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

Large N approach for the Kondo Lattice

Supported by the National Science Foundation

• P. Coleman

(CMT, Rutgers) Hvar, Oct 3rd 2005.

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.

Pc P AFM

• F. Liquid

Heavy Fermion Quantum Criticality

Collapse in TF* of heavy FL. – divergence of m*,

• divergence of T2

coefficient in resistivity Need to develop a single controlled “mean field” theory that connects local moment magnetism with the heavy fermion paramagnet. Strange Universal Metal:

• sub-quadratic

resistance

• γ~1/T0 ln( T0/T)

(Sereni)

• E/T scaling in

χ’’(Qo,E,T)

• anomalous

exponents

Gegenwart et al (2002) Custers et al (2003).

YbRh2Si2 Field tuned QCP

T T T2 T2 T2

E/T Scaling:

Locality? Physics Below the upper Critical Dimension.

CeCu6-xAux (x=0.1)

Schroeder et al, Nature 407, 351 (2000). 10

• 1

100 101 102

E/T E/T

CeCu6-xAux (x=0.1)

Schroeder et al, Nature 407,351 (2000).

Bilayer He-3 on Graphite – 2D Heavy Fermion system.

• M. Neumann, J. Nyéki, A.

Casey,

• B. Cowan & John Saunders,

SCES/LTP 05. Unpublished. “valence” fluctuations into upper layer initially melt the lower layer. As upper layer fills, QPT where lower layer solidifies into a spin liquid or AFM. upper layer ~ “conduction” fermions lower layer ~ almost localized To be published.

Pc P AFM

• F. Liquid

Spins ordered Spins form composite fermions Critical spin and charge modes ?

Kondo Lattice Model (Kasuya, 1951)

Large N: controlled route to Mean Field Theories.

• spherical model: N component spin.

(Sx, Sy, Sz) -> (S1, S2, …SN) 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. Sαβ = f†

αfβ , nf= 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. Sαβ = b†

αbβ , nb= 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:

M 2S/N 0.2 (Anderson ’53, Arovas Auerbach ’87)

Spin liquid 2D Heisenberg Antiferromagnet

The Dream:

M 2S/N 0.2

Spin liquid Antiferromagnet

TK/JH

The Dream:

M 2S/N 0.2

Spin liquid Antiferromagnet

TK/JH

Heavy Fermion Paramagnet.

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. K > 2S overscreened Kondo effect. Singular ground-states with finite entropy. 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=nb=2S to impose perfect screening.
• Antiferromagnetic interaction has Sp(N), N=2j+1 even.

(Read and Sachdev, 90).

Spinon Holon (fermion!) – mediates Kondo interaction. Spinon pairing -> RVB order.

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.

“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.) Single Impurity Calculation.

Lattice

Arovas Auerbach Spinon: 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 pairing solutions. )

• “Holons” deconfine at the

QCP.

• k-dependence of holons

appears to be important in 3D, where with the current approx, we have an annoying 1st order QCP.

• In Full solution with k-

dependence of holon propagators, Mass divergence

• nly possible if Holons become

gapless at each point on the Fermi surface.

• 1/N corrections – U(1) gauge

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 > 0.4

Outlook :

Spin liquid Antiferromagnet

TK/JH

Heavy Fermion Paramagnet with strong magnetic correlations

? 2D

Local Fermi liquid (Calculations in progress) T0 Deconfined Holons and spinons.

M 2S/N 0.4

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 2nd order

QCP.

• Possible extension to one band models ?

Standard Model: QSDW

• Moriya, Doniach,Schrieffer (60s)
• Hertz (76)
• Millis (93)

If deff> 4, f4 terms “irrelevent” Critical modes are Gaussian.

vertex non- singular

F.S. instability NO E/T SCALING , NO MASS DIVERGENCE IN 3D

Trovarelli et al (2000).

YbRh2Si2