P. Coleman (CMT, Rutgers) Hvar, Oct 3 rd 2005. Supported by the - PowerPoint PPT Presentation

Large N approach for the Kondo Lattice P. Coleman (CMT, Rutgers) Hvar, Oct 3 rd 2005. Supported by the National Science Foundation

Jerome Rech CMT Rutgers/CEA Saclay Indranil Paul Argonne National Laboratory Olivier Parcollet CEA Saclay Gergely Zarand Budapest University of Technology and Economics Coleman, Paul and Rech cond-mat

1. Motivation: quantum criticality. 2. Large N: past work. 3. Schwinger boson approach to the fully screened two-impurity Kondo model. 4. Kondo lattice: work in progress. 5. Discussion: Physics Nobel Prizes 2005.

Strange Universal Heavy Fermion Metal: Quantum Criticality - sub-quadratic resistance AFM F. Liquid - γ ~1/T 0 ln( T 0 /T) (Sereni) - E/T scaling in Collapse in T F * of χ ’’ (Q o ,E,T) P c P heavy FL. - anomalous exponents – divergence of m*, Need to develop a single - divergence of T 2 controlled “mean field” theory coefficient in that connects local moment resistivity magnetism with the heavy fermion paramagnet.

Gegenwart et al (2002) Custers et al (2003). Field tuned QCP YbRh 2 Si 2 T T T 2 T 2 T 2

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

Bilayer He-3 on Graphite – 2D Heavy Fermion system. upper layer ~ “conduction” fermions lower layer ~ almost localized To be published. “valence” fluctuations into upper M. Neumann, J. Nyéki, A. layer initially melt the lower layer. Casey, B. Cowan & John Saunders, As upper layer fills, QPT where SCES/LTP 05. lower layer solidifies into a spin liquid or AFM. Unpublished.

Kondo Lattice Model (Kasuya, 1951) Critical spin and charge modes ? AFM F. Liquid P c P Spins ordered Spins form composite fermions

Large N: controlled route to Mean Field Theories. • spherical model: N component spin. (Sx, Sy, Sz) -> (S 1 , S 2 , …S N ) Major role in development of Wilson-Fisher theory of classical criticality.

Large N: controlled route to Mean Field Theories. • Large N theory of heavy fermions: N-component fermions. S αβ = f † α f β , n f = Q “Abrikosov” fermion. N component fermion. α = 1, 2… N. Q/N fixed. (Read and Newns ’83, PC ’84,’87, Auerbach Levin ’86, Millis Lee ‘87) Collective effects of antiferromagnetism essentially absent from the theory.

Large N: controlled route to Mean Field Theories. • Theory of low dimensional local moment antiferromagnetism. S αβ = b † α b β , n b = 2S “Schwinger” Boson N component boson. α = 1, 2… N. 2S/N fixed. (Arovas Auerbach 1988, Read Sachdev 1990). Fully screened Kondo effect, with its Fermi liquid ground-state has remained an unsolved

The Dream: 2S/N 2D Heisenberg Antiferromagnet M 0.2 Spin liquid (Anderson ’53, Arovas Auerbach ’87)

The Dream: 2S/N Antiferromagnet M 0.2 Spin liquid T K /J H

The Dream: 2S/N Antiferromagnet M 0.2 Spin liquid Heavy Fermion Paramagnet. T K /J H

The Difficulty. Bosons can’t antisymmetrize: only one boson enters the Kondo singlet to produce a partially screened moment. 2S-1

The Solution. Introduce K-screening channels. Let 2S/N and K/N remain fixed as N-> infinity. Parcollet and Georges (1996). Now the Kondo effect appears in the large N limit. 2S-K K-channels K < 2S underscreened Kondo effect. Singular ground-states with finite entropy. K > 2S overscreened Kondo effect. No Fermi liquid behavior.

• O. Parcollet and A. Georges (96). K> 2S. Non Fermi liquid physics of overscreened Kondo Model. • C. Pepin and PC (‘03). K=1 Underscreened KM. • I. Paul and PC (’04). K=1 Underscreened but finite phase shift using boson replicas. • A. Posazhennikova and PC (’05) underscreened

The unsolved snag. Introduce K-screening channels. Let 2S/N and K/N remain fixed as N-> infinity. Parcollet and Georges (1996). Now the Kondo effect appears in the large N limit. 2S = K K-channels How can you tune K/N = 2S/N in the ground-state? Phase shift δ = π /N, so won’t the Kondo resonance vanish as N-> infinity, with negligible contribution to Free energy?

• � K=n b =2S to impose perfect screening. • � Antiferromagnetic interaction has Sp(N), N=2j+1 even. (Read and Sachdev, 90). Holon (fermion!) – Spinon mediates Kondo interaction. Spinon pairing -> RVB order.

0

Entropy Formula Quark Gluon Plasmas! Blaizot et al (03). General derivation: (PC, I. Paul and J. Rech, 05). Exact in large N limit – enables us to compute thermodynamics directly from spectral functions, in single impurity, or in lattice. At low temperatures – if the holons and spinons are gapped, the Fermi liquid develops.

Single Impurity Calculation. “Spinons” and “Holons” are confined, with a gap given by the Kondo scale. Fermi liquid physics emerges at lower energy. (1/N) phase shift * N spin channels * K screening channels = O(N)

Single Impurity Calculation. Interacting Fermi liquid formed with a Wilson ratio W = 1+k. (consistent with Bethe Ansatz Nozieres-Blandin, Ward Ids.)

Lattice Spinon: Arovas Auerbach Holon ~ Mobile Kondo singlet.

Two Impurity Kondo model J. Rech, PC, ). O. Parcollet & G. Zarand (2005)

Kondo Lattice (very preliminary results in 2D – neglecting k-dependence of holon self-energies – ignoring possibility of • “Holons” deconfine at the pairing solutions. ) QCP. 2S/N= K/N > 0.4 Magnetic instability. 2S/N =K/N < 0.4 Spin liquid – Fermi liquid. very similar to Varma-Jones Fixed point. •k-dependence of holons appears to be important in 3D, where with the current approx, k > 0.4 we have an annoying 1 st order QCP. • In Full solution with k- dependence of holon propagators, Mass divergence only possible if Holons become gapless at each point on the Fermi surface. • 1/N corrections – U(1) gauge

Outlook : (Calculations in progress) 2S/N Antiferromagnet Deconfined Holons and spinons. T 0 M 0.4 ? 2D Heavy Fermion Paramagnet with Spin liquid strong magnetic Local Fermi liquid correlations T K /J H

Conclusions Schwinger Bosons can be used to unify antiferromagnetism and • fully screened Fermi liquid in the Kondo lattice model. Holons and spinons gapped in Fermi liquid, but unconfined once • bosons pair, at large N. Simple example of method – two impurity • KM exhibits Jones-Varma quantum phase transition. Spinon and holon gap must close simultaneously at a 2 nd order • QCP. Possible extension to one band models ? •

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

YbRh 2 Si 2 Trovarelli et al (2000).