1. The quest to discover the self-organizing principles that govern collective behavior in matter is a new frontier, A New Frontier 12 34 10 ELEMENTS BINARYTERTIARY QUATERNARY SIMPLEST BIOLOGICAL MOLECULES CMR materials Anomalous normal state of cuprates T(K)  ab (m  cm) 0 200 400 600 800 1000 0.0 0.2 0.4 0.6 0.8 1.0 Takagi et al, PRL ‘92 Drop snapping singularity. Avalanches in granular matter.

3. Heavy Electrons and The Riddle of Quantum Criticality Piers Coleman, Rutgers CMT, NJ, USA 1.Local moments and the Kondo effect 2.Heavy fermion metals, superconductors and magnets. 3.Break-up of the Heavy Electron at a quantum critical point. Supported by NSF DMR Cargese, August 2005. Lectures will be given on the blackboard These viewgraphs are supplementary.

4. e-e- I. Local moments and the Kondo effect. “Confinement” Ott et al 1983 II. Heavy fermion materials T T2T2 T2T2 Custers et al (2004) III. HF Quantum Criticality. “Deconfinement” Heavy Electrons and The Riddle of Quantum Criticality Piers Coleman, Rutgers CMT, NJ, USA Cargese, August 2005.

5. Detailed notes: “Local moment physics in heavy electron systems”, P, Coleman, Lectures in the pysics of highly correlated electron systems VI, editor F. Mancini, AIP (2002), 79-160, cond-mat/0206003. General reading: “Many Body Physics: unfinished frontier”, P. Coleman, cond-mat 2002, Ann Henri Poincare 4, S559-S580 (2003), cond-mat/0307004. Quantum Criticality, P. Coleman and Andrew Schofield, Nature, 433, 226- 229 (2005), cond-mat/0503002. Quantum Criticality: “How do Fermi liquids get heavy and die?”, P.C, C. Pépin, Q.Si,R. Ramazashvili J. Phys Cond Matt. 13. R723 (2001), cond-mat/020?001 “Schwinger boson approach to the fully screened Kondo model”, J. Rech et al, cond-mat/0507001.

6. Local Moment Formation and the resistance minimum

7. Mathias and Clogston (1962)

8. Anderson Model of Moment Formation Local Moment Forms

9. Physics of Local Moments e-e- Kondo interaction. 100 200 300 100 200 T(K) CeAl 3 UBe 13  -1 (emu/mol )-1 0 0

10. E N(E) Kondo Physics Kondo Model S=1/2 S=0

11. The Dense Kondo Lattice and the development of coherence.

13. UBe 13 100 200 300 100 200 T(K) CeAl 3 UBe 13  -1 (emu/mol )-1 0 0 Heavy Fermion Metals: Extreme Limit of Mass Renormalization. UBe 13 100 200 300 100 200 T(K) CeAl 3 UBe 13  -1 (emu/mol )-1 0 0 UBe 13 100 200 300 100 200 T(K) CeAl 3 UBe 13  -1 (emu/mol )-1 0 0

14. Interactions in the Lattice. e-e- e-e- Kondo interaction. S. Doniach, 1978.

15. Kondo Lattice Physics in the 1-1-5 Compounds. (Courtesy, J. Sarrao)

16. Frustration induced d-electron heavy fermion system. (Urano et al ’00, Hopkinson and PC, ’02)

17. Beyond the comparison of Energy Scales: what do we know about the detailed physics of the Doniach phase diagram? How does the spin form the heavy quasiparticles- (and what about the “Kondo screening cloud” ? ) When pairing develops: how does the spin participate in the superconducting order parameter? What other forms of exotic order are possible? What is the physics of the Quantum Critical Point?

18. How does the spin form the heavy quasiparticles- and is there a Kondo screening cloud?

22. What kind of Fermi surface transformation accompanies the development of antiferromagnetism?

23. When pairing develops: how does the spin participate in the superconducting order parameter?

24. T(K) (Sarrao et al Nature, ’02) 250 50 T C =18K 50 100 150

25. Composite Pairing. (Andrei and PC, ’88; Miranda, PC, Tsvelik ’94; Balatsky and Abrahams ’95, Andrei, PC, Kee,Tsvelik ’97, Senthil et al ‘02) Kondo Effect in the presence of a spin liquid may drive superconductivity.

26. What is the physics of the Quantum Critical Point?

27. Landau, JETP 3, 920 (1957) Landau: interactions can be turned on adiabatically, preserving the excitation spectrum. Interactions adiabatically - - “Quasiparticle”

29. Non Fermi Liquid T When the interaction becomes too large…. …what happens? QCP Hertz, 1976 c 0 Ordered State. Fermi Liquid Naked singularity in the Phase Diagram

30. PcPc P AFM metal John Hertz: Critical droplet is Quantum if P.C and A. Schofield, Nature (2005) D A N

31. Anomalous normal state of cuprates T(K)  ab (m  cm) 0 200 400 600 800 1000 0.0 0.2 0.4 0.6 0.8 1.0 Takagi et al, PRL ‘92 Heavy Electron Quantum Critical Point. Quantum Criticality in 2 and 3 dimensions (Gegenwart et al, 2001)

32. Gegenwart et al (2002) YbRh 2 Si 2 Field tuned QCP T T T2T2 T2T2 T2T2

33. PcPc P AFM metal Heavy Fermion Materials H. Von Lohneyson (1996) Quantum Criticality: divergent specific heat capacity Quantum Critical Point

34. E/T Scaling: Locality Physics Below the upper Critical Dimension. CeCu 6-x Au x (x=0.1) Schroeder et al, Nature 407, 351 (2000). 10 -1 10 0 10 1 10 2 E/T CeCu 6-x Au x (x=0.1) Schroeder et al, Nature 407,351 (2000).

35. Standard Model: QSDW Moriya, Doniach,Schrieffer (60s) Hertz (76) Millis (93) If d eff > 4,  4 terms “irrelevent” Critical modes are Gaussian. vertex non- singular F.S. instability NO E/T SCALING, NO MASS DIVERGENCE IN 3D

36. Local quantum criticality (1/d- dmft) (Si et al, Nature 2001): Spin is the critical mode, critical fluctuations are only critical in time. Deconfined Criticality ( Pepin et al, JCM 2001, Senthil et al, Science 2004): QCP may involve two length-scales, two fixed points. (1)Crossover to fluid of free spinons at ξ 1 (2)Confinement of spinons at ξ 2 >> ξ 1 (to form a valence bond fluid) by tunneling through monopole gauge fields configurations What is the appropriate gauge theory. What is the appropriate mean field theory? (Spherical model 1/N?) t H Critical Confinement Free spinons ξ2ξ2 ξ1ξ1

37. PcPc P AFM metal Heavy Fermion Materials Spins condensedSpins bound to charge as composite fermions Deconfinement of spin: fundamentally new kind of zero mode. Kondo Lattice Model (Kasuya, 1951)

38. Questions on QCP Assuming that the WFK paradigm still applies – then what is the right mean-field theory to expand around? Assuming that locality is always unstable – how does the electron mass diverge over the entire Fermi surface? What is the nature of the strange metal – are the charge carriers electrons and are the charge carriers degenerate and quantum, or classical and Boltzmanian?

40. PPCPC P PCPC e-e- h+h+

41. T. Namiki, H. Sato, J. Urakawa, H. Sugawara, Y. Aoki, R. Settai and Y. Onuki, Physica B 281&282 359-360 (2000).

42. What new forms of order become possible?

43. U Si Marston and Affleck (89) Marginal Fermi liquid Quantum critical point Pseudogapmetalx Superconductivity Fermi liquid T T* AFM Hidden order? (Chakravarty et al., Varma ’01) Bernal et al, 2001 29 Si NMR Chandra et al (2002) New Forms of Order

44. Open Questions: How do quasiparticles form from spins? How does spin form a superconducting order parameter? What is the nature of the QCP: The Fermi surface transformation? What are the virtual quantum fluctuations that produce the mass renormalization? The nature of the coupling to magnetism? What new forms of order become possible in Kondo spin systems?