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1
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
7
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
8
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
9
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
10
• 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
11
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?
12
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:
4
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
13
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:
14
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
15
Other consequences
• renK
renK TTP ~
• Low temperature spin-glass instability:
Estimated from
J
Tg
12
• Resistivity from the decoupled part: marginal Fermi liquid
~/22*
KJ
Large window with marginal behavior above Tg
cdc WWAn /exp
16
Pr2Ir2O7 (S. Nakatsuji et al., preprint)
Pirochlore lattice of Pr ions• Very frustrated• Large residual resistivity
17
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.