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Qimiao Si Rice University

Quantum Criticality and Fermi Surfaces. Qimiao Si Rice University. KITPC, July 6, 2007. Stefan Kirchner, Seiji Yamamoto, (Rice University) Silvio Rabello, J. L. Smith Lijun Zhu (UC Riverside) Eugene Pivovarov (UC San Diego) Kevin Ingersent (Univ. of Florida)

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Qimiao Si Rice University

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  1. Quantum Criticality and Fermi Surfaces Qimiao Si Rice University KITPC, July 6, 2007

  2. Stefan Kirchner, Seiji Yamamoto, (Rice University) Silvio Rabello, J. L. Smith Lijun Zhu (UC Riverside) Eugene Pivovarov (UC San Diego) Kevin Ingersent(Univ. of Florida) Daniel Grempel(CEA-Saclay) Jian-Xin Zhu (Los Alamos) Jianhui Dai (Zhejiang U.) S. Paschen P. Gegenwart R. Küchler T. Lühmann S. Wirth Y. Tokiwa,N. Oeschler T. Cichorek K. Neumaier O. Tegus O. Trovarelli C. Geibel J. A. Mydosh F. Steglich P. Coleman E. Abrahams

  3. Materials(possibly)showing Quantum Criticality • Insulating Ising magnet • LiHoF4: transverse field Ising model • Heavy fermion metals • ‘‘Simple’’ magnetic metals • Cr1-xVx, Sr3Ru2O7, MnSi (1st order, but …) … • High Tc superconductors • Mott transition • V2O3, …: QCP? (magnetic ordering intervenes at low T!) • ultracold atoms: 2nd order? • Field-driven BEC of magnons • MIT/SIT/QH-QH in disordered electron systems

  4. Heavy fermions near an antiferromagnetic QCP: (modern era of the heavy fermion field) H. v. Löhneysen et al, PRL 1994 CeCu6-xAux TN AF Metal Linear resistivity TN

  5. Heavy fermions near an antiferromagnetic QCP: (modern era of the heavy fermion field) N. Mathur et al, Nature 1998 CePd2Si2 H. v. Löhneysen et al, PRL 1994 CeCu6-xAux TN TN AF Metal Supercond. AF Metal Linear resistivity YbRh2Si2 TN J. Custers et al, Nature 2003

  6. T=0 spin-density-wave transition order parameter fluctuations in space and (imaginary) time

  7. T=0 spin-density-wave transition order parameter fluctuations in space and (imaginary) time

  8. Dynamical and Static Susceptibilities in CeCu5.9Au0.1 • /T scaling • Fractional exponent a=0.75 • a=0.75 `everywhere’ in q. A. Schröder et al., Nature ’00; PRL ’98; O. Stockert et al., PRL ’98; M. Aronson et al., PRL ‘95 1/c(q) q=0 . . . q=Q T0.75 INS @ q=Q INS and M/H

  9. Quantum critical heavy fermions • prototype for quantum criticality • local quantum criticality – destruction of Kondo entanglement • Fermi surfaces of Kondo lattice • Global phase diagram • Extended-DMFT of Kondo Lattice • Some recent experiments • Bose-Fermi Kondo Model • Kondo-destroying QCP

  10. Single-impurity Kondo model • Kondo temperature TK0 • Kondo singlet • Kondo resonance: spin ½ charge e

  11. Kondo lattices: Historical development of heavy Fermi liquid: Single-impurity: Anderson, Wilson, Nozières, Andrei, Wiegmann, Coleman, Read & Newns, … Lattice: Varma, Doniach, Auerbach & Levin, Millis & Lee, Rice & Ueda, …

  12. Local Quantum Critical Point • Destruction of Kondo screening (entanglement): energy scale Eloc*  0 marks an electronic slowing down at the magnetic QCP QS, S. Rabello, K. Ingersent, & J. L. Smith, Nature 413, 804 (2001) QS, J. L. Smith, and K. Ingersent, IJMPB 13, 2331 (1999)

  13. Local Quantum Critical Point • Destruction of Kondo screening (entanglement): energy scale Eloc*  0 marks an electronic slowing down at the magnetic QCP • Spin damping: anomalous exponent and /T scaling • Fermi surface jumps from “large” to “small” QS, S. Rabello, K. Ingersent, & J. L. Smith, Nature 413, 804 (2001) QS, J. L. Smith, and K. Ingersent, IJMPB 13, 2331 (1999)

  14. Quantum critical heavy fermions • prototype for quantum criticality • local quantum criticality – destruction of Kondo entanglement • Fermi surfaces of Kondo lattice • Global phase diagram • Extended-DMFT of Kondo Lattice • Some recent experiments • Bose-Fermi Kondo Model • Kondo-destroying QCP

  15. Fermi surface inside the antiferromagnetic part of the Kondo lattice phase diagram

  16. Kondo lattice JK<<Irkky<<W JK=0 as the reference point of expansion

  17. Local Moments:Quantum non-Linear Sigma Model Heisenberg model + coherent spin path integral QNLM Haldane (1983) Affleck (1985) : Chakravarty, Halperin, Nelson PRB 39, 2344 (1989) Not important inside ordered phase Conduction electrons Effective fermi-magnon coupling N.B.: coupling irrelevant (see next slide) S. Yamamoto & QS, PRL (July 5, 2007); cond-mat/0610001

  18. At JK=0: Kondo lattice Local-moment antiferromagnetism Conduction electrons ω k q 0 Q

  19. Effective Kondo coupling w/ magnons Q Large momentum transfer scattering NOT allowed kinematically. with AF order Forward scattering ALLOWED. RELEVENT ???

  20. note: remember for forward scattering Combined bosonic/fermionic* RGTree-level scaling (*ferminoc RG: Shankar, RMP ’94) marginal! From NLM From electrons near FS

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

  22. RG at 1-loop & beyond: ~(dΛ)3/2

  23. RG at 1-loop & beyond: Marginal even at 1-Loop and beyond. AFphase w/ “small” Fermi surface! S. Yamamoto & QS, PRL (July 5, 2007); cond-mat/0610001

  24. RG at 1-loop & beyond: Marginal even at 1-Loop and beyond. Large N: = AFphase w/ “small” Fermi surface! ~ No pole in self energy. Fermi surface remains “small”. S. Yamamoto & QS, PRL (July 5, 2007); cond-mat/0610001

  25. RG at 1-loop & beyond: Marginal even at 1-Loop and beyond. Side: AFS: “hybridization” is finite at finite energies; it only goes to zero upon reaching the fixed point. Implications for the choice of variational wavefunctions [H. Watanabe and M. Ogata, arXiv:0704.1722] AFphase w/ “small” Fermi surface!

  26. Quantum critical heavy fermions • prototype for quantum criticality • local quantum criticality – destruction of Kondo entanglement • Fermi surfaces of Kondo lattice • Global phase diagram • Extended-DMFT of Kondo Lattice • Some recent experiments • Bose-Fermi Kondo Model • Kondo-destroying QCP

  27. Kondo lattice Kondo coupling JK JK JK=0

  28. Kondo lattice G ~ frustration, reduced dimensionality, etc. G Local-moment magnetism, Irkky Kondo coupling JK JK JK=0

  29. JK<<Irkky<<W G AFS JK Néel, without Kondo screening

  30. JK >>W>>Irkky paramagnet, w/ Kondo screening G PML JK Cf. A. C. Hewson, The Kondo Problem to Heavy Fermions (Cambridge Univ. Press, 1993)

  31. JK >>W>>Irkky • xNsite tightly bound local singlets (cf. If x were =1, Kondo insulator) • (1-x)Nsite lone moments:

  32. JK >>W>>Irkky • xNsite tightly bound local singlets (cf. If x were =1, Kondo insulator) • (1-x)Nsite lone moments: • projection: • (1-x)Nsite holes with U=∞

  33. JK >>W>>Irkky • xNsite tightly bound local singlets (cf. If x were =1, Kondo insulator) • (1-x)Nsite lone moments: • projection: • (1-x)Nsite holes with U=∞ • Luttinger’s theorem: (1-x) holes/site in the Fermi surface (1+x) electrons/site ---- Large Fermi surface!

  34. Global phase diagram G I PML II AFS AFL JK SDW of PML QS, Physica B 378, 23 (2006); QS, J.-X. Zhu, D. Grempel, JPCM ‘05

  35. Type II transition Hertz-Moriya-Millis fixed point for T=0 SDW transition Type I transition Second order if Destruction of Kondo screening where magnetism sets in

  36. Type I quantum transition (cont’d) G I PML AFS AFL

  37. Quantum critical heavy fermions • prototype for quantum criticality • local quantum criticality – destruction of Kondo entanglement • Fermi surfaces of Kondo lattice • Global phase diagram • Extended-DMFT of Kondo Lattice • Some recent experiments • Bose-Fermi Kondo Model • Kondo-destroying QCP

  38. Kondo lattices:

  39. Extended-DMFT* of Kondo Lattice (* Smith & QS; Chitra & Kotliar) • Mapping to a Bose-Fermi Kondo model: + self-consistency conditions • Electron self-energy Σ () G(k,ω)=1/[ω – εk - Σ(ω)] • “spin self-energy” M () (q,ω)=1/[ Iq + M(ω)]

  40. E-DMFT solution to the Kondo lattice • The self-consistent fluctuating field bath: • Destruction of Kondo screening: Kondo JK Critical Divergent χloc(ω) locates the local problem on the critical manifold g QS, S. Rabello, K. Ingersent, & J. L. Smith, Nature 413, 804 (2001)

  41. Kondo lattice with Ising anisotropy: Extended-DMFT Kondo JK Critical g

  42. Kondo lattice with Ising anisotropy: Quantum Monte Carlo of EDMFT Kondo JK • At the QCP, I ≈ Ic cloc(wn) Critical g wn D. Grempel and QS, Phys. Rev. Lett. ’03

  43. Kondo lattice with Ising anisotropy: Quantum Monte Carlo of EDMFT Kondo JK • At the QCP, I ≈ Ic cloc(wn) Critical g c -1(Q,wn) wn wn D. Grempel and QS, Phys. Rev. Lett. ’03 a = 0.72

  44. Kondo lattice with Ising anisotropy Quantum Monte Carlo d ≡ IRKKY / TK0 J.-X. Zhu, D. Grempel, and QS, PRL (2003)

  45. EDMFT of Anderson lattice with Ising anisotropy EDMFT of Jc1 Jc2 d ≡ IRKKY / TK0 P. Sun and G. Kotliar, Phys.Rev.Lett. ’03

  46. Avoiding double-counting in EDMFT Quantum transition is second order only whenΔIq is kept the same on the paramagnetic and magnetic sides QS, J.-X. Zhu, & D. Grempel, J. Phys.: Condens. Matter 17, R1025 (2005) see also: P. Sun & G. Kotliar, Phys. Rev. B71, 245104 (2005)

  47. Kondo lattice with Ising anisotropy Numerical RG Quantum Monte Carlo (T=0) (T=0.01 TK0) d ≡ IRKKY / TK0 J.-X. Zhu, S. Kirchner, QS, & R. Bulla, cond-mat/0607567 (2006); See also, M. Glossop & K. Ingersent, cond-mat/0607566 (2006) J.-X. Zhu, D. Grempel, and QS, Phys. Rev. Lett. (2003)

  48. Quantum critical heavy fermions • prototype for quantum criticality • local quantum criticality – destruction of Kondo entanglement • Fermi surfaces of Kondo lattice • Global phase diagram • Extended-DMFT of Kondo Lattice • Some recent experiments • Bose-Fermi Kondo Model • Kondo-destroying QCP

  49. dHvA across M agnetic Transition in CeRhIn5 T. Park et al., Nature 440, 65 (‘06); G. Knebel et al., PRB74, 020501 (‘06) H. Shishido, R. Settai, H. Harima, & Y. Onuki, JPSJ 74, 1103 (2005) _

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