Quantum Criticality in Heavy Fermion Metals. Qimiao Si. Rice University

Transcription

1 Quantum Criticality in Heavy Fermion Metals Qimiao Si Rice University CM Seminar, Texas A&M U, College Station, Oct. 2, 2009

2 Pallab Goswami, Jed Pixley Seiji Yamamoto Stefan Kirchner Jian-Xin Zhu, Lijun Zhu Kevin Ingersent Daniel Grempel Ralf Bulla Jianhui Dai (Rice University) (NHMFL, FSU) (MPI-PKS) (Los Alamos) (Univ. of Florida) (CEA-Saclay) (U. Cologne) (Zhejiang U.) S. Friedemann C. Krellner Y. Tokiwa P. Gegenwart S. Paschen S. Wirth N. Oeschler T. Westerkamp R. Küchler T. Lühmann T. Cichorek K. Neumaier O. Tegus O. Trovarelli C. Geibel J. A. Mydosh F. Steglich P. Coleman E. Abrahams

3 H = -I < ij> σ z i σ z j temperature T B T=0 A A: every spin (spontaneously) points up Order parameter: m= lim B: every microstate equally probable: m=0 h 0 lim + Nsite M /N site = 1

4 H = -I < ij> σ z i σ z j - (I δ ) σ i x i temperature T T=0 A B C control parameter δ A: every spin (spontaneously) points up Order parameter: m= lim B: every microstate equally probable: m=0 h 0 lim + Nsite M /N site = 1 C: every spin points along the transverse field: m=0

5 Quantum Phase Transition H = -I < ij> σ z i σ z j - (I δ ) σ i x i temperature T T=0 A B ordered state Quantum Critical QCP C control parameter δ A: every spin (spontaneously) points up Order parameter: m= lim B: every microstate equally probable: m=0 h 0 lim + Nsite M /N site = 1 C: every spin points along the transverse field: m=0

6 temperature T order QCP δ -- control parameter Following Landau -- fluctuations of order parameter, but in d+z dimensions

7 Chakravarty, Halperin, & Nelson, PRB 39, 2344 (1989) T Following Landau -- fluctuations of order parameter, but in d+z dimensions QCP δ

8 Heavy fermion metals as prototype quantum critical points CeCu 6-x Au x H. v. Löhneysen et al CePd 2 Si 2 Linear resistivity N. Mathur et al T N YbRh 2 Si 2 J. Custers et al

9 A. Schröder et al., Nature 00; PRL 98; O. Stockert et al., PRL 98; M. Aronson et al., PRL 95 Spin Dynamics in CeCu 5.9 Au 0.1 ω/t scaling α=0.75 `everywhere in q. Fractional exponent α=0.75 1/χ(q) q=0. q=q AF Q ω/t INS and M/H T 0.75

10 How can the order-parameter-flucutation description break down? Quantum temporal fluctuations may not simply be extra dimensions of classical order-parameter fluctuations: Z ~ Z ( config config in (x, τ) ) The partition function for an individual configuration in space and time may not be positive semi-definite (cf, the minus sign problem of quantum Monte Carlo).

11 Beyond the order-parameter fluctuations: Inherent quantum modes may be important -- need to identify the additional critical modes before constructing the critical field theory.

12 Kondo lattices:

13 Kondo lattices: heavy Fermi liquid: Kondo singlet Kondo resonance Historical development of heavy Fermi liquid: Single-impurity: Anderson, Wilson, Nozières, Andrei, Wiegmann, Coleman, Read & Newns, Affleck & Luwdig, Lattice: Varma, Doniach, Auerbach & Levin, Millis & Lee, Rice & Ueda, Fulde & Zwicknagl,

14 Pre-History: Kondo resonance (one local moment in a conduction electron bath) Kondo temperature: Singlet ground state: Kondo resonance: Kondo-screened ground state yields an electronic resonance local moment acquires electron quantum number due to Kondo entanglement

15 Cond. electron band ε (k) Pre-History: Heavy Fermi Liquid Heavy electron bands E 1,2 (k) Kondo resonance Large Fermi surface

16 Heavy fermions meet quantum criticality Kondo breakdown at AF heavy fermion QCP Kondo breakdown in AF heavy fermion phase Experiments

17 Critical Kondo Destruction Kondo Destruction (f-electron Mott localization) at the T=0 onset of antiferromagnetism Cf. QS, S. Rabello, K. Ingersent, & J. L. Smith, Nature 413, 804 (2001); Phys. Rev. B68, (2003) P. Coleman et al, JPCM 13, R723 (2001)

18 Heavy Fermi Liquid (Kondo Lattice) Kondo resonance G c 1 ( k, ω) = ω-ε - Σ( k, ω) k pole in Σ heavy electron bands

19 Heavy Fermi Liquid (Kondo Lattice) Kondo resonance G c 1 ( k, ω) = ω-ε - Σ( k, ω) k k-independent pole in Σ heavy electron bands

20 Extended-DMFT* of Kondo Lattice (* Smith & QS; Chitra & Kotliar) Mapping to a Bose-Fermi Kondo model: + self-consistency conditions Electron self-energy Σ (ω) G(k,ω)=1/[ω ε k - Σ(ω)] spin self-energy M (ω) χ(q,ω)=1/[ I q + M(ω)]

21 Extended-DMFT of Kondo Lattice Kondo Lattice Bose-Fermi Kondo Local moment J k g + self-consistency fermion bath fluctuating magnetic field QS, Rabello, Ingersent, Smith, Nature 01; PRB 03; J-X Zhu, D. Grempel, QS, PRL 03; P. Sun, G. Kotliar, PRL 03 Zhu, Kirchner, Bulla, QS, PRL 07; M. Glossop, K. Ingerset, PRL 07; Glossop, Zhu, Kirchner et al, to be published ( 09)

22 Kondo lattice with Ising anisotropy Quantum Monte Carlo (T=0.01 T K0 ) δ I RKKY / T K 0 J.-X. Zhu, D. Grempel, and QS, Phys. Rev. Lett. (2003)

23 Kondo lattice with Ising anisotropy Transition 1 st order? Important to avoid double counting in EDMFT: QS, J.-X. Zhu, & D. Grempel, JPCM (2005); P. Sun & G. Kotliar, PRB (2005) Quantum Monte Carlo (T=0.01 T K0 ) δ I RKKY / T K 0 J.-X. Zhu, D. Grempel, and QS, Phys. Rev. Lett. (2003)

24 Kondo lattice with Ising anisotropy Numerical RG (T=0) Quantum Monte Carlo (T=0.01 T K0 ) J.-X. Zhu, S. Kirchner, R. Bulla & QS, PRL 99, (2007); See also, M. Glossop & K. Ingersent, PRL 99, (2007) δ I RKKY / T K 0 J.-X. Zhu, D. Grempel, and QS, Phys. Rev. Lett. (2003)

25 Dynamical spin susceptibility of the Local QCP

26 Kondo lattice with Ising anisotropy QMC χ -1 (Q,ω n ) α = 0.72 J.-X. Zhu, D. Grempel and QS, Phys. Rev. Lett. 03 ω n

27 Kondo lattice with Ising anisotropy QMC χ -1 (Q,ω n ) α = 0.72 J.-X. Zhu, D. Grempel and QS, Phys. Rev. Lett. 03 ω n NRG α= 0.83 α= 0.78 J.-X. Zhu, S. Kirchner, R. Bulla & QS, PRL 07 M. Glossop & K. Ingersent, PRL 07

28

29 Cond. electron band ε (k) Pre-History: Heavy Fermi Liquid Heavy electron band E 1,2 (k) Kondo resonance Large Fermi surface

30 Kondo effect inside the antiferromagnet?

31 Heavy fermions meet quantum criticality Kondo breakdown at AF heavy fermion QCP Kondo breakdown in AF heavy fermion phase Experiments

32 Kondo lattice J K << I rkky << t J K =0 as the reference point of expansion: f- local moments: AF, QNLσM conduction electrons: Fermi volume x S. Yamamoto & QS, PRL 99, (2007)

33 Representing the local moments via Quantum non-linear Sigma Model Heisenberg model + coherent spin path integral QNLσM Not important deep Inside ordered phase Effective fermi-magnon coupling

34 Kondo lattice J K << I rkky << t J K =0 as the reference point of expansion: f- local moments: AF, QNLσM conduction electrons: Fermi volume x S. Yamamoto & QS, PRL 99, (2007) J K EXACTLY MARGINAL; AFs small Fermi surface stable

35 Global phase diagram Yamamoto & QS, to be published ( 09); Yamamoto & QS, PRL 07; QS, Physica B 378, 23 (2006) Watanabe & Ogata, PRL 07; Martin & Assaad, PRL 08; Vojta, PRB 08

36 Type II transition: Second order: Destruction of Kondo screening where magnetism sets in (localization-delocalization of f-electrons)

37 Heavy fermions meet quantum criticality Kondo breakdown at AF heavy fermion QCP Kondo breakdown in AF heavy fermion phase Experiments

38 Spin dynamics Dynamical spin susceptibility: cf. CeCu 6-x Au x (A. Schröder et al 98; O. Stockert et al. 98) UCu 5-x Pd x (?) (M. Aronson et al. 95; D. MacLaughlin et al. 00) Static bulk spin susceptibility: cf. CeCu 6-x Au x (A. Schröder et al. 00) YbRh 2 Si 2 (P. Gegenwart et al. 02) CeCoIn 5 (?) (J. Thompson et al. 02, S. Nakatsuji et al. 02) NMR relaxation rate: cf. YbRh 2 Si 2 (K. Ishida et al. 03) CeCu 6-x Au x (R. Walstedt et al. 03): (1/T 1 ) Cu ~ T α [sum over generic q transfer hyperfine coupling?]

39 Hall Effect Evidence for Fermi Surafce Jump in YbRh 2 Si 2 crossover width decreases as T is lowered T=0 (extrapolation): sharp QCP S. Paschen, T. Lühmann, S. Wirth, P.Gegenwart, O.Trovarelli, C. Geibel, F. Steglich, P.Coleman, & QS, Nature 432, 881 (2004)

40 Hall Effect Evidence for Fermi Surafce Jump in YbRh 2 Si 2 crossover width decreases as T is lowered T=0 (extrapolation): sharp QCP S. Paschen, T. Lühmann, S. Wirth, P.Gegenwart, O.Trovarelli, C. Geibel, F. Steglich, P.Coleman, & QS, Nature 432, 881 (2004)

41 dhva measurement of Fermi Surface across Magnetic Transition in CeRhIn 5 T. Park et al., Nature 440, 65 ( 06); G. Knebel et al., PRB74, ( 06) H. Shishido, _ R. Settai, H. Harima, & Y. Onuki, JPSJ 74, 1103 (2005)

42 Multiple Energy Scales of QCP P. Gegenwart, T. Westerkamp, C. Krellner, Y. Tokiwa, S. Paschen, C. Geibel, F. Steglich, E. Abrahams, & QS, Science 315, 969 (2007)

43 SUMMARY Heavy fermions -- prototype quantum critical points Kondo destruction at antiferromagnetic QCP collapsing Kondo scale, on approach from paramagnetic side Kondo destruction inside the antiferromagetic region Jump of Fermi surface across QCP Experiments: Dynamical scaling, multiple energy scales, Fermi surface YbRh 2 Si 2, CeCu 6-x Au x, CeRhIn 5, etc.

44 Bose-Fermi Kondo Model (Smith & QS ; Sengupta) Local moment J k g fermion bath fluctuating magnetic field

45 ε-expansion of Bose-Fermi Kondo model: SU(2) & XY Kondo J K p δ ( ω w p ) ~ ω 1 ε Critical g 0<ε<1: sub-ohmic dissipation

46 ε-expansion of Bose-Fermi Kondo model: SU(2) & XY Kondo J K p δ ( ω w Critical p ) ~ ω Kondo J K 1 ε Ising Critical g g 0<ε<1: sub-ohmic dissipation

47 ε-expansion of Bose-Fermi Kondo model: SU(2) & XY Kondo J K p δ ( ω w Critical p ) ~ ω Kondo J K 1 ε Ising Critical g g Critical: Crucial for LQCP solution 0<ε<1: sub-ohmic dissipation

48 Role of Berry phase S. Kirchner & QS, arxiv: is a geometrical phase and equals the area on the unit sphere enclosed by For ½<ε<1: Retaining Berry phase violates quantum-to-classical mapping Dropping Berry phase restores quantum-to-classical mapping

49 Role of Berry phase S. Kirchner & QS, arxiv: is a geometrical phase and equals the area on the unit sphere enclosed by For ½<ε<1: Retaining Berry phase violates quantum-to-classical mapping Dropping Berry phase restores quantum-to-classical mapping Various other approaches: Large N, NRG, Monte Carlo Zhu, Kirchner, QS, Georges, PRL 04; Glossop, Ingersent, PRL 05 S. Kirchner, QS, PRL 08 S. Kirchner, QS, Ingersent, PRL 09

50 E-DMFT solution to the Kondo lattice The self-consistent fluctuating field bath: =1 ε p δ ( ω w p )~ ω 1 ε 1 ~Im ln1/( iω ) Destruction of Kondo screening: χ loc ( τ ) ~ 1 τ ε ~ 1 τ Kondo J K Critical Divergent χ loc (ω) locates the local problem on the critical manifold g

51 Kondo lattice J K <<I rkky <<W J K =0 as the reference point of expansion: f- local moments: AF, QNLσM conduction electrons: Fermi volume x simplest case: NOT intersecting AF zone boundary

52 Kondo lattice At J K =0: Local-moment antiferromagnetism Conduction electrons ω k 0 Q q

53 Representing the local moments via Quantum non-linear Sigma Model Heisenberg model + coherent spin path integral QNLσM Not important deep Inside ordered phase Effective fermi-magnon coupling

54 Effective Kondo coupling w/ magnons with AF order q Q Large momentum transfer scattering NOT allowed kinematically. Q S( x, τ) m+ ne i Q x q 0 Forward scattering ALLOWED. S. Yamamoto & QS, PRL 99, (2007)

55 RG for mixed bosons and fermions with a Fermi surface All directions scales Only 1 direction scales (Shankar) S. Yamamoto & QS, arxiv:

56 Combined bosonic/fermionic* RG Tree-level scaling (*ferminoc RG: Shankar, RMP 94) marginal! From NLσM From electrons near FS note: remember for forward scattering

57 RG at 1-loop & beyond:

58 RG at 1-loop & beyond: Marginal even at 1-Loop and beyond.

59 RG at 1-loop & beyond: Marginal even at 1-Loop and beyond. AF phase w/ small Fermi surface! S. Yamamoto & QS, PRL 99, (2007)

60 RG at 1-loop & beyond: Marginal even at 1-Loop and beyond. Large N: = ~ AF phase w/ small Fermi surface! No pole in self energy. Fermi surface remains small. S. Yamamoto & QS, PRL 99, (2007)

61 Effective Kondo coupling w/ magnons with AF order q Q Large momentum transfer scattering NOT allowed kinematically. Q S( x, τ) m+ ne i Q x q 0 Forward scattering ALLOWED. S. Yamamoto & QS, PRL 99, (2007) MARGINAL

62 J K <<I rkky <<W Néel, without Kondo screening G AF S J K

63 J K >>W>>I rkky G paramagnet, w/ Kondo screening PM L J K

64 J K >>W>>I rkky xn site tightly bound local singlets (1-x)N site lone moments: (cf. If x were =1, Kondo insulator)

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

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

67 J K <<I rkky <<W Néel, without Kondo screening G Differs from PML phase in two ways: order parameter Fermi surface AF S J K

68 Global phase diagram G I AF S II PM L AF L J K SDW of PM L

69 Kondo-screened AF phases

70 Kondo-breakdown AF phase

71 Representing the local moments via Quantum non-linear Sigma Model Heisenberg model + coherent spin path integral QNLσM

72 Global phase diagram G??? I AF S AF L II PM L J K