Qimiao Si Rice University Kondo Lattices: What do we learn from microscopics? Qimiao Si Rice University 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) KIAS, Oct 24, 2005

A: every spin (spontaneously) points up B temperature T C T=0 A control parameter  A: every spin (spontaneously) points up Order parameter: B: every microstate equally probable: m=0 C: every spin points along the transverse field: m=0

Quantum Phase Transition QCP Quantum Critical ordered state control parameter  temperature T T=0 A B C A: every spin (spontaneously) points up Order parameter: B: every microstate equally probable: m=0 C: every spin points along the transverse field: m=0

Heavy fermion metals near a magnetic QCP: YbRh2Si2 Linear resistivity TN TN J. Custers et al, Nature 2003

T=0 SDW Transition order parameter fluctuations in space and (imaginary) time

T=0 SDW Transition order parameter fluctuations in space and (imaginary) time fermions are integrated out

T=0 SDW Transition order parameter fluctuations in space and (imaginary) time fermions are integrated out

Quantum Critical Electron Systems temperature T Non-Fermi Liquid magnetic order T=0 QCP control parameter  Do non-Fermi liquid electronic excitations in turn change the nature of quantum criticality?

+ Kondo Lattice Model a lattice of s=1/2 local moments, one per site a conduction-electron band

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

Pre-History II: Heavy Fermi Liquid (Kondo Lattice) Slave fermions: w/ constraint: Slave boson:

Pre-History II: Heavy Fermi Liquid (Kondo Lattice) Mean field theory … k-independent pole in Σ

Pre-History II: Heavy Fermi Liquid (Kondo Lattice) Mean field theory … k-independent pole in Σ … heavy electron band Beyond mean field: gauge theory in its Higgs phase

Pre-History II: Heavy Fermi Liquid (Kondo Lattice) Mean field theory … k-independent pole in Σ … heavy electron band Beyond mean field: gauge theory in its Higgs phase Magnetic ordering: SDW out of the heavy quasiparticles

DMFT* of Kondo Lattice Mapping to a self-consistent Kondo model (* Georges and Kotliar, Metzner and Vollhardt, … ) Mapping to a self-consistent Kondo model + self-consistency conditions Correctly describes Kondo screening: heavy fermion phase But: no competing mechanism against Kondo effect: Kondo screening is too robust No dynamical competition between Kondo and RKKY

Extended-DMFT* of Kondo Lattice (* Smith & QS; Chitra & Kotliar; Sengupta & Georges ) 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)]

Extended-DMFT of Kondo Lattice Bose-Fermi Kondo fermion bath Jk Local moment fluctuating magnetic field g + self-consistency Cf. QS, S. Rabello, K. Ingersent and J.L.Smith, Phys. Rev. B ’03 for details

ε-expansion of Bose-Fermi Kondo model: JK Critical g LM Order ε: J. L. Smith & QS ’97; A. M. Sengupta ’97; Higher orders in ε and spin anisotropies: L. Zhu & QS ’02; G. Zarand & E. Demler ’02 J K = 0: S. Sachdev & J. Ye ’93 (large N); M. Vojta, C. Buragohain & S. Sachdev ‘00

ε-expansion of Bose-Fermi Kondo model: Ising SU(2) & XY Kondo Kondo JK JK Critical Critical g g Critical: Crucial for LQCP solution Order ε: J. L. Smith & QS ’97; A. M. Sengupta ’97; Higher orders in ε and spin anisotropies: L. Zhu & QS ’02; G. Zarand & E. Demler ’02 J K = 0: S. Sachdev & J. Ye ’93 (large N); M. Vojta, C. Buragohain & S. Sachdev ‘00

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)

Local Quantum Critical Point Destruction of Kondo screening (Eloc*  0) at the QCP Critical Kondo screening characterizes non-Fermi liquid excitations QS, S. Rabello, K. Ingersent, & J. L. Smith, Nature 413, 804 (2001) QS, J. L. Smith, and K. Ingersent, IJMPB 13, 2331 (1999)

Local Quantum Critical Point Destruction of Kondo effect (Eloc*  0) at the QCP Local susceptibility also diverges: where “spin self-energy” has anomalous exponent QS, S. Rabello, K. Ingersent, & J. L. Smith, Nature 413, 804 (2001)

Kondo lattice with Ising anisotropy EDMFT of (Quantum Monte Carlo algorithm of Grempel and Rozenberg ’99) Eloc* TN d ≡ IRKKY / TK0 The destruction of Kondo resonances (Eloc*  0) meets with the vanishing of the Néel temperature J.-X. Zhu, D. Grempel, & QS, Phys.Rev.Lett. ’03

EDMFT of Anderson lattice with Ising anisotropy ( P. Sun and G. Kotliar, Phys.Rev.Lett. ’03 ) EDMFT of Jc1 Jc2 d ≡ IRKKY / TK0 First order transition results from double-counting of RKKY interaction: QS, J-X Zhu, & D. R. Grempel, Journ. Phys. Cond. Matter ‘05 P. Sun and G. Kotliar, Phys.Rev. B ‘05

Kondo lattice with Ising anisotropy: Evidence for 2nd-order transition at T=0 (cont’d) mAF Eloc* @ T=0.01TK0 d ≡ IRKKY / TK0 mAF  0: continuous AF transition Eloc*  0: destruction of Kondo resonances

Quantum Critical Dynamics Local spin susceptibility at I ≈ Ic ≈ 1.2 T0K : cloc (wn) @ T=0.01TK0 wn Calculated  ≈ 0.7 D. Grempel and QS, Phys. Rev. Lett. ’03

Fractional exponent in the dynamics Inverse peak susceptibility at I ≈ Ic D. Grempel and QS, Phys. Rev. Lett. ’03 c --1(Q,wn) c --1(Q) a = 0.72 a = 0.72 d wn (T, Ic)  T; (T = 0)  (Ic – I)

Fermi Surface Evolution

In what sense is the QCP local? 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.

M. E. Fisher, S-K Ma, & B. G. Nickel, PRL ,76 Inherent quantum nature of the Kondo-destruction critical point (single-impurity Bose-Fermi Kondo model) Order parameter fluctuations: local Φ4 theory with ε=0.5 would be the upper critical “dimension” … M. E. Fisher, S-K Ma, & B. G. Nickel, PRL ,76 J. M. Kosterlitz, PRL ‘76 … for ε>0.5, the QCP would be Gaussian; should see violation of ω/T scaling

Inherent quantum nature of critical Kondo effect (S. Kirchner, T-H Park, QS, & D. R. Grempel, to be published ’05) ε=0.8 Related observations in related models: L. Zhu, S. Kirchner, QS, & A. Georges, Phys. Rev. Lett. ’04; M. Vojta, N-H Tong, & R. Bulla, Phys. Rev. Lett. , ’05 M. Glossop and K. Ingersent, cond-mat/0501601

Dynamical large-N limit of Bose-Fermi Kondo (Parcollet & Georges, PRL ‘97; Cox & Ruckenstein, PRL ‘93) Leading term: T(,T) = f(/T), with f(0) ≠f(∞) Cf. f(0) =f(∞) for Gaussian f.p. (Damle & Sachdev ’97) L. Zhu, S. Kirchner, QS, & A. Georges, Phys. Rev. Lett. ’04; S. Kirchner, L. Zhu, QS, & D. Natelson, cond-mat/0507215

Beyond microscopcs What is the field theory? For a<1, Smag is Gaussian; the q-dependence of M(q,ω) would be smooth. The coupling to Scritical-kondo makes a contribution to M(q,ω) which is presumably also smoothly q-dependent. The spatial anomalous dimension ηspatial=0.

Kondo Lattice in One Dimension (E. Pivovarov, & QS, Phys.Rev. B ’04) Earlier work on spin gap of the Kondo phase: O. Zachar, & A. M. Tsvelik, Phys. Rev. B ’01; E. Sikkema, I. Affleck, & S. R. White, Phys. Rev. Lett. ’97; O. Zachar, S. A. Kivelson, & V. J. Emery, Phys. Rev. Lett. ’96

SUMMARY Microscopic results of Kondo lattices: two types of quantum critical points T=0 SDW transition (Gaussian) Locally quantum-critical: destruction of Kondo effect exactly at the magnetic QCP (interacting) Plausible argument for robustness What is the field theory?