1. Electronic Griffiths phases and dissipative spin liquids
- Campinas, Brazil
E. M.
Darko Tanasković
Vlad Dobrosavljević
- Magnet Lab/FSU
Complex Behavior in Correlated Electron Systems
Lorentz Center – Leiden – August 11, 2005

2. 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))
UCu5-xPdx
La1-xCexCu2.2Si2
M1-xUxPd3 (M=Y,Sc)
Andrade et al., PRL 1998
Bernal et al., PRL 1985
Aronson et al., PRL 2001
UCu4Pd

3. Theoretical scenarios
Phenomenological Kondo disorder model (Bernal et al., PRL `95; E.M., V. Dobrosavljević, G. Kotliar, PRL `97): distribution of Kondo temperatures Þ P(TK)
Magnetic Griffiths phase (Castro Neto, Castilla, Jones, PRL `98, PRB `00): distribution of fluctuating locally ordered clusters of size N Þ P(N)
Spin glass critical point (Sengupta, Georges, PRB `95; Rozenberg, Grempel, PRB `99)
Dominated by low TK spins if P(TK=0) ¹0
Kondo disorder model
Form of P(TK) is assumed: is there a microscopic mechanism?

4. Electronic Griffiths phase (E.M., V. Dobrosavljević, PRL `01)
Local DOS at the Fermi level (wave function amplitude) fluctuates spatially ® Anderson localization effects
TK is exponentially sensitive to the local DOS
(Dobrosavljevic, Kirkpatrick, Kotliar, `92)
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)
Green’s function of conduction electrons with site “j” removed
Each f-site gives rise to a local self-energy Sj (wn) for the lattice problem, which is numerically solved

5. Electronic Griffiths phase (E.M., V. Dobrosavljević, PRL `01)
Power law distribution of Kondo temperatures at moderate disorder
a=a(W) is tunable with disorder strength
(Broad) Griffiths phase induced by the proximity to an Anderson transition
NFL if a<1

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;....)
(Poisson)
For example, for a fluctuating ferromagnetic droplet of size V
(tunneling)
Power-law distribution of energy scales (tunneling rates)
From this, the usual phenomenology follows, quite independent of the nature of the fluctuators

7. What is the origin of the electronic GP?
Effective model (D. Tanasković, V. Dobrosavljević, E.M., PRB `04)
Model with c-site (diagonal) disorder only and Gaussian distribution
Infinite coordination limit (z ® ¥) (Dynamical Mean Field Theory)
No DOS fluctuations (no Anderson localization effects)!
Fixed conduction electron bath

8. What is the origin of the electronic GP?
When ej ® ¥,
Usual Griffiths phase behavior!
Since
disorder W
MIT W*
insulator
Fermi liquid
EGP with NFL behavior
a<1

9. How to justify the effective model?
In a real lattice, the conduction bath is not fixed but fluctuates randomly
To leading order, ReDj(0) fluctuations are gaussian and µ W2
Even if P(ej) is bounded P(ejren) is not!
Good agreement between statDMFT and effective model

10. Problems with the usual scenario
Thermodynamic divergences are too strong a»1/W2; experiments show near log behavior (a»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?
RKKY interactions between (distant) low-TK (unscreened) spins:
oscillatory with distance Þ 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/ )
Self-consistency:
Local action: “Bose-Fermi Kondo model”
Related work: Burdin, Grempel, Georges, PRB ´02

11. Question: Will a positive e be self-consistently generated?
Single-impurity Bose-Fermi Kondo model (Q. Si, J. L. Smith, EPL `99, A. M. Sengupta, PRB `00)
g (RKKY coupling)
rcJK (Kondo coupling)
~ e
Kondo screened
No Kondo effect
One spin subject to a fermionic bath and a fluctuating magnetic field (bosonic bath).
For c(t)~1/t2-e with e>0, 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 e>0, then some spins will decouple and not be Kondo quenched Þ two fluid behavior
Question: Will a positive e 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)
Thus, e0 > 0 (sub-Ohmic dissipation of spins) if a < 2. We saw that:
For strong enough disorder, the “bare” theory leads to a sub-Ohmic bath
disorder W
MIT W*
insulator
Fermi liquid
“bare EGP”
e0 > 0
a < 2
a < 1
Wc0
Two-fluid behavior

13. How will full self-consistency change this?
Additive contributions from each fluid:
Suppose the self-consistent bath goes like
Decoupled spins: (Sengupta, `00; Zhu, Si, PRB `02; Zaránd, Demler, PRB ´02)
Quenched spins:
where n(e) is the “correlation time exponent” of the Bose-Fermi transition
Clearly, edc>eK Þ decoupled spins dominate at low frequencies

14. Self-consistency Imposing self-consistency:
Þ Sachdev-Ye spin liquid (PRL `93)
Numerical results using large-N methods to solve the single-impurity problems:
Marginal behavior over many decades
disorder W
MIT W*
insulator
Fermi liquid
“bare EGP”
e0 > 0 W1
NFL spin liquid
Two-fluid behavior

15. Other consequences n(e) » e/2 + O(e2) »1/2 Þ
Resistivity from the decoupled part: marginal Fermi liquid
Low temperature spin-glass instability:
Estimated from
Large window with marginal behavior above Tg

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 Þ appearance of two fluids, Kondo quenched and spin liquid in a broad range of temperatures.
Spin liquid Þ local c(w) is log-divergent.
Kondo quenched Þ Power-law distribution of TK with a»0.5 (but c is non-singular, c~w0.5 ).
Linear resistivity.
Ultimately unstable towards spin-glass ordering at the lowest T.