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