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Studies of Antiferromagnetic Spin Fluctuations in Heavy Fermion Systems.

G. Kotliar

Rutgers University.

Collaborators:

Ping Sun, Sergej Pankov, Antoine Georges, Serge Florens, Subir Sachdev

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.

Motivation. Spin fermion model of Rosch et. al. does it

describe the data ? ( S. Pankov, S. Florens, A. Georges )

EDMFT-QMC calculations for the Anderson Lattice model ( P. Sun).

Conclusion.

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Local moments + Conduction Electrons.

High temperatures local moments and conduction electrons.

Low temperatures, TK >> JRKKY ,

a heavy Fermi liquid forms. The quasiparticles are composites of conduction electrons and spins.

Heavy quasiparticles absorb the spin entropy. Low temperatures TK << JRKKY the moments order. AF

state. Spin ordering absorbs the spin entropy. What happens in between? 2 impurity mode,

Varma and Jones (PRL 1989)

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Early Treatments: Slave Bosons. Link and Bond variables. Crossover from weak to strong coupling as Jrkky/Tk increase. [M.

Grilli G. Kotliar and A. MillisMean Field Theories of Cuprate Superconductors: A Systematic Analysis, M. Grilli, G. Kotliar and A. Millis, Phys. Rev. B. 42, 329-341 (1990).

Analogy with bose condensation. Strong Correlation Transport

and Coherence, G. Kotliar, Int. Jour. of Mod. Phys. B5 (1991) 341-352.

Two states: one with doubled unit cell, one with Luttinger fermi surface (no AF) Mean Field Phase Diagram of

the Two Band  Model for CuO Layers, C. Castellani, M. Grilli and G. Kotliar, Phys. Rev. B43, 8000-8004, (1991).

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Early Treatments: Slave Bosons. Link and Bond variables.<b> coherence order parameter. Crossover from weak to strong coupling as the bqndwith of the

conduction band is varied. Jrkky/Tk increase. [A., M. Grilli, G. Kotliar and A. Millis, Phys. Rev. B. 42, 329-341 (1990).

Analogy with bose condensation., G. Kotliar, Int. Jour. of Mod.

Phys. B5 (1991) 341-352. Finite temperature study, within large N. Bourdin Grempel and

Georges PRL (2000). N. Andrei and P. Coleman, staggered flux vs Kondo state.

Two states: one with doubled unit cell, one with Luttinger fermi surface (no AF) C. Castellani, M. Grilli and G.

Kotliar, Phys. Rev. B43, 8000-8004, (1991).

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Renewed interest: CeCu6-xAux YbRh2Si2

Schroeder et.al. Nature (2000)

Functional form for DMFT, cf marginal fermi liquid.

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Renewed interest: YbRh2Si2 Linear resisitivity = a Log[b/T] T> T*

= 1/T.3 T<T*

Kadowaki Woods ratioA/ 2=const (x-xc) > e

A/ 2=1/(x-xc) .3 (x-xc)<e

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YbRh2Si2, Gegenwart et. al. Susceptility C = 14 times the Yb moment. T0.=-.3 K

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Gegenwart

1 T Cac - = -

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Critical Point

Can one integrate the Fermions? Is the Kondo-RKKY transition relevant to the

magnetic critical point? Rosch et. al. 2d spin fluctuations and 3d

electrons. Motivated by experiments. Explain linear

resistivity, logarithimic enhancement of specific heat, Kadowaki Woods ratio ?

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Critical Point Almost local self energy. Internal consistency: vertex corrections are finite

I Paul and GK Phys. Rev. B 64, 184414 (2001)

Internal consistency: boson and fermion self energy scale the same way.

Thermoelectric power. [Indranil Paul and GK

S (T) /T scales with Obeyed in CeCuAu J. Benz et. al. Physica B

259-261, 380 (1999).

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Does the 2d spin+ 3d fermion model account for the anomalous damping of the spin fluctuations?

ds+z=4 marginally irrelevant coupling. Strictly speaking no E/T scaling, and Asymptotically scaling functions are all mean field

like but can the corrections to scaling mimmick and effective exponent ?

Answer: S. Pankov, S. Florens A. Georges and GK NO. The leading correction to scaling produce an effective exponent eff 1

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Corrections to scaling

[ ]1 2( ) (1 ( ))cLogw ww

- LC = +

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Spin self energy in a self consistent large N solution of the EMDFT equations of the spin fermion model. [Pankov Florens Georges and GK 2003]

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Introduction to DMFT.

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1

10

1( ) ( )

( )n nn k nk

G i ii t i

w ww m w

-

-é ùê ú= +Sê ú- + - Sê úë ûå

DMFT Impurity cavity construction: A. Georges, G. Kotliar, PRB, (1992)]

†

0 0 0

[ ] ( )[ ( , ')] ( ')o o o oS Go c Go c U n nb b b

s st t t t ¯= +òò ò

† †

, ,

( )( )ij ij i j j i i ii j i

t c c c c U n n

0

†( )( ) ( ) ( )L n o n o n S GG i c i c iw w w=- á ñ

10 ( ) ( )n n nG i i iw w m w- = + - D

0

1 † 10 0 0 0 ( )( )[ ] ( ) [ ( ) ( ) ]n n n n S Gi G G i c i c iw w w w- -S = + á ñ

Weiss field

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1

10

1( ) ( )

( )n nn k nk

G i ii t i

w ww m w

-

-é ùê ú= +Sê ú- + - Sê úë ûå

DMFT Impurity cavity construction

1

10

1( ) ( )

V ( )n nk nk

D i ii

w ww

-

-é ùê ú= +Pê ú- Pê úë ûå 0

1 † 10 0 ( )( )[ ] ( ) [ ( ) ( ) ]n n n n S Gi G G i c i c ia bw w w w- -S = + á ñ

†

0 0

( ) ( , ') ( ') ( , ') o o o o o oc Go c n n U n nb b

s st t t t d t t ¯ ¯+òò

† †

, ,

( )( )ij ij i j j i i ii j i

t c c c c U n n

()

1 100 0 0( )[ ] ( ) [ ( ) ( ) ]n n n n Si G D i n i n iw w w w- -P = + á ñ

,ij i j

i j

V n n

( , ')Do t t+

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DMFT: Effective Action point of view.R. Chitra and G. Kotliar Phys Rev. B.(2000), (2001).

Identify observable, A. Construct an exact functional of <A>=a, [a] which is stationary at the physical value of a.

Example, density in DFT theory. (Fukuda et. al.) When a is local, it gives an exact mapping onto a local

problem, defines a Weiss field. The method is useful when practical and accurate

approximations to the exact functional exist. Example: LDA, GGA, in DFT.

It is useful to introduce a Lagrange multiplier conjugate to a, [a,

It gives as a byproduct a additional lattice information.

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Example: DMFT for lattice model (e.g. single band Hubbard).

Observable: Local Greens function Gii ().

Exact functional [Gii () DMFT Approximation to the functional.

[ , ] log[ ] ( ) ( ) [ ]DMFT DMFTij ii iin n niG Tr i t Tr i G i Gw w w-G S =- - S - S +Få

[ ] Sum of 2PI graphs with local UDMFT atom ii

i

GF = Få

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Example: EDMFT

Observable: Local Greens function Gii (). Local spin spin or charge charge correlation P ().

Exact functional [Gii () P (). EDMFT Approximation by keeping only local

graphs in the Baym Kadanoff functional. “Best” “local “ approximation, targeted to the

observable that one wants to compute. Natural extension to treat phases with long

range order. [Chitra and Kotlar PRB 2000]

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DMFT

Top to bottom approach. Captures the physics of Kondo and the

magnetism. To treat the dispersion of the spin fluctuations,

add Bose field. DMFT in the Bose field. Functional formulation, ordered and disordered

phases. “Optimal Choice of local spin and electron self energies”.

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EDMFT Application to the Kondo lattice. Q. Si S Rabello K Ingersent and J Smith

Nature 423 804 (2001).

Remarkable agreement with the experimental observation of a quantum critical point with non trivial Landau damping.

P. Sun and GK: approach the problem from high temperatures, with a different model (Anderson model ).

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Model and parameters

U = 3:0,V = 0:6, Ef = -0:5

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EDMFT equations.

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EDMFT equations

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Phase Diagram. (P . Sun )

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Phase diagram. First order of the transition. At high

temperatures, artifact of EDMFT, Pankov et. al.

PRB 2002.

At low temperatures ? Fluctuation driven First order transition in CeIn3 ?

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Local susceptibility

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Evolution of the magnetic structure.

In this parameter regime, the QP are formed

Before the magnetic transition?

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Size of the jump

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Evolution of the quasiparticles parameters. (P. Sun 2003)

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Evolution of the electronic structure

System becomes more incoherent as the transition is approached.

On the antiferromagnetic side : Majority spins are more incoherent than the minority spins.

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F electron Weiss field (P. Sun 2003)

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Spin self energy .

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Conclusion

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Extended DMFT electron phonon

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Extended DMFT e.ph. Problem

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E-DMFT classical case, soft spins

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E-DMFT classical case Ising limit

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Advantage and Difficulties of E-DMFT

The transition is first order at finite temperatures for d< 4

No finite temperature transition for d less than 2 (like spherical approximation)

Improved values of the critical temperature

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E-DMFT test in the classical case[Bethe Lattice, S. Pankov 2001]