1. Philipp Gegenwart Max-Planck Institute for Chemical Physics of Solids, Dresden, Germany Experimental Tutorial on Quantum Criticality in strongly correlated electron systems: E.g. Reviews on quantum criticality in strongly correlated electron systems: E.g. G.R. Stewart, Rev. Mod. Phys. 73, 797 (2001). G.R. Stewart, Rev. Mod. Phys. 73, 797 (2001). H. v. Löhneysen, A. Rosch, M. Vojta, P. Wölfle, cond-mat/0606317 H. v. Löhneysen, A. Rosch, M. Vojta, P. Wölfle, cond-mat/0606317 Outline of : Outline of this talk: Introduction Introduction Quantum criticality in some antiferromagnetic HF systems (mainly those studied in Dresden) Quantum criticality in some antiferromagnetic HF systems (mainly those studied in Dresden) Ferromagnetic quantum criticality Ferromagnetic quantum criticality First part Second part

2. T. Westerkamp, J.-G. Donath, F. Weickert, J. Custers, R. Küchler, Y. Tokiwa, T. Radu, J. Ferstl, C. Krellner, O. Trovarelli, C. Geibel, G. Sparn, S. Paschen, J.A. Mydosh, F. Steglich K. Neumaier 1, E.-W. Scheidt 2, G.R. Stewart 3, A.P. Mackenzie 4, R.S. Perry 4,5, Y. Maeno 5, K. Ishida 5, E.D. Bauer 6, J.L. Sarrao 6, J. Sereni 7, M. Garst 8, Q. Si 9, C. Pépin 10 & P. Coleman 11 1 Walther Meissner Institute, Garching, Germany 2 Augsburg University, Germany 3 University of Florida, Gainesville FL, USA 4 St. Andrews University, Scotland 5 Kyoto University, Japan 6 Los Alamos National Laboratory, USA 7 CNEA Bariloche, Argentina 8 University of Minnesota, Minneapolis, USA 9 Rice University, Texas, USA 10 CEA-Saclay, France 11 Rutgers University, USA Collaborators

3. Lattice of certain f-electrons (most Ce, Yb or U) in metallic environment La 3+ : 4f 0, Ce 3+ : 4f 1 (J = 5/2), Yb 3+ : 4f 13 (J = 7/2), Lu 3+ : 4f 14 (6s 2 5d 1,l=3) partially filled inner 4f/5f shells  localized magnetic moment CEF splitting  effective S=1/2 f-electron based Heavy Fermion systems T T* ~ 5 – 50 K localized moments + conduction electrons moments bound in spin singlets

4. Microscopic model: Kondo effect (Jun Kondo ´63) local moment conduction el J: hybridization between local moments and conduction el.  AF coupling J < 0  lnT Kondo- minimum TKTK T5T5 T K : characteristic „Kondo“-temperature T < T K : formation of a bound state between local spin and conduction electron spin  local spin singlet

5. Anderson Impurity Model cond.- el f-elhybridization V sf on-site Coulomb repulsion U ff Formation of an (Abrikosov-Suhl) resonance at E F of width k B T*  extremely high N(E F )  heavy fermions

6. Landau Fermi liquid Lev Landau ´57 Excitations of system with strongly interacting electrons Free electron gas 1:1 correspondence

7. Magnetic instability in Heavy Fermion systems Fermi-surface: Doniach 1977

8. Itinerant (conventional) scenario Moriya, Hertz, Millis, Lonzarich, … g T TNTN gcgc TKTK NFL FL SDW OP fluctuations in space and time AF: z=2 (d eff = d+z) Heavy quasiparticles stay intact at QCP, scattering off critical SDW  NFL “unconventional” quantum criticality (Coleman, Pépin, Senthil, Si): Internal structure of heavy quasiparticles important:  4f-electrons localize Energy scales beyond those associated with slowing down of OP fluctuations

9. CeCu 6-x Au x : x c =0.1 inelastic neutron scattering O. Stockert et al., PRL 80 (1998): critical fluctuations quasi-2D ! A. Schröder et al., Nature 407 (2000): E/T S(q,  )T 0.75  0 T 0.75 H/T 1/  (q) T 0.75 non-Curie-Weiss behavior q-independent  local !! CeCu 6-x Au x

10. FLAF  = p, x, B NFL T Thermal expansion  = –1/V ∂S/∂p  = V -1 dV/dT Specific heat: C/T = ∂S/∂T Itinerant theory:  ~ T  z ~ T -1 (L. Zhu, M. Garst, A. Rosch, Q. Si, PRL 2003) Grüneisen ratio analysis   Resolution: < 0.01Å  l/l = 10 -10 ( l = 5 mm) for T  20 mK, B  20 Tesla

11. Experimental classification:conventional CeNi 2 Ge 2 CeIn 3-x Sn x CeCu 2 Si 2 CeCoIn 5 … unconventional CeCu 6-x Au x YbRh 2 Si 2…

12. CeNi 2 Ge 2 : very clean system close to zero-field QCP P. Gegenwart, F. Kromer, M. Lang, G. Sparn, C. Geibel, F. Steglich, Phys. Rev. Lett. 82, 1293 (1999) See also: F.M. Grosche, P. Agarwal, S.R. Julian, N.J. Wilson, R.K.W. Haselwimmer, S.J.S. Lister, N.D. Mathur, F.V. Carter, S.S. Saxena, G.G. Lonzarich, J. Phys. Cond. Matt. 12 (2000) L533–L540 T K = 30 K, paramagnetic ground state

13. ~ aT 1/2 +bT CeNi 2 Ge 2 : thermal expansion R. Küchler, N. Oeschler, P. Gegenwart, T. Cichorek, K. Neumaier, O. Tegus, C. Geibel, J.A. Mydosh, F. Steglich, L. Zhu, Q. Si, Phys. Rev. Lett. 91, 066405 (2003) ~ aT  1/2 +b In accordance with prediction of itinerant theory

14.   for T  0 CeNi 2 Ge 2 : specific heat R. Küchler et al., PRL 91, 066405 (2003). T. Cichorek et al., Acta. Phys. Pol. B34, 371 (2003).

15. CeNi 2 Ge 2 : Grüneisen ratio  cr (T) ~ T −1/( z) prediction: = ½, z = 2  x = 1   observations in accordance with itinerant scenario INS: no hints for 2D critical fluct. Remaining problem: QCP not identified (would require negative pressure) critical components:  cr =  (T)−bT C cr =C(T)−  T  cr = V mol /  T  cr /C cr  cr ~ 1/T x with x=1 (−0.1 / +0.05)

16. Cubic CeIn 3-x Sn x N.D. Mathur et al., Nature 394 (1998) CeIn 3 R. Küchler, P. Gegenwart, J. Custers, O. Stockert, N. Caroca-Canales, C. Geibel, J. Sereni, F. Steglich, PRL 96, 256403 (2006) Increase of J by Sn substitution Increase of J by Sn substitution Volume change subdominant Volume change subdominant T N can be traced down to 20 mK ! T N can be traced down to 20 mK !

17. CeIn 3-x Sn x R. Küchler, P. Gegenwart, J. Custers, O. Stockert, N. Caroca-Canales, C. Geibel, J. Sereni, F. Steglich, PRL 96, 256403 (2006) Thermodynamics in accordance with 3D-SDW scenario Thermodynamics in accordance with 3D-SDW scenario Electrical resistivity:  (T) =  0 + A’T, however: large  0 ! Electrical resistivity:  (T) =  0 + A’T, however: large  0 !

18. CeCu 6-x M x C/T ~ log T (universal!) H.v. Löhneysen et al., PRL 1994, 1996 A. Rosch et al., PRL 1997 O. Stockert et al., PRL 1998 2D-SDW scenario ? A. Schröder et al., Nature 2000 E/T scaling in  “(q,  )  (q) ~ {T  (q)}  0.75 for all q   locally critical scenario could we disprove 2D-SDW scenario thermodynamically?

19. CeCu 6-x Ag x E.-W. Scheidt et al., Physica B 321, 133 (2002). AF QCP

20. CeCu 5.8 Ag 0.2 R. Küchler, P. Gegenwart, K. Heuser, E.-W. Scheidt, G.R. Stewart and F. Steglich, Phys. Rev. Lett. 93, 096402 (2004).

21. CeCu 5.8 Ag 0.2 R. Küchler et al., Phys. Rev. Lett. 93, 096402 (2004) Incompatible with itinerant scenario!

22. YbRh 2 Si 2 : a clean system very close to a QCP P. Gegenwart et al., PRL 89, 056402 (2002).

23. =Bc=Bc C/T ~ T -1/3 0(b)0(b) J. Custers et al., Nature 424, 524 (2003) YbRh 2 (Si 0.95 Ge 0.05 ) 2

24. Stronger than logarithmic mass divergence ~b  1/3 b=b= 00 YbRh 2 (Si.95 Ge.05 ) 2 stronger than logarithmic mass divergence incompatible with itinerant theory T/b scaling FLAF  NFL T 1 2 J. Custers et al., Nature 424, 524 (2003)

25. Thermal expansion and Grüneisen ratio R. Küchler et al., PRL 91, 066405 (2003) Prediction:  cr (T) ~ T −1/( z) (L. Zhu, M. Garst, A. Rosch, Q. Si, PRL 2003) = ½, z=2 (AF)  x = 1 = ½, z=3 (FM)  x = ⅔

26. AF and FM critical fluctuations P. Gegenwart, J. Custers, Y. Tokiwa, C. Geibel, F. Steglich, Phys. Rev. Lett. 94, 076402 (2005). B // c

27. Pauli-susceptibility P. Gegenwart et al., PRL 2005

28. 29 Si – NMR on YbRh 2 Si 2 K. Ishida et al. Phys. Rev. Lett 89, 107202 (2002): Knight shift K ~  ’(q=0) ~  bulk Saturation in FL state at B > B c Spin-lattice relaxation rate 1/T 1 T ~ q-average of  ’’(q,  ) At B > 0.15 T: Koringa –relation S  1/T 1 TK 2 holds with dominating q=0 fluct. B  0.15 T: disparate behavior  Competing AF (q  0) and FM (q=0) fluctuations   ’’(q,  ) has a two component spectrum

29. Comparison: YbRh 2 Si 2 vs CeCu 5.9 Au 0.1 q q  q Q Q 0 CeCu 5.9 Au 0.1 YbRh 2 Si 2 AF and FM quantum critical fluct. YRS Spin-Ising symmetry Easy-plane symmetry

30. Hall effect evolution S. Paschen et al., Nature 432 (2004) 881: P. Coleman, C. Pépin, Q. Si, R. Ramazashvili, J. Phys. Condes. Matter 13 R723 (2001). Large change of  H though tiny  ordered ! SDW: continuous evolution of  H

31. Thermodynamic evidence for multiple energy scales at QCP Fermi surface change  clear signatures in thermodynamics Multiple energy scales at QCP P. Gegenwart et al., cond-mat/0604571.

32. Conclusions of part 1 There exist HF systems which display itinerant (conventional) quantum critical behavior: CeNi 2 Ge 2, CeIn 3-x Sn x, … YbRh 2 Si 2 : incompatible with itinerant scenario: - - Stronger than logarithmic mass divergence - - Grüneisen ratio divergence ~ T  0.7 - - Hall effect change - - Multiple energy scales vanish at quantum critical point QC fluctuations have a very strong FM component: - - Divergence of bulk susceptibility - - Highly enhanced SW ratio, small Korringa ratio, A/  0 2 scaling - - Relation to spin anisotropy (easy-plane)?

33. Metallic ferromagnetic QCPs ? Itinerant ferromagnets: QPT becomes generically first-order at low-T Experiments on ZrZn 2, MnSi, UGe 2, … M. Uhlarz, C. Pfleiderer, S.M. Hayden, PRL ´04 D. Belitz and T.R. Kirkpatrick, PRL ´99 1) 1)New route towards FM quantum criticality: metamagnetic QC(E)P e.g. in URu 2 Si 2, Sr 3 Ru 2 O 7, … 2) 2)What happens if disorder broadens the first-order QPT?

34. Layered perovskite ruthenates Sr n+1 Ru n O 3n+1 n=1: unconventional superconductor n=2: strongly enhanced paramagnet (SWR = 10) metamagnetic transition! n=3: itinerant el. Ferromagnet (T c = 105 K) n=  : itinerant el. Ferromagnet (T c = 160 K)

35. Field angle phase diagram on “second-generation” samples (RRR ~ 80) 0 20 40 60 80 100 0 200 400 600 800 1000 1200 1400 5 6 7 8 F i e l d [ t e s l a ] T e m p e r a t u r e [ m K ] a n g l e f r o m a b [ d e g r e e s ] S.A. Grigera et al. PRB 67, 214427 (2003) QCEP @ 8 T // c-axis Evidence for QC fluctuations: Diverging A(H) at H c (S.A. Grigera et al, Science 2001)

36. Thermal expansion P. Gegenwart, F. Weickert, M. Garst, R.S. Perry, Y. Maeno, Phys. Rev. Lett. 96, 136402 (2006) Calculation for itinerant metamagnetic QCEP

37. Behavior consistent with 2D QCEP scenario P. Gegenwart, F. Weickert, M. Garst, R.S. Perry, Y. Maeno, Phys. Rev. Lett. 96, 136402 (2006)

38. Thermal expansion on Sr 3 Ru 2 O 7 Compatible with underlying 2D QCEP at H c = 7.85 T  =0 marks accumulation points of entropy

39. 6.57.07.58.08.59.0 1.2 1.6 2.1 T (K) 0.1 0.3 0.6 0.9 1.2 0.2 0.4 0.5 0.7 0.8 1.0 1.1 1.3  cm) B (T) Dominant elastic scattering  Formation of domains! Fine-structure near 8 Tesla S.A. Grigera, P. Gegenwart, R.A. Borzi, F. Weickert, A.J. Schofield, R.S. Perry, T. Tayama, T. Sakakibara, Y. Maeno, A.G. Green and A.P. Mackenzie, SCIENCE 306 (2004), 1154.

40. Thermodynamic analysis of fine-structure 1) 1)No clear phase transitions 2) 2)Signatures of quantum criticality survive in QC regime also: 1/(T 1 T)~1/T @7.9T down to 0.3K!! (Ishida group) 3)First-order transitions have slopes pointing away from bounded state Clausius-Clapyeron:   Enhanced entropy in bounded regime!

41. Conclusion Sr 3 Ru 2 O 7 Quantum criticality in accordance with itinerant scenario for metamagentic quantum critical end point (d=2) Fine-structure close to 8 Tesla due to domain formation Formation of symmetry-broken phase (Pomeranchuk instability)? Unlikely because of enhanced entropy   Real-space phase separation? (C. Honerkamp, PRB 2005) liquid gas two- phase

42. Smeared Ferromagnetic Quantum Phase Transition Theoretical prediction: FM QPT generically first order at T = 0 [D. Belitz et al, PRL 1999] QCEP Sharp QPT can be destroyed by disorder exponential tail [T. Vojta, PRL 2003] [M. Uhlarz et al, PRL 2004 ]

43. The Alloy CePd 1-x Rh x   Orthorhombic CrB structure   CePd is ferromagnetic with T C = 6.6 K   CeRh has an intermediate valent ground state c Ce Pd,Rh   High T measurements suggested quantum critical point (dotted red line)   Detailed low T investigation: tail

44. AC Susceptibility in the Tail Region Crossover transition for x > 0.6 indicated by sharp cusps in  AC ‘ down to mK temperatures Frequency dependence at low frequencies and high sensitivity on tiny magnetic DC fields no long range order Maxima of  ‘(T) in phase diagram  ‘(T) in DC field

45. Spin Glass-like Behavior   Frequency shift (e.g. x=0.85:  T C /[T C  log( )] of 5%)   Spin glass-like behavior No maximum in specific heat but NFL behavior for x ≥ 0.85

46. Grüneisen parameter shows no divergence

47. ”Kondo Cluster Glass“   Strong increase of T K for x ≥ 0.6 indicated by Weiss temperature  P, evolution of entropy and lattice parameters Possible reason for spin glass-like state: Variation of T K for Ce ions depending on Rh or Pd nearest neighbors leading distribution of local Kondo temperatures ”Kondo cluster glass“

48. Conclusion & Outlook Classification of different types of QCPs in HF systems (conventional vs unconventional) Importance of frustration in the spin interaction? Role of disorder? – e.g.: smearing of sharp 1 st order trans.