1. Quantum Griffiths Phases of Correlated Electrons Collaborators: Eric Adrade (Campinas) Matthew Case (FSU) Eduardo Miranda (Campinas) REVIEW: Reports on Progress in Physics 68, 2337–2408 (2005) Vladimir Dobrosavljevic Department of Physics and National High Magnetic Field Laboratory Florida State University,USA Funding: NSF grants: DMR-9974311 DMR-0234215 DMR-0542026

2. Disorder and QCP: The Cold War Era (1960-1990) Criticality and Collectivization There is not, nor should there be, an irreconcilable contrast between the individual and the collective, between the interests of an individual person and the interests of the collective.” (Joseph Stalin) Collectivization: Theory Ken Wilson’s RG Long wavelength modes rule! GRIFFITHS singularities, Harris criterion: “Weak” disorder corrections

3. Trouble Starts (circa 1990) Dissidents run away over the Berlin Wall Weak coupling RG finds run away flows for QCPs with disorder (Sachdev,...,Vojta,...)

4. Quantum Griffiths phases and IRFP (1990s) D. Fisher (1991): new scenario for (insulating) QCPs with disorder (Ising) Griffiths phase g T ordered phase clean phase boundary Griffiths phase (Till + Huse): Rare, dilute magnetically ordered cluster tunnels with rate Δ(L) ~ exp{-AL d } P(L) ~ exp{-ρL d } P(Δ) ~ Δ α-1 ; χ ~ T α-1 α → 0 at QCP (IRFP)

5. General classification for single-droplet dynamics (T. Vojta) Large droplets: SEMICLASSICAL! L τ Droplet dynamics (all symmetry classes) Classical spin chain (Kosrerlitz, 1976) LGW action (L > ξ) 2

6. Symmetry and dissipation (SINGLE DROPLET) 1. Insulating magnets (z=1) – short-range interaction (in time) Ising: at “LCD” – tunneling rate Δ(L) ~ τ ξ -1 ~ exp{-π|r|L d } Heisenberg: below LCD – powerlaw only no QGP! 2.Metallic magnets (z=2) – long-range 1/τ 2 interaction (dissipation) Ising: above “LCD” – dissipative phase transition Large droplets (L > L c ) freeze!! (Caldeira-Leggett, i.e. K-T) “ROUNDING” of QCP (T. Vojta, Hoyos) Heisenberg: at “LCD” Δ(L) ~ τ ξ -1 ~ exp{-π|r|L d } QGP ??? (Vojta-Schmalian) * SDRG theory (Vojta+Hoyos, 2008)

7. How RKKY affect the droplet dynamics?? RKKY-interacting droplets? (Dobrosavljevic, Miranda, PRL 2005) 1 2 3 4 L1L1 J 14 R 12 J RKKY random sign NOTE: Droplet-QGP – all dimensions! Strategy: integrate-out “other” droplets δS RKKY =J 2 ∑ ∫ dτ ∫dτ’ φ(τ) χ av (τ-τ’) φ(τ’) χ av (ω n ) = ∫dΔ P(Δ) χ(Δ; ω n ) = ~ ∫dΔ Δ α-1 [iω n + Δ] -1 = χ(0) - |ω| α-1 additional dissipation due to spin fluctuations non-Ohmic (strong) dissipation for α < 2!!

8.    g T     ordered phase clean phase boundary Griffiths (non-Ohmic dissipation) cluster glass Cluster-glass phase (“foot”): generic case of QGP in metals (Dobrosavljevic, Miranda, PRL 2005) Fluctuation-driven first-order glass transition Matthew Case & V.D. (PRL, 2007)

9. Cluster-glass Quantum Criticality: large N solution (M. Case + V.D.; PRL 99, 147204 (2007)) Uniform droplet case (QSG): P(r i )=  (r-r i )  no Griff. fluctuations, conventional QCP Distribution of droplet sizes: P(r i )  exp[-2  (r i -r)/u] Novel fluctuation-driven first order transition!! full self-consistency:  av (  )   0 - |  |  ’-1

10. Experimental candidates? Double-Layered Ruthenate alloys (Sr 1-x Ca x ) 3 Ru 2 O 7 [Z. Mao (Tulane) + V.D., Phys. Rev. B (Rapid Comm.) 78, 180407 (2008)]

11. Disorder near Mott transitions: generic phase diagram Previous studies (motivated by Si:P): –Milovanović, Sachdev, and Bhatt, PRL 63, 82 (1989). Mean-field theory of the disordered Hubbard model –V. Dobrosavljević and G. Kotliar, PRL78, 3943 (1997). “statDMFT” on Bethe lattice (finite coordination, D=inf.!!) Disorder MIT Insulator Non-Fermi-liquid metal Fermi-liquid metal  (T=0)=  o  (T=0)=  Electronic Griffiths Phases?

12. statDMFT: local (though spatially non-uniform) self-energies Local renormalizations Each local site is governed by an impurity action: : hybridization to the neighboring sites statDMFT in D=2 Eric Andrade, E. Miranda, V.D., cond-mat/0808.0913

13. Clean case (W=0): Mott metal-insulator transition at U=U c, where Z  (Brinkman and Rice, 1970). Fermi liquid approach in which each fermion acquires a quasi- particle renormalization and each site-energy is renormalized: Physical content of statDMFT

14. For U  U c (W), all Z i   vanish (disordered Mott transition) If we re-scale all Z i by Z 0 ~ U c (W)-U, we can look at P(Z i /Z 0 ) For D =∞ (DMFT), P(Z/Z 0 ) - universal form at U c. Z0Z0 gap Results in D = ∞ (D. Tanasković et al., PRL 2003; M. C. O. Aguiar et al., PRB 2005)

15. In D=2, the environment of each site (“bath”) has strong spatial fluctuations New physics: rare evens due to fluctuations and spatial correlations Distribution P(Z/Z 0 ) acquires a low-Z tail : DMFT Results in D=2 (E. C. Andrade, Eduardo Miranda, V. D., arXiv:0808.0913v1)

16. Remembering that the local Kondo temperature and Singular thermodynamic response The exponent  is a function of disorder and interaction strength.  =1 marks the onset of singular thermodynamics. Quantum Griffiths phase (see, e.g., E. Miranda and V. D., Rep. Progr. Phys. 68, 2337 (2005); T. Vojta, J. Phys. A 39, R143 (2006)) Results: Thermodynamics

17. Z typ = exp{ } Phase Diagram (U > W)

18. Most characterized Quantum Griffiths phases are precursors of a critical point where the effective disorder is infinite (D. S. Fisher, PRL 69, 534 (1992); PRB 51, 6411 (1995); ….) Compatible with infinite randomness fixed point scenario  -1 – variance of log(Z) Infinite randomness at the MIT?

19. Replace the environment of given site outside square by uniform (DMFT-CPA) effective medium. Reduce square size down to DMFT limit. Rare evrents due to rare regions with weaker disorder U=0.96U c ;W=2.5D Rare event! The rare event is preserved for a box of size l > 9: rather smooth profile with a characteristic size. “Size” of the rare events? Typical sample

20. Killing the Mott droplet “Size” of the rare events: a movie

21. The effective disorder at the Fermi level is given by the distribution of: This quantity is strongly renormalized close to the Mott MIT Width of the v i distribution is pinned to Fermi level (Kondo resonance) Spectrosopic signatures: disorder screening

22. However, the effect is lost even slightly away from the Fermi energy: U=0.96U c ;W=0.375D E=E F E=E F - 0.05D The strong disorder effects reflect the wide fluctuations of Z i Similar to high-T c materials, as seen by STM, tunneling (e.g. K. McElroy et al. Science 309, 1048 (2005)) Generic to the strongly correlated materials? Tallahassee Aspen Energy-resolved inhomogeneity!

23. Mottness-induced contrast

24. An electronic Griffiths phase emerges as a precursor to the disordered Mott MIT (non-Fermi liquid metal) Strong-correlation-induced healing of disorder at low energies, but very inhomogeneous away Infinite randomness fixed point: new type of critical behavior for disordered MIT??? (Weak) localization vs. interactions: is there a true 2D metal in D=2??? Conclusions and...

25. The physical picture Localization-induced electronic Griffiths phase (Miranda & Dobrosavljevic, 2001)

26. EGP sets in for W > W * = (  t 2  av J K ) 1/2 W*W* WcWc W MIT at W = W c ~ E F EGP always comes BEFORE the MIT Electronic Griffiths Phase in Kondo Alloys (Tanaskovic, Dobrosavljevic, Miranda; also Grempel et al.)

27. Similar non-Ohmic (strong) dissipation Quantum (S=1/2) spin dynamics (Berry phase) Local action: “Bose-Fermi (BF) Kondo model” (“E-DMFT”; A. Sengupta, Q. Si, Grempel,...) EGP +RKKY interactions: beyond semi-classical spins! (Tanaskovic, Dobrosavljevic, Miranda)!

28. BF model has a (local) phase transition for a sub-Ohmic dissipative bath (  > 0 ) g (RKKY coupling)  c J K (Kondo coupling) ~  Kondo screened spin fluid EGP model: distribution of Kondo couplings all the way to zero! A finite fraction of spins fall on each side of the critical line Kondo effect destroyed by dissipation on a finite fraction of spins Decoupled spins J K flows to zero; they form a spin fluid (Sachdev-Ye) (frustrated insulating magnet) disorder W MIT W*W* W1W1 insulator Fermi liquid NFL - spin fluid  > 0 “bare EGP” Destruction of the Kondo effect and two-fluid behavior

29. χ (T) ~ ln(T o /T) for spin fluid (decoupled spins) Finite (very low!!) temperature SG instability as soon as spins decouple Quantitative (numerical) results: fermionic large N approach ( Grempel et al.) - disorder Spin-glass (SG) instability of the EGP

30. Conclusions: In metals dissipation destroys QGP at lowest T → (quantum) glassy ordering Magnetic (QCP) QGP: → semi-classical dynamics at T > T G Fluctuation–driven first-order QCP of the “cluster glass” Spin liquid in EGP at T > T G