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Electronic Griffiths phases and dissipative spin liquids

E. M.

Darko Tanasković

Vlad Dobrosavljević

Complex Behavior in Correlated Electron SystemsLorentz Center – Leiden – August 11, 2005

- Campinas, Brazil

- Magnet Lab/FSU

2

Andrade et al., PRL 1998

1~/ TTTC

Non-Fermi Liquid behavior in Kondo systems• Many disordered heavy fermion systems show anomalous properties, inconsistent with Landau’s Fermi liquid theory (see, e.g., G. Stewart, RMP 73, 797 (2001), E.M., V. Dobrosavljević, to appear in Rep. Prog. Phys. (2005))

Bernal et al., PRL 1985Aronson et al., PRL 2001

UCu4Pd

UCu5-xPdx

La1-xCexCu2.2Si2M1-xUxPd3 (M=Y,Sc)

1

0/ln~/

TTT

T

TTC or

ATT 0~ A 1

01 ~

3

Theoretical scenarios• Phenomenological Kondo disorder model (Bernal et al., PRL `95; E.M., V. Dobrosavljević, G. Kotliar, PRL `97): distribution of Kondo temperatures P(TK)• Magnetic Griffiths phase (Castro Neto, Castilla, Jones, PRL `98, PRB `00): distribution of fluctuating locally ordered clusters of size N P(N)• Spin glass critical point (Sengupta, Georges, PRB `95; Rozenberg, Grempel, PRB `99)

TTTTCT /ln/ 0

Dominated by low TK spins if P(TK=0) 0

Kondo disorder model

Form of P(TK) is assumed: is there a microscopic mechanism?

4

Electronic Griffiths phase (E.M., V. Dobrosavljević, PRL `01)

• Statistical Dynamical Mean Field Theory (for the Anderson lattice) (Dobrosavljević, Kotliar, PRL `97)

• A local correlated action at each f-site (U Anderson single-impurity model)

KK JxDxT /1exp

• TK is exponentially sensitive to the local DOS(Dobrosavljevic, Kirkpatrick, Kotliar, `92)

2xx

FE

• Local DOS at the Fermi level (wave function amplitude) fluctuates spatially Anderson localization effects

Green’s function of conduction electrons with

site “j” removed

• Each f-site gives rise to a local self-energy j n for the lattice problem, which is numerically solved

5

1 KK TTP

Electronic Griffiths phase (E.M., V. Dobrosavljević, PRL `01)

• Power law distribution of Kondo temperatures at moderate disorder

1 KK TTP W is tunable with disorder strength

• (Broad) Griffiths phase induced by the proximity to an Anderson transition

1

1

TT

dTT

T

TnT T

KK

NFL if <1

1 KK TTP

6

Generic mechanism of quantum Griffiths phases• Exponentially rare events with exponentially low energy scales, e. g., in a random field Ising model (D. Fisher, PRL `92, PRB `95) but also in other systems (Senthil, Sachdev, PRL `96; Castro Neto, Jones, PRB `00; T. Vojta, Schmalian, PRB `05;....)

bVE

E NN

ii

exp~~~10

(tunneling)

• From this, the usual phenomenology follows, quite independent of the nature of the fluctuators

1

T

T

dEE

T

TnT T

1

TdEEdT

d

dT

Tdn

dT

dS

T

TC

T

1/0 ~expexp~ bcEbVEEcVdVE

Power-law distribution of energy scales (tunneling rates)

cVVP exp~ (Poisson)

• For example, for a fluctuating ferromagnetic droplet of size V

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What is the origin of the electronic GP?

• Infinite coordination limit (z ) (Dynamical Mean Field Theory)• No DOS fluctuations (no Anderson localization effects)!

njnfjfnnjj

eff fEifSn

,,

, ncjnnfj i

V

2

1

22

knn

jkncnj GtG

z

t

Fixed conduction electron bath

• Effective model (D. Tanasković, V. Dobrosavljević, E.M., PRB `04)

222/12 2/exp2 WWP ii

iε

)( U

Model with c-site (diagonal) disorder only and Gaussian distribution

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What is the origin of the electronic GP?

0Im2exp

fj

fKj

EDT

2

22 0Im0Im

j

cfj

ncjnnfj

V

i

Vj

• When j ,

0Imexp

2

0c

jKKj TT

222/12 2/exp2 WWP ii

• Since

2

1

0 2

0Im

W

J

T

TTP c

K

KK

where,

Usual Griffiths phase behavior!

disorder WMITW*

insulatorFermi liquid EGP with NFL behavior

<1

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How to justify the effective model?In a real lattice, the conduction bath is not fixed but fluctuates randomly

z

kn

jknjnc Gt

1

2

njjnnfj i

V

2

0Re jjrenj

• To leading order, Rej(0) fluctuations are gaussian and W2

• Even if P(j) is bounded P(jren) is not!

Good agreement between statDMFT and effective model

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• RKKY interactions between (distant) low-TK (unscreened) spins:oscillatory with distance random in magnitude and sign

• Expect quantum spin-glass dynamics at low T (D. MacLaughlin et al. PRL`01)• (E)DMFT formulation: infinite-range spin glass interactions (paramagnetic phase)(Tanasković, Dobrosavljević, E.M., cond-mat/0412100)

Problems with the usual scenario• Thermodynamic divergences are too strong W; experiments show near log behavior (1).• Proliferation of “free” spins: entropy expected to be quenched by interactions at low T, (probably spin-glass, D. MacLaughlin et al. PRL `01)

What is missing?

;RKKYj

effj

eff SSS ´´´2

2

ffRKKY SSddJ

S

• Self-consistency:

• Local action: “Bose-Fermi Kondo model”

´´ ff SS Related work: Burdin, Grempel, Georges, PRB ´02

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Single-impurity Bose-Fermi Kondo model(Q. Si, J. L. Smith, EPL `99, A. M. Sengupta, PRB `00)

g (RKKY coupling)

cJ

K (K

ondo

cou

plin

g)

~

Kondo screened

No Kondo effect

• One spin subject to a fermionic bath and a fluctuating magnetic field (bosonic bath).• For ~1/ with , there is a lot of dissipation by the bosonic bath:

For weak enough JK, the Kondo effect is destroyed by dissipation. For strong JK, the spin is Kondo quenched.

If there is a wide distribution of Kondo temperatures and , then some spins will decouple and not be Kondo quenched two fluid behavior

Question: Will a positive be self-consistently generated?

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The leading order effect of the boson bath (instability analysis)

• Ignore self-consistency and calculate the spin response of the “bare” theory (limit of arbitrarily weak RKKY)

,00 KKK TTPdT

2

0

/1~ 01

0

201

0

0

withor

nnKK Ci

TTP

• Thus, (sub-Ohmic dissipation of spins) if . We saw that:

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0Im2

2

0Im02

JWW

W

J cc

c

for

For strong enough disorder, the “bare” theory leads to a sub-Ohmic bath

disorder WMITW*

insulatorFermi liquid “bare EGP” > 0

Wc0

Two-fluid behavior

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How will full self-consistency change this?

Suppose the self-consistent bath goes like 10 nn Ci

20

1

dcndcndcdcCi with

• Decoupled spins: (Sengupta, `00; Zhu, Si, PRB `02; Zaránd, Demler, PRB ´02)

Clearly, dc>K decoupled spins dominate at low frequencies

• Quenched spins:

/120

1

KnKnKKCi with

where is the “correlation time exponent” of the Bose-Fermi transition

Kren

K JT ~

1/1~

renK

renKK TTPTP

nKdcndcdcn inini 1Additive contributions from each fluid:

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Self-consistency

Sachdev-Ye spin liquid (PRL `93)

1ln~

Numerical results using large-N methods to solve the single-impurity problems:• Marginal behavior over many decades

12 dc

Imposing self-consistency: ndcn ii

disorder WMITW*

insulatorFermi liquid

“bare EGP” > 0W1

NFL spin liquid

Two-fluid behavior

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Other consequences

• renK

renK TTP ~

• Low temperature spin-glass instability:

Estimated from

J

Tg

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• Resistivity from the decoupled part: marginal Fermi liquid

~/22*

KJ

Large window with marginal behavior above Tg

cdc WWAn /exp

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Pr2Ir2O7 (S. Nakatsuji et al., preprint)

Pirochlore lattice of Pr ions• Very frustrated• Large residual resistivity

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Conclusions• Clarification of the mechanism of the electronic Griffiths phase.• With the inclusion of spin-spin interactions:

• For W>Wc appearance of two fluids, Kondo quenched and spin liquid in a broad range of temperatures.• Spin liquid local is log-divergent.• Kondo quenched Power-law distribution of TK with 0.5 (but is non-singular, 0.5 ).• Linear resistivity.• Ultimately unstable towards spin-glass ordering at the lowest T.