Heavy fermions : condensed matter theory perspective

Transcription

1 Heavy fermions : condensed matter theory perspective C. Pépin (IPhT, CEA-Saclay + IIP, Natal) M. Norman (Argonne) I. Paul (U. Paris VI) A. Benlagra (Dresden) K-S. Kim (Pohang, Korea) KITP, Oct. 13th 2011

2 Materials in the Revolutionary Wars.! courtesy P. Coleman 1779: end of war of! independence.! "For God's sake and our countries- send copper bottomed ships to relieve the foul and crippled ones."! Copper plated Hull! IRON BOLTS : RAPIDLY CORRODED! 2

3 Heavy Fermion Metals: Extreme Limit of Mass Renormalization. 300 CeAl 3 UBe 13 c -1 (emu/mol ) UBe 13 UBe 13 * m m ~ T(K) 3

4 Ce : Yb : U : S=1/2 L=3 Spin Orbit : J= L-S = 5/2 S=1/2 L=3 S O : J= L+S = 7/2 S=1 L=3+2 S O : J= L-S = 4 Crystal Electric Field effects split the big moments and compete with Hunds rules Ferromagnetic fluctuations valence fluctuations multiple stage screening? Miyake Valence fluctuations at pc Multiple Kondo screening Jaccard 03 Nakatsuji 03

5 Quantum criticality T - ONLY ENERGY SCALE T AF Spin Gap Pair fluctuations? SC High Tc: QCP may play a vital role in setting the normal state properties. Fermi Liquid? QCP? x Do we understand QCPs where the order parameter is uncontraversial? 5

6 Lhoneysen, (1999) 10 CePd 2 Si 2 magnetic order heavy Fermi liquid Custers et al (CP), Nature 2003 Temperature (K) T N antiferromagnetic state superconducting state 3T c Pressure (kbar) Resistivity linear!

7 What is critical and what is not? Clear NFL in transport and specific heat Explained by the standard theory of itinerant magnetism? Compound H c /P c /x c C v T!1? T a Reference YbRh 2 (Si 1 x Ge x ) 2 x c =0.05 Hc kc H?c c =0.66T =0.06T T 0.34 T Dresden, Grenoble CeCoIn 5 H c =5T T T Los Alamos, Grenoble Ce(Cu 1 x Au x ) 6 x c =0.016 Log To T T Karlsruhe CeCu 6 x Ag x x c =0.2 Log To T T 1.1 Gainesville CeNi 2 Ge 2 P c =0 Log To T T 1.4 Karlsruhe, Cambridge U 2 Pt 2 In P c =0 Log To T T Leiden CeCu 2 Si 2 Pc =0 Log To T T 1.5 Dresden, Grenoble Ce(Ni 1 x Pd x )Ge 2 x = T 1/2 0 + T 3/2 Los Alamos Y bagge H =4T Log To T T Ames, Grenoble CeIn 3 x Sn x p c = 26kbar? T 1.6 Dresden U 2 Pd 2 In P c < 0? T Leiden Is the anomaly due to Quantum Criticality? CePd 2 Si 2 P c > 0? T 1.2 Karlsruhe, Dresden CeRhIn 5 P c 1.6GP a? T Los Alamos, Grenoble CeIn 3 P c > 0? T 1.5 Dresden Ce 1 x La x Ru 2 Si 2 x c =0.1 no? Grenoble U 3 Ni 3 Sn 4 P c > 0 no? Leiden p

8 Two scenarios Spin Density Wave Kondo Breakdown T CeNi2Ge2 T YbRh2Si2 AF HF x AF CeCoIn5 HF E* x SDW scenario: big Fermi surface at the QCP QCP with fractionalization

9 Theoretical approaches

10 Low energy properties Universality High energy physics microscopic hamiltonian UV Universality Integrate out the fast degrees of freedom IR Low energy, slow, universal part

11 Low energy properties Universality High energy physics microscopic hamiltonian UV Universality Integrate out the fast degrees of freedom IR Low energy, slow, universal part What is observed around some QCP in heavy fermions Universal Too! Landau Fermi liquid theory verified by ``all conductors above 1D Universal exponents

12 Can we integrate the fermions out of the partition function? effective bosonic theory For example z=2 fermions are mass-less but fast compared to bosons? fermions bosons k

13 Can we integrate the fermions out of the partition function? effective bosonic theory For example z=2 fermions are mass-less but fast compared to bosons? fermions bosons k transverse modes are slow! (controlled by boson fluctuations) Two types of modes cannot be separated at the level of the action k-kf ~ T radial modes are fast

14 The Spin Density Wave Scenario S John A. Hertz. r Hertz-Millis-Moryia-Beal- Monod q 1/τ

15 The spin-fermion model A Abanov,A. Chubukov, RMP 2003 Belitz, Kirkpatrick, Vojta, RMP 05 J Rech, CP, A Chubukov. 3D Spin Density Wave Rosch, 98 Ex: CeNi2Ge2... Q 2D Spin Density Wave into 3D metal Rosch ( 98), Georges, Kotliar, Paul ( 03) hot regions Ex: CeCu6Au, or CeCu6Ag... No anomalous exponent in spin susceptibility

16 Eliashberg theory around itinerant ferromagnetism, or U(1) gauge theory coupled to matter H sf = k c k, c k, + 1 s,0 q S q S q k, q + g c k, c k+q, S q, k,q,, s,0 q, = q 2 + A 2 + O q 4, 4.

17 Eliashberg theory around itinerant ferromagnetism, or U(1) gauge theory coupled to matter H sf = k c k, c k, + 1 s,0 q S q S q k, q + g c k, c k+q, S q, k,q,, k, ω k, ω d=2 1/3 0 2/3 0 Rech, CP, Chubukov (06) 2/3 1/3 0 1/3 s,0 q, = q 2 + A 2 + O q 4, 4. The bare power counting diverges in d apple 3

18 Eliashberg theory around itinerant ferromagnetism, or U(1) gauge theory coupled to matter H sf = k c k, c k, + 1 s,0 q S q S q k, q + g c k, c k+q, S q, k,q,, k, ω k, ω d=2 1/3 0 2/3 0 Rech, CP, Chubukov (06) 2/3 1/3 0 1/3 s,0 q, = q 2 + A 2 + O q 4, 4. The bare power counting diverges in d apple 3 neglect vertex corrections dressed propagators (self-energy) α ḡ2 γv 3 F ḡ 1 β mḡ NE F γv F m B Nm 1. (5.3

19 Eliashberg theory around itinerant ferromagnetism, or U(1) gauge theory coupled to matter H sf = k c k, c k, + 1 s,0 q S q S q k, q + g c k, c k+q, S q, k,q,, k, ω k, ω d=2 1/3 0 2/3 0 Rech, CP, Chubukov (06) 2/3 1/3 0 1/3 s,0 q, = q 2 + A 2 + O q 4, 4. The bare power counting diverges in d apple 3 neglect vertex corrections dressed propagators (self-energy) a) q, 0 b) q, 0 c) q, 0 d) q, 0 α ḡ2 γv 3 F ḡ 1 β mḡ NE F γv F m B Nm 1. (5.3 BKV type singularity Belitz, Vojta, Kirkpatrick(03), Chubukov, Maslov (07) Green, ben Simon(11)

20 And the culprit is...

21 2k F - scattering processes the back -scattering affecting AFM, nematic, and Ferro FS deformed at the hot spots anomalous exponents 16

22 2k F - scattering processes the back -scattering SS Lee (2010) k q k + q l k + l l q p + l p p + q affecting AFM, nematic, and Ferro FS deformed at the hot spots anomalous exponents 16

23 2k F - scattering processes the back -scattering SS Lee (2010) k q k + q l k + l l q p + l p p + q affecting AFM, nematic, and Ferro FS deformed at the hot spots anomalous exponents Metlitski, Sachdev (2010) 16

24 Recent susy-bosonization in high dimensions Hendrik Meier,CP, Efetov Re-summation of the BS processes Curvature effects : charge and spin channels are coupled Re-summation of all non analyticities for the FL theory δω = ζ(3)t 3 πv 2 F { ln 2 (1 + γ I π L) L 2 } +3 ln2 (1 + γπ IIL) L 2 (5 I = c 3 s II = c + s * K.B. Efetov, C. Pepin, H. Meier, Exact bosonization for an interacting Fermi gas in arbitrary dimensions Phys. Rev. Lett. 103, (2009); PRB 82, (2010), preprint

25 Strong(er) coupling

26 Spinon Fermi surface Small electron fermi surface Fractionalized Spin liquid Fermi Liquid Hot Fermi surface Cold Fermi surface Fermi Liquid Quantum Critical Heavy fermi liquid Senthil, Sachdev, Vojta -PRL2003 PRB 2004

27 Entropic considerations Ce : Yb : U : S=1/2 L=3 Spin Orbit : J= L-S = 5/2 S=1/2 L=3 S O : J= L+S = 7/2 S=1 L=3+2 S O : J= L-S = 4

28 Entropic considerations Ce : Yb : U : S=1/2 L=3 Spin Orbit : J= L-S = 5/2 S=1/2 L=3 S O : J= L+S = 7/2 S=1 L=3+2 S O : J= L-S = 4 Kondo screening

29 Entropic considerations Ce : Yb : U : S=1/2 L=3 Spin Orbit : J= L-S = 5/2 S=1/2 L=3 S O : J= L+S = 7/2 S=1 L=3+2 S O : J= L-S = 4 Spin Liquid Kondo screening

30 Entropic considerations Ce : Yb : U : S=1/2 L=3 Spin Orbit : J= L-S = 5/2 S=1/2 L=3 S O : J= L+S = 7/2 S=1 L=3+2 S O : J= L-S = 4 Spin Liquid Kondo screening Cooper pairs

31 Entropic considerations Ce : Yb : U : S=1/2 L=3 Spin Orbit : J= L-S = 5/2 S=1/2 L=3 S O : J= L+S = 7/2 S=1 L=3+2 S O : J= L-S = 4 AF singlets Spin Liquid Kondo screening Cooper pairs

32 Entropic considerations Ce : Yb : U : S=1/2 L=3 Spin Orbit : J= L-S = 5/2 S=1/2 L=3 S O : J= L+S = 7/2 S=1 L=3+2 S O : J= L-S = 4 AF singlets Spin Liquid Kondo screening Cooper pairs Yang, Pines, Fisk, Thompson, 2008 RKKY quenching KB Kondo quenching T0 Spin liquid heavy Fermi liquid Fermi liquid QCP V

33 Competition between Coulomb and Kinetic energy H Coulomb = U i n f n f H Kinetic = ij f i t ijf j

34 Competition between Coulomb and Kinetic energy H Coulomb = U i n f n f H Kinetic = ij f i t ijf j T Spin liquid Super conductor Fermi liquid x Anti ferromagnet Mott transition High Tc

35 Competition between Coulomb and Kinetic energy H Coulomb = U i n f n f H Kinetic = ij f i t ijf j T T Spin liquid Spin liquid Anti ferromagnet Super conductor Fermi liquid Mott transition x Anti ferromagnet Vc Heavy Fermi liquid Mott transition V High Tc Anderson lattice

36 Breakdown of the Kondo effect associated with a Mott transition on the f-electrons P. Coleman (Schroder 2000) deconfinement, fractionalization Burdin, Grempel, Georges (98) breakdown by exhaustion B. Jones (2010) RG on Kondo Breakdown Zhu, Martin, PNP (09) modulations in Kondo breakdown Q. Si, Nature (02-) S. Kirchner (06,08) locally quantum critical Pines, Zhang, Fisk (08) S. Kirchner (06,08) two fluids model Continentino (09) Vekhter + Seo+ CP (10) Paul,Norman (10) SC quantum critical point CP, Norman,Paul (07) selective Mott transition, z=3 regime of fluctuations Senthil, Sachdev,Vojta (04) model for fractionalization, spin liquid

37 Breakdown of the Kondo effect associated with a Mott transition on the f-electrons P. Coleman (Schroder 2000) deconfinement, fractionalization Burdin, Grempel, Georges (98) breakdown by exhaustion B. Jones (2010) RG on Kondo Breakdown Zhu, Martin, PNP (09) modulations in Kondo breakdown Q. Si, Nature (02-) S. Kirchner (06,08) locally quantum critical Pines, Zhang, Fisk (08) S. Kirchner (06,08) two fluids model Continentino (09) Vekhter + Seo+ CP (10) Paul,Norman (10) SC quantum critical point CP, Norman,Paul (07) selective Mott transition, z=3 regime of fluctuations Senthil, Sachdev,Vojta (04) model for fractionalization, spin liquid

38 T T0 SPIN LIQUID AF LRO CeRhIn5 heavy Fermi liquid CeCoIn5 QCP x CeCu5.9Au0.1 Localised Itinerant URu2Si2 Fr YRh2Si2 QCP!line

39 Doping plays the role of pressure Yb(Rh 1-x M x ) 2 Si 2 T (K) T (K) Ir, x = 0.06 T* T N? T LFL? x = 0 T* T LFL T N Is there a spin liquid on the left of T*? Differentiate the scenarios where the KB is tight to AFM transition (Si et al.) from the ones where the KB is alone T (K) Co, x = 0.07 T N T* T LFL? B (T) Dresden group (10)

40 Evidence for localization in 115. Localized AF moments T * (K) Co SC AFM SC Rh X Ir 2 itinerant f moments FIG. 1: In-plane resistivity of CeCoIn 5 plotted (a) vs. T at Paglione 04 0T,and(b)vs.T 2/3 at 12 T. Data for CeRhIn 5 at 0 T and 21 kbar applied pressure is also shown (from Ref. 20). (c) H-T phase diagram of CeCoIn 5, with evolution of the exponent n (in ρ T n ) observed in the non-fermi liquid (NFL) In CeCoIn 5, the evolution of the exponent n, from linear to sub-linear with increasing field, is unusual. Furthermore, it is non-monotonic, with the smallest value observed in a small region near the critical field (shaded area in Fig. KB 1(c)), where + Cross-over n 0.45 between 0.5 K3D and 8 K at 5.1 T. Near this field, there is a qualitative change in transport properties which appears as a sign change in the magnetoresistance (MR), from positive at low H to negative at high H (see data in Fig. 1(c)). The field evolution of the transport power law, together with the MR crossover, are indicative of competing energy scales, and may indeed be a consequence of the presence of two QCPs. A scenario involving two distinct QCPs has recently been identified in the heavy-fermion superconductor CeCu 2 Si 2, where competing magnetic and mixed-valence states give rise to two separate superconducting phases as a function of lattice density [26]. In the case of CeCoIn 5, the two QCPs may involve two distinct groups of Fermi surfaces, for example. It is instructive to compare CeCoIn 5 to its close cousin CeRhIn 5. At ambient pressure and zero field, the former is a superconductor with T c =2.3Kand the latter is an antiferromagnet with T N =3.8 K. Under an applied pressure of 21 kbar, CeRhIn 5 develops a superconducting state [20], with a T c similar to that of CeCoIn 5 but with a zero-field resistivity that is not linear in temperature. Rather, as shown in Fig. 1(b), CeRhIn 5 displays a T 2/3 dependence, with a prefactor and range that are comparable to that of CeCoIn 5 at high field. 2D sub-linear exponents CeCoIn5 Curro 07 Pagliuso Saitovich Contientino (10)

41 Study under Pressure TTAI, Pressure ARTICLE induced superconductivity in 115 series IN PRESS Hisatomo HARIMA1 and Yoshichika O NUKI J. Phys. Soc. Jpn., Vol. 74, No. 4, April, 2005 LETTERS nce, Osaka University, Toyonaka, Osaka H. Shishido et al. / Journal of Magnetism and Magnetic Materials (2004) aculty of Science, Kobe University, Kobe CeRhIn5 60 H // [001] (a) (b) CeRhIn 2.9GPa(a) (a) 5 CeCoIn5 80mK CeRhIn5CeRhIn5 at high hen (dhva) experiment for an antiferromagnet β2 β2 T N due to main dhva branches! and β2 al areas of the Fermi surfaces 40 i to UGe2 explained by two kinds of nearly cylindrical Fermi surfaces ofsimilar a β2 Tsc the corresponding 2 up to about 2.3 GPa, while are unchanged (Lonzarich stalk) 20 y above a pressure P! ¼ 1:6 GPa where pressure-induced 82kOe but AFM 169kOe vs FM H 0 at ambient pressure, 20 m0 at 1.6 GPa and 60 m0 at 2.2 GPa GPa, new dhva branches appear, which are in good agreement 2A CeRhIn (b) 2 0 es of a 4f-itinerant heavy fermion superconductor CeCoIn Pressure (GPa) Pressure (GPa) 5, 2.9GPa (b) CeCoIn5 to the volume n5 becomes itinerant and significantly contributes Fig. 2. Pressure dependence of cyclotron mass of branch b2 in 2 Tsc (a) CeRhIn5 and (b) CeCoIn5 :The inset shows the Fermi haracter is thus changed from localized to itinerant at a critical rconducting transition temperature becomes a maximum. surface corresponding branch b2 : α3 These results indicate that CeRhIn5 approaches the 0 conductivity, quantum critical point A α2α1 critical 4A quantum region with increasing pressure, while H. S HISHIDO et al. P* mber 22, 2004; accepted February 8, 2005) 7 dhva Frequency (x10 Oe) Temperature (K) m*c (m0) 4 Pressure (GPa) Fig. 1. Fig. 1(a) shows pressure dependence of TN and Tsc of CeRhIn5 ; obtained from the resistivity data ðwþ [3] and NMR fermions and ac-susceptibility data &Þ [4]. (b) shows TscCeCoIn of CeCoIn 5: 5 (b) (a) ðj; LaRhIn 5 CeCoIn5 deviates from it with increasing pressure Pc 8 CeRhIn5 6 β2 α1 (a) α1 α2 α3 α2,3 4 a A 2 b c 0 This α work was financially supported by the Grant-inAid for1 COE Research (c) (10CE 2004) from the Ministry CeCoIn 5 of Education, Culture, Sports, Science and Technology α 3 of Japan. 60 CeRhIn5 (b) m c* (m0) mpetition (CeRhIn5) of branch b2 ðmc! ¼ 5:7m0 Þ at ambient pressure reaches (RKKY) about 40m0 at 2 GPa: β2 echnique α2 β2 β2 Fig. 2(b) shows the pressure βdependence of the 2 2α References 2α3 2 mpounds. cyclotron mass mc! of branch b2 in CeCoIn5 which was 40 determined at H ¼ 14 T: The cyclotron mass decreases d cerium β [1] H. Hegger, et al., Phys. Rev. Lett. 84 (2000) α3 with increasing pressure. The cyclotronβmass of branch 1 [2] V.A. Sidorov, et al., Phys. Rev. Lett. 89 (2002) ure Tmag b2 ðm! c ¼ 58m0 Þ at ambient pressure decreases down to et al., J. Phys. 4[3] T. Muramatsu, 10 Soc. Jpn. 70 (2001) pressure 38m0 at 3 GPa: We note that the cyclotron mass is [4] S. Kawasaki, 7 et al., Phys. Rev. B 65 (2002) (R). Oe) dhva Frequency (x10 strongly field-dependent at ambient pressure [7], but the H. Shishido, et al., J. Phys. Soc. Jpn. 71 (2002) d around α1 band 14 [5] - electron 14 - electron cyclotron massband is unchanged on the field at about [6] H. Shishido, et al., Phys. Rev. B 66 (2002) romagnet Fig. 3. (a) dhva oscillation at 2.9 GPa in CeRhIn and (b) its FFT 3 GPa: [7] R. Settai, et al., J. Phys. Condens. Matter 13 (2001) L αspectrum, α3 together with (c) the FFT spectrum in CeCoIn5. 3 α2,3 α1 α1 etragonal 0 α2 α2 pe layers " 7 5:56! 10 Oe (mc ¼ 15 m0 ) in!1, 4:53! 10 Oe (18 m0 ) in lly along Pressure (GPa)!2 and 4:24! 107 Oe (8:4 m0 ) in!3, as shown in Fig. 3(c).10) n5 reveal The origin of branch is not clear, but branch A is close to Fig. 4. Pressure dependence of the dhva frequency and cyclotron mass in banda15 - electron band 15 - electron CeRhIn5 branch " with F ¼ 1! 107 Oe in CeCoIn5, which was, CeRhIn5. With infig. 1. Theoretical Fermi surfaces (a) LaRhIn (b) H k ½001% in however, not inobserved experimentally along 5 (CeRhIn5 ) and 26 Onuki s group (05) CeRhIn5 10) surfaces not shown. CeCoIn5. Small Fermi CeCoIn branch " in CeCoIn5 is identified as a band 5. arehere, ases with 13-hole nearly spherical Fermi Fermi surface, not shown in almost unchanged up to 1.63 GPa,9) but are not clear at indicates Fig. 1(b). higher pressures. however, are properties kinds of 11 cylindrical Fermi asurfaces CeCoIn vendredi two 14 octobre To elucidate change ofin the Fermi5 surface Furthermore, we note that the present dhva experiment

42 Multiple Energy Scales in Quantum Critical Regime Finite energy scales T SF, T QP in QC regime. J. Paglione et al, cond-mat/ Finite T FL at QCP from resistivity. Courtesy J. Flouquet (unpublished) Finite low-energy scale near Kondo breakdown QCP Gegenwart et al, cond-mat/

43 Conclusions Strong experimental evidence for anomalous quantum criticality in HF compounds Breakdown of the conventional techniques which integrate out the fermions for (almost all?) models below d=3. Fractionalization-deconfinement and emerging spin liquid represent the state of the art to explain the data Better theories (and methods) needed... for example Ads/CMT or... a new bosonization technique?