear resistivity
A. T. Holmes, D. Jaccard & K.M., PRB 69 (2004) 024508
S. Watanabe & K.M., arXiv:0906.3986
C/T ~ -lnTSpecific heat:
v
y1=1 y1=1
Ueda & Moriya: JPSJ 39 (1975) 605