1. Qimiao Si Rice University KIAS, Oct 29, 2005 Heavy fermion metals: Global phase diagram, local quantum criticality, and experiments

2. S. PaschenP. GegenwartR. Küchler T. LühmannS. WirthN. Oeschler T. CichorekK. NeumaierO. Tegus O. TrovarelliC. GeibelJ. A. Mydosh F. SteglichP. Coleman Lijun Zhu, Stefan Kirchner, Tae-Ho Park, Eugene Pivovarov, (Rice University) Silvio Rabello, J. L. Smith Kevin Ingersent (Univ. of Florida) Daniel Grempel (CEA-Saclay) Jianxin Zhu (Los Alamos)

3. Quantum Critical Point QCP: existence itself is conceptually simple… … but, can be elusive (required parameter tuning beyond practical range, order hidden, too many competing phases, 1 st order along the physical axis, etc.) and, the nature of the QCP seems to be exceedingly rich.

4. Insulating Ising magnet –LiHoF 4 : transverse field Ising model Heavy fermion magnetic metals ‘‘Simple’’ magnetic metals –Cr 1-x V x, Sr 3 Ru 2 O 7, MnSi (1 st order, but …) … High Tc superconductors (?) Mott transition –V 2 O 3, …: QCP? (magnetic ordering intervenes at low T!) –cold atoms: 2 nd order? Frustrated magnets (?) Field-driven BEC of magnons Materials (possibly) showing Quantum Criticality

5. Metal-insulator transition in 3D Si:P, … –Many theoretical questions remain (Finkelstein scaling theory? local moments?...) MIT in 2DEG of Si-MOSFETs,... (?) –Phase diagram? (Experiments seeing a genuine metal phase?) Superconductor-insulator transitions in films –Phase diagram? (Intermediate metal?) QH-QH and QH-Insulator transitions –2 nd order? Materials (possibly) showing Quantum Criticality

6. Early part of the heavy fermion field Heavy electron mass Unconventional superconductivity Kondo screening  Kondo resonances  – Fermi liquid of heavy quasiparticles On the theory front: Single-impurity: Anderson, Wilson, Nozières, Andrei, Wiegmann, Coleman, Read & Newns, … Lattice: Varma, Doniach, Auerbach & Levin, Millis & Lee, Rice & Ueda, …

7. Past decade of the heavy fermion field Non-Fermi Liquid Behavior Quantum Criticality New focus, perhaps due to cross-fertilization w/ high Tc & other correlated systems

8. Heavy fermions near a magnetic QCP: CeCu 6-x Au x H. v. Löhneysen et al, PRL 1994 AF Metal TNTN TNTN

9. Heavy fermions near a magnetic QCP: CeCu 6-x Au x YbRh 2 Si 2 H. v. Löhneysen et al, PRL 1994 J. Custers et al, Nature 2003 CePd 2 Si 2 N. Mathur et al, Nature 1998 AF Metal TNTN TNTN TNTN Supercond. Linear resistivity

10. Heavy fermions near a magnetic QCP: –YbRh 2 Si easy-plane spin-anisotropy; T K 0 ≈ 25 K –Ce(Cu 1-x Au x ) 6 Ising anisotropy; T K 0 ≈ 6 K –CePd 2 Si 2, CeIn 3 ( first order?--NQR ), CeNi 2 Ge 2, CeCu 2 Si 2 –YbAgGe [frustrated (hexagonal) lattice] –CeMIn 5, –URu 2 Si 2 (?)

11. Kondo Lattice Model I: RKKY interaction; AF G=I nnn /I nn etc.

12. Kondo Lattice Model Bandwidth W Kondo coupling J K I: RKKY interaction; AF G=I nnn /I nn etc.

13. Kondo Lattice Model Bandwidth W Kondo coupling J K Fixed I and W with I<<W, varying G and J K I: RKKY interaction; AF G=I nnn /I nn etc.

14. Kondo lattices JKJK G G ~ frustration, reduced dimensionality, etc. Local moment magnetism, I rkky Kondo coupling J K Bandwidth W

15. JKJK G J K >>W>>I rkky

16. xN site tightly bound local singlets (cf. If x were =1, Kondo insulator) (1-x)N site lone moments:

17. J K >>W>>I rkky xN site tightly bound local singlets (cf. If x were =1, Kondo insulator) (1-x)N site lone moments: –projection: –(1-x)N site holes with U=∞

18. J K >>W>>I rkky xN site tightly bound local singlets (cf. If x were =1, Kondo insulator) (1-x)N site lone moments: –projection: –(1-x)N site holes with U=∞ Luttinger’s theorem: (1-x) holes/site in the Fermi surface (1+x) electrons/site ---- Large Fermi surface!

19. JKJK G PM L paramagnet, w/ Kondo screening J K >>W>>I rkky

20. JKJK G J K <<I rkky <<W Local moment magnetism, I rkky

21. J K <<I rkky <<W With Ising anisotropy, the magnetic spectrum of the local moment component is gapped. J K is irrelevant!

22. J K <<I rkky <<W With Ising anisotropy, the magnetic spectrum of the local moment component is gapped. J K is irrelevant! Local moments stay charge neutral, and do not contribute to the electronic excitations. Fermi surface is small

23. JKJK G AF S QS, J.-X. Zhu, & D. Grempel, cond-mat /0506207 J K <<I rkky <<W Néel, without Kondo screening

24. JKJK G AF S PM L AF L QS, J.-X. Zhu, & D. Grempel, cond-mat /0506207 I II Global phase diagram

25. Type I transition Type II transition Hertz fixed point for T=0 SDW transition Second order if Destruction of Kondo screening at the magnetic QCP

26. Local Quantum Critical Point Fluctuations of the magnetic order parameter are slow at the magnetic QCP The slow magnetic fluctuations decohere the Kondo screening Kondo effect is critical, which is in addition to the critical fluctuations of magnetic order parameter

27. Local Quantum Critical Point QS, S. Rabello, K. Ingersent, & J. L. Smith, Nature 413, 804 (2001) Anomalous spin dynamics Destruction of Kondo effect (E loc *  0) at the QCP

28. Nature of the phases TNTN T 2 resistivity on both sides of the QCP

29. Nature of the phases (cont’d) TNTN Small Fermi surface in the AF metal phase TNTN S. Araki, R. Settai, T. C. Kobayashi, H. Harima, & Y. Onuki, Phys Rev. B 64, 224417 (2001) CeRh 2 Si 2

30. Localization of f-electrons –Reconstruction of the Fermi surface across  QCP –m*  ∞ over the entire Fermi surface as    QCP Anomalous spin dynamics. Destruction of Kondo effect –Non-Fermi liquid excitations part of the quantum-critical spectrum. In what sense is the QCP local?

31. CeCu 6-x Au x (x c ≈0.01) H. v. Löhneysen et al, PRL 1994 AF Metal TNTN TNTN

32. Dynamical and Static Susceptibilities in CeCu 5.9 Au 0.1 A. Schröder et al., Nature ’00; PRL ’98; O. Stockert, H. v. Löhneysen, A. Rosch, N. Pyka, & M. Loewenhaupt, PRL ’98 E/T scaling Fractional exponent  =0.75  =0.75 `everywhere’ in q. INS @ q=Q E/T q=Qq=Q q=0 INS and M/H T 0.75 1/  (q)......

33. Dynamics of the quantum critical CeCu 5.9 Au 0.1 Frequency and temperature dependences of the dynamical spin susceptibility: –an anomalous exponent  < 1 –  /T scaling implying non-Gaussian fixed point The anomalous exponent  is seen essentially `everywhere’ in the momentum space

34. O. Stockert, H. v. Löhneysen, A. Rosch, N. Pyka, & M. Loewenhaupt, Phys. Rev. Lett. ’98

35. TNTN Ce(Ru 1-x Rh x ) 2 Si 2 (x c ≈0.04) H. Kadowaki, Y. Tabata, M. Sato, N. Aso, S. Raymond, & S. Kawarazaki, cond-mat/0504386

36. Ce(Ru 1-x Rh x ) 2 Si 2 H. Kadowaki, Y. Tabata, M. Sato, N. Aso, S. Raymond, & S. Kawarazaki, cond-mat/0504386 TNTN  0 =350mJ/K 2 for x=0 C/T=  0 -a T 1/2

37. Localization of f-electrons –Reconstruction of the Fermi surface across  QCP –m*  ∞ over the entire Fermi surface as    QCP Anomalous spin dynamics everywhere in q. Destruction of Kondo effect –Non-Fermi liquid excitations part of the quantum-critical spectrum. In what sense is the QCP local?

38. Hall Effect in YbRh 2 Si 2 : probing the Fermi-surface change TNTN Linear resistivity

39. S. Paschen, T. Lühmann, S. Wirth, P.Gegenwart, O.Trovarelli, C. Geibel, F. Steglich, P.Coleman, & QS, Nature 432, 881 (2004) Hall Effect in YbRh 2 Si 2

40. S. Paschen, T. Lühmann, S. Wirth, P.Gegenwart, O.Trovarelli, C. Geibel, F. Steglich, P.Coleman, & QS, Nature 432, 881 (2004) Hall Effect in YbRh 2 Si 2

43. Finite T crossover width  T 0.5±0.1 T=0 (extrapolation): sharp jump @ QCP Hall Effect in YbRh 2 Si 2 S. Paschen, T. Lühmann, S. Wirth, P.Gegenwart, O.Trovarelli, C. Geibel, F. Steglich, P.Coleman, & QS, Nature 432, 881 (2004)

44. Finite T crossover width  T 0.5±0.1 T=0 (extrapolation): sharp jump @ QCP Hall Effect in YbRh 2 Si 2 S. Paschen, T. Lühmann, S. Wirth, P.Gegenwart, O.Trovarelli, C. Geibel, F. Steglich, P.Coleman, & QS, Nature 432, 881 (2004)

45. H. Shishido, R. Settai, H. Harima, & Y. Onuki, JPSJ 74, 1103 (2005) _ dHvA in CeRhIn 5

46. Divergence of the Grüneisen Ratio L. Zhu, M. Garst, A. Rosch, and QS, Phys. Rev. Lett. ’03 with

47. Divergence of the Grüneisen Ratio L. Zhu, M. Garst, A. Rosch, and QS, Phys. Rev. Lett. ’03 R. Küchler et al., Phys. Rev. Lett. ’03 with

48. LQCP: x loc ≈ 0.66 to 2 nd order in ε-expansion for the XY case Cf. AF-SDW: x = 1 / z = 1 R. Küchler et al., Phys. Rev. Lett. ’03 Grüneisen exponent in Ge-doped YbRh 2 Si 2

49. Spin-glass QCP in heavy fermions? JKJK G paramagnet, w/ Kondo screening SG, without Kondo screening Type I: interacting f.p. –UPd x Cu 5-x (?):  /T scaling (M. Aronson et al ’95; D. MacLaughlin et al) –Sc 1-x U x Pd 3 (?):  /T scaling (P. Dai et al’04) Type II: Gaussian f.p. –Fluctuation of the spin glass order parameter (Sachdev et al, ’95; Sengupta and Georges ‘95) I II

50. Spin-glass QCP in heavy fermions? S. Wilson, P. Dai et al, Phys. Rev. Lett. ’05 D. Gajewski, R. Chau, and M. B. Maple, Phys. Rev. B (’00)

51. SUMMARY Global phase diagram of the magnetic heavy fermion metals Two types of quantum critical metals –T=0 SDW transition (Gaussian) –Locally quantum-critical: destruction of Kondo effect exactly at the magnetic QCP (interacting) Evidence from magnetic dynamics, Fermi surface evolution, and thermodynamic ratio. Relevance to other strongly correlated metals?