Heavy Fermion Superconductivity: Competition and Cooperation of Spin Fluctuations 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-T c cuprates % Signatures of valence transition or crossover in Fermi surface change of CeRhIn 5 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 CeCu 2 Si 2 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) 10 3 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 La 1-x Gd x TcTc 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 T c is high enough to be observable, Pairing and heaviness of quasi-particles should be the same origin: magnetic fluctuations, quadruple fluctuations, etc. k -k k’ -k’ (weight of quasi-particle in electron)

Electronic (spin-fluctuation or charge-fluctuation) mechanism Nakajima: Prog. Theor. Phys. 50 (1973) Anderson & Brinkman: Phys. Rev. Lett. 30 (1973) 1108 p -p p’ -p’ pairing interaction cf. Kohn-Luttinger ： PRL 15 (1965) 524 finite T c “in principle” even for purely repulsive interaction Theory for superfluid 3 He Pairing interaction in triplet channel Para-magnon = ferromagnetic spin fluctuations Spin triplet P-wave pairing

Anderson-Brinkman: Phys. Rev. Lett. 30 (1973) Ferromagnetic spin-fluctuation mechanism was so successful for understanding the existence of 3 He-A phase (ABM phase) The same process is valid also for antiferromagnetic spin fluctuations KM, Schmitt-Rink, Varma: PRB 34 (1986) Scalapino, Loh, Hirsch: PRB 34 (1986) spin triplet spin singlet Pairing interaction in singlet channel Kuroda: Prog. Theor. Phys. 51 (1974) 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”

CeIn 3 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) 50 CePd 2 Si 2 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) CePd 2 Si 2 : Grosche et al, Physica B 223/224 (1996) 50 CeIn 3 : Mathur et al, Nature 394 (1998) 39 CeCu 2 Si 2 : Steglich et al, PRL 43 (1979) 1892 CeRhIn 5 : Heeger et al, PRL 84 (2000) 4986 CeCu 2 Ge 2 : Jaccard et al, IRAPT’97; Physica B 259/261 (1996) 1 CeCu 2 Si 2 : Thomas et al, J. Phys. Condens. matter 8 (1996) L51 CeCu 2 Si 2 : 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)

QCP of Valence transition in D. Jaccard et al, Physica B (1999) 1 cf. Gutzwiller arguments Rapid decrease of Rapid change of n f (valence of Ce) M. Rice & K. Ueda, PRB 34 (1986) 6420 K.M. et al, PhysicaB (1999) 676 Ce Cu Ge Kadowaki & Woods: SSC 58 (1986) 507

T-linear resistivity (T>T c ) in CeCu 2 Ge 2 near P~P v Jaccard et al: Physica B (1997) 297

H. Q. Yuan et al, Science 302 (2003) 2104 CeCu 2 (Si 1-x Ge x ) 2 x=0 (Thomas et al ’96) x=1 (Jaccard et al ’97) x=0.1 (Yuan et al ’03) Two distinct T c 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 P c =0 SCES’02 Krakow  (T) -  0 ∝ T 

Holmes, Jaccard, KM: PRB 69 (2004) ) 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 P v H. Q. Yuan et al, Science 302 (2003) 2104 Two separate domes of Tc and 2) & 3) Pressure PVPV

Fujiwara, Kobayashi et al: JPSJ 77 (2008) NQR relaxation rates (1/T 1 T) ∝ A ∝ γ 2 Line nodes: T 3 – law at all pressures

K. Fujiwara: SCES suggesting sharp valence crossover

Another SC mechanism for heavy fermions, enhanced valence fluctuations: Theoretical attempts.

(Kondo regime)

r ～ a : lattice const. (if k F ~  /a) Real-space picture of pairing potential  (0) (q) Strong on-site repulsion short-range attraction d-wave pairing

Extended periodic Anderson model (PAM) with f-c Coulomb repulsion U fc % 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 crossover Kondo Mixed Valence  +n U ~  ffcc super- conductivity Watanabe et al.: JPSJ 75 (2006) 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 → P V Charge distribution around impurity at far from P~P V Charge distribution around impurity at around P~P V f-electron impurity cond-electron f-electron impurity V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) CeCu 2 Si 2 Self-energy due to critical valence fluctuations Umklapp scattering 0<q<3k F /2 Peak structure of effective mass at P=P v 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 plane depends on the details of materials parameters Variety of valence transition Typical example of 1 st order valence transition Assumption: there exist compounds such that Tvc~0 or Tvc << E F *

S. Watanabe, A. Tsuruta, KM, J. Flouquet: PRL 100 (2008) & JPSJ 78 (2009) 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 T c <0 (i.e., in the crossover regime) ? BB Tc

Kondo Mixed valence QCP Field-induced VQCP Metamagnetic jump appears in crossover regime 3 dimension U fc ff Field induced VQCPReduction of T K Metamagnetism S. Watanabe et al, PRL 100 (2008) at h =0.01 at h =0.02 cf. A. J. Millis, A. J. Schofield, G. G. Lonzarich & S. A. Grigera, PRL 88 (2002)

E. C. Palm, et al: Physica B (2003) 587 1st-order transition at h~42T S. Kawasaki, et al: PRL 96(2006) CeIrIn 5 T. Takeuchi, et al: JPSJ 70(2001) 877  f (P) U fc V-QCP Crossover of valence 1 st order V-T In-NQR Q starts to change at P ~ 2.1GPa Yashima & Kitaoka (2010 )

Raymond & Jaccard J. Low Temp. Phys. 120 (2000) 107 Jaccard et al: Physica B (1999) 1 Promising candidate for magnetic field induced QCP-VT CeCu 6  f (P) U fc V-QCP Crossover of valence 1 st 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 CeCu 6 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 CeRhIn 5 S. Watanabe & K.M. JPSJ 79 (2010)

Knebel et al: JPSJ 77 (2008) NK Sato Group (Nagoya)

dHvA result in CeRhIn 5 H. Shishido, R. Settai, H. Harima & Y. Onuki, J. Phys. Soc. Jpn. 74 (2005) 1103 “Small” Fermi surfaces similar to those of LaRhIn 5 for P<P c “Large” Fermi surfaces similar to those of CeCoIn 5 for P>P c drastic change at P c =2.35 GPa PcPc Knebel et al: JPSJ 77 (2008) cf. Park et al: Nature 440 (2006) 65 What is the nature of transition?

Transport anomalies in CeRhIn 5 G. Knebel et al., JPSJ 77 (2008) PcPc T. Muramatsu et al., JPSJ 70 (2001) 3362 T-linear resistivity emerges most prominently near P=P c  P (GPa) PcPc T. Park et al., Nature 456 (2008) 366 “residual resistivity” has a peak at P=P c  = 1 Signature of sharp valence crossover

scaling under pressure Cyclotron mass of  2 branch by dHvA at H=12~17 T at H=15 T G. Knebel et al., JPSJ 77 (2008) Shishido et al., JPSJ 74 (2005) 1103 m * scales with Mass enhancement near P=P c not from AF QCP but from band effect PcPc cf. K.M. et al., Solid State Commun. 71 (1989) 1149

Extended periodic Anderson model c f Kondo Mixed Valence c f c f first-order transition k E(k)E(k) 2D-like  2 branch square lattice filling: (cf. half filling n = 1) S. Watanabe, A. Tsuruta, K. Miyake & J. Flouquet, JPSJ 78 (2009)

Slave-boson mean field theory G. Kotliar & A. E. Ruckenstein, PRL 57 (1996) 1362 : probability for empty, singly-, & doubly-occupied states , ’,  : Lagrange multipliers : mas renormalization factor 7 equations are solved self-consistently Q = ( ,  ): AF-ordered vector

At quantum critical end point (QCP) of first-order valence transition, valence fluctuation diverges: Ground state phase diagram ff U fc ff vv U fc =1.0 n = ff nfnf U fc (QCP) Kondo Mixed Valence ff msms U fc =0.5 AF-paramagnetic boundary almost coincides with valence transition & valence crossover line Suppression of AF order by valence fluctuations in CeRhIn 5 t = 1, V = 0.2, U =∞ H=0

Drastic change of Fermi surface h=0.005 t = 1, V = 0.2, U =∞, U fc =0.5, n = 0.9, “Small” Fermi surface changes to large Fermi surface at AF to paramagnetic transition discontinously :k F for conduction band  k at n c =0.8 kFkF S. Watanabe et al., JPSJ 79 (2010)

Mass enhancement h=0.005 t = 1, V = 0.2, U =∞, U fc =0.5, n = 0.9, Gap between original lower hybridized band & the folded band increases f-dominant flat part of the folded band approaches Fermi level  Mass enhancement by band effect This explains scaling G. Knebel et al., JPSJ 77 (2008) As  f increases toward, Z   increases E E S. Watanabe et al., JPSJ 79 (2010)

Comparison with dHvA measurement For  f =-0.4, D(  )=0.84 is about 10 times larger than D c (  )=0.092 at n c =0.8 H. Shishido et al., JPSJ 74 (2005) 1103 At P = 0  = 50 mJ/molK 2 in CeRhIn 5 is about 10 times larger than  = 5.7 mJ/molK 2 in LaRhIn 5 missing  2 branch for P >P c Larger D  for  f > h makes D  small CeCoIn 5 PcPc

(A) (B) (C) (C’) V CeCu 2 (Si,Ge) 2 β-YbAlB 4 etc. CeRhIn 5 under H Effect of hybridization strength on P-T phase diagram S. Watanabe & KM: J. Phys. Condens. Matter, 23 (2011) CeCoIn 5, CeIrIn 5

New universality class of QCP: Critical end point of valence transition in heavy fermions S. Watanabe, KM: PRL 105 (2010)

Unconventional criticality in  -YbAlB 4 T(K) Y. Matsumoto et al., arXiv: S. Nakatsuji et al., Nature Phys. 4 (2008) 603 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/T 1 T YbRh 2 Si 2 T -lnT T -0.6 T -0.5  -YbAlB 4 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/T 2D 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-T 1/4 T -3/2 C.W. T -3/4

c f Quantum criticality of VQCP Construction of mode-coupling theory for valence fluctuations starting from H PAM cf. Hubbard model SCR theory for spin fluctuations Local correlation effect by U K. M, J. Phys.:Condens. Matter 19 (2007) S. Watanabe & K. M., Phys. Status Solidi B 247 (2010) 490 P T P T P T Ce VQCP 1st order valence transition valence crossover critical end point Ce, Yb compounds Most of large Usmall U  -YbAlB 4 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/(T 1 T ) ~  (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 R W >>2 Unified understanding expected for unconventional NFL in  -YbAlB 4, YbRh 2 Si 2, YbAuCu 4 (J. L. Sarrao et al 1999) Ce 0.9-x La x Th 0.1 (J. C. Lashley et al 2006) YbCu 5-x Al x (E. Bauer et al 1997), etc.

Connection to high-T c cuprates

Mukuda et al: JPSJ 77 (2008) True phase diagram in high-T c cuprates free from disorder cf. Varma et al: Solid State Commun. 62 (1987) 681 p-d charge transfer mechanism for cuprates due to U dp

S. Shimizu et al: JPSJ 80 (2011) layered high-T c 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-T c cuprates & Critical valence transition or crossover seems to be crucial for understanding CeRhIn 5, 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 RPA: At critical end point as well as QCP,

YbCu 5-x Al x C/T ~ -logT E. Bauer et al, PRB 56 (1997) /3  (T ) ~ T T (K) x =1.5: Yb valence x x x T VQCP 0 K 300 K 10 K x =1.5 Yb valence crossover occurs near x ~ 1.5 cf. K. Yamamoto et al, JPSJ 76 (2007) H. Yamaoka et al, PRB 80 (2009) X-ray L III absorption edge measurements

CeIrIn 5 12T 15T 17T C. Capan et al PRB 70 (2004) R Residual resistivity increases T-linear resistivity Valence crossover line in T-H phase diagram Q-CEP of 1st-order valence transition S. Watanabe et al, PRL 100 (2008) convex shape FL region

CeIrIn 5 C. Capan et al., PRB 80 (2009) H c =28 T  0 has a peak at H c Fermi surface volume does not change at H=H c Consistent with field-induced V-QCP Watanabe, Tsuruta, K.M. & Flouquet, PRL 100 (2008) ; JPSJ 78 (2009) Kondo MV Fermi surface is always large i.e., c-f hybridization is always finite (f electrons are always itinerant) Adiabatic continuation holds with Luttinger’s sum rule satisfied S. Watanabe et al., JPSJ 75 (2006)

Fermi surface in AF phase dHvA: H=15T h=0.005 Contour plot of lower hybridized band :Fermi surface of  conduction band  k at n c =0.8 Small Fermi surface 10.8 i.e., V =0 in H PAM = == Fermi surface in AF phase for V=0.2 is nearly the same as small Fermi surface for V=0 For P <P c Fermi surface in CeRhIn 5 is very similar to LaRhIn 5 h=0.005 t = 1, V = 0.2, U =∞, U fc =0.5, n = 0.9, E E

Knebel et al: JPSJ 77 (2008)  f (P) U fc V-QCP Crossover of valence 1 st order V-T Expected QCP-VT  f (P) U fc V-QCP Crossover of valence 1 st order V-T H=0 H cf. Park et al: Nature 440 (2006) 65 AF Watanabe (Next Talk)

1st-order valence transition surface valence crossover surface 0K VQCP U fc T ff Quantum valence criticality  v ~  (T) ~ T -  0.5 <  < 0.7 ~~ 1/(T 1 T ) ~ T -  T-linear resistivityC/T ~ -logT Large Wilson ratio R W >>2 S. Watanabe & K.M., PRL 105 (2010) Mode-coupling theory for valence fluctuations gives: YbRh 2 Si 2 T -logT T -0.6 T -0.5  -YbAlB 4 T 1.5 T -logT T -0.5 exp. desired P. Gegenwart et al, Nature 4 (2008) 186 S. Nakatsuji et al, Nature Phys. 4 (2008) 603  C/T  1/(T 1 T ) S. Watanabe et al, JPSJ 78 (2009)

Quantum criticality in YbRh 2 Si 2 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 YbRh 2 Si 2 shows magnetic transition at T N = 65 mK  ~ T -logT  ~ T -0.6

YbRh 2 Si 2 K. Ishida et al., PRL 89 (2002) /(T 1 T ) ~ T -1/2

Lagrange multiplier i m = 1,..., N To construct mode-coupling theory for valence fluctuations by U fc after taking account of local correlations by U, we employ slave-boson large-N expansion framework: H PAM : Lagrangian b i : slave-boson operator

For with, we introduce identity For, saddle point solution is obtained by stationary condition  S 0 =0: “perturbed” “unperturbed” U fc q =  q, b q =b  q

Gaussian fixed point Renormalization group analysis: d = 3, z =3 J. A. Hertz, PRB 14 (1976) 1165 A. J. Millis, PRB 48 (1993) 7183 Higher order terms than Gaussian term are irrelevant Gaussian fixed point q’=sq,  ’=s z  q  qcqc qc/sqc/s cc c/szc/sz 0

Mode coupling theory of valence fluctuations Construct best Gaussian taking account of mode couplings for up to j=4 terms Feynman’s inequality: Self-consistent equation for  l i : mean free path by impurity scattering K.M., O. Narikiyo & Y. Onishi, Physica B (1999) 676 Variational principle

Divergence of uniform spin susceptibility Critical valence fluctuations are qualitatively described by RPA framework with respect to U fc Dynamical f-spin susceptibility has common structure to  v (q,i  l )  =  At V-QCP, renormalized valence fluctuation  v (0,0) diverges (0,0) diverges Uniform spin susceptibility diverges DMRG S. Watanabe & K.M., arXiv: result Watanabe et al.: JPSJ 75 (2006)

Almost flat dispersion of valence fluctuation mode D=1, V=0.5, U= -D-D D  ffcc (q,0) 0  cfcf (q,0) 0 { Almost q-independent dispersion emerges in Kondo regime & also in valence-fluctuation regime Local correlation effect U A: extremely small !!  f =-1.0  f =-0.5  f = 0.0

Unconventional criticality by valence fluctuations In clean system C q =C/q in d=3, for Aq B 2 <  ~ A A when at V-QCP (y 0 =0)  v (0,0) =  -1 Now we consider low-T regime (T<<T* F ) T 0 is extremely small due to small A (V-QCP) t =T/T 0 is enlarged even for low T Least square fit of y(t) for 5<t<100  v ~  (t) ~ t -  0.5 <  < 0.7 ~~ 1/(T 1 T ) ~  (t) ~ t -  shown below y 1 =1

T-linear resistivity In y > 1 ( t > 5 ) regime, T-linear resistivity appears ~ ~ Cq=C/qCq=C/q If A is extremely small, dynamical exponent z may be regarded as z = in C q =C/q z-2 When 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) S. Watanabe & K.M., arXiv: C/T ~ -lnT Specific heat: vv y 1 =1 Ueda & Moriya: JPSJ 39 (1975) 605