Quantum Griffiths Phases of Correlated Electrons

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Quantum Griffiths Phases of Correlated Electrons
Vladimir Dobrosavljevic Department of Physics and
National High Magnetic Field Laboratory Florida
State University,USA
Funding NSF grants DMR-9974311 DMR-0234215 DMR
-0542026
Collaborators Eric Adrade (Campinas) Matthew
Case (FSU) Eduardo Miranda (Campinas)
REVIEW Reports on Progress in Physics 68,
23372408 (2005)
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Disorder and QCP The Cold War Era (1960-1990)
Long wavelength modes rule! GRIFFITHS
singularities, Harris criterion Weak disorder
corrections
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Trouble Starts (circa 1990)
Dissidents run away over the Berlin Wall
Weak coupling RG finds run away flows for QCPs
with disorder (Sachdev,...,Vojta,...)
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Quantum Griffiths phases and IRFP (1990s)

• D. Fisher (1991) new scenario for (insulating)
  QCPs with disorder (Ising)

Griffiths phase (Till Huse)
Rare, dilute magnetically ordered cluster
tunnels with rate ?(L) exp-ALd
P(L) exp-?Ld P(?) ?a-1 ? Ta-1 a ?
0 at QCP (IRFP)
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General classification for single-droplet
dynamics (T. Vojta)

• Large droplets SEMICLASSICAL!

L
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Symmetry and dissipation (SINGLE DROPLET)

• Insulating magnets (z1) short-range
  interaction (in time)
• Ising at LCD tunneling rate ?(L) t? -1
  exp-prLd
• Heisenberg below LCD powerlaw only no QGP!
• Metallic magnets (z2) long-range 1/t2
  interaction (dissipation)
• Ising above LCD dissipative phase transition
  
• Large droplets (L gt Lc) freeze!!
  (Caldeira-Leggett, i.e. K-T)
• ROUNDING of QCP (T. Vojta, Hoyos)
• Heisenberg at LCD
• ?(L) t? -1 exp-prLd
• QGP ??? (Vojta-Schmalian)
• SDRG theory (VojtaHoyos, 2008)

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RKKY-interacting droplets? (Dobrosavljevic,
Miranda, PRL 2005)

• How RKKY affect the droplet dynamics??

random sign

• NOTE Droplet-QGP all dimensions!
• Strategy integrate-out other droplets

dSRKKYJ2 ? ?dt ?dt f(t) ?av(t-t)
f(t) ?av(?n) ?d? P(?) ?(? ?n)
?d? ?a-1 i?n ?-1 ?(0) - ?a-1
additional dissipation due to spin fluctuations
non-Ohmic (strong) dissipation for a lt 2!!
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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)
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Cluster-glass Quantum Criticality large N
solution (M. Case V.D. PRL  99, 147204 (2007))

• Uniform droplet case (QSG) P(ri)?(r-ri)

? no Griff. fluctuations, conventional QCP

• Distribution of droplet sizes
  P(ri)?exp-2??(ri-r)/u
• Novel fluctuation-driven first order transition!!

full self-consistency
??????av(?) ? ?0 - ??-1
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Experimental candidates?
Double-Layered Ruthenate alloys
(Sr1-xCax)3Ru2O7 Z. Mao (Tulane) V.D., Phys.
Rev. B (Rapid Comm.) 78, 180407 (2008)
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Electronic Griffiths Phases?

• Disorder near Mott transitions generic phase
  diagram
• Previous studies (motivated by SiP)
• Milovanovic, Sachdev, and Bhatt, PRL 63, 82
  (1989). Mean-field theory of the disordered
  Hubbard model
• V. Dobrosavljevic and G. Kotliar,
  PRL78, 3943 (1997). statDMFT on Bethe lattice
  (finite coordination, Dinf.!!)

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statDMFT in D2 Eric Andrade, E. Miranda, V.D.,
cond-mat/0808.0913 

• statDMFT local (though spatially non-uniform)
  self-energies

Each local site is governed by an impurity action
Local renormalizations
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Physical content of statDMFT

• Clean case (W0) Mott metal-insulator transition
  at UUc, where Z 0 (Brinkman and Rice, 1970).
• Fermi liquid approach in which each fermion
  acquires a quasi-particle renormalization
  and each site-energy is renormalized

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Results in D 8 (D. Tanaskovic et al., PRL 2003
M. C. O. Aguiar et al., PRB 2005)

• For U Uc(W), all Zi 0 vanish (disordered Mott
  transition)
• If we re-scale all Zi by Z0 Uc(W)-U, we can
  look at P(Zi /Z0)
• For D 8 (DMFT), P(Z/Z0) - universal form at Uc.

Z0
gap
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Results in D2 (E. C. Andrade, Eduardo Miranda,
V. D., arXiv0808.0913v1)

• In D2, the environment of each site (bath) has
  strong spatial fluctuations
• New physics rare evens due to fluctuations and
  spatial correlations

DMFT
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Results Thermodynamics

• Remembering that the local Kondo temperature
  and

Singular thermodynamic response
The exponent a is a function of disorder and
interaction strength. a1 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))
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Phase Diagram (U gt W)
Ztyp exp lt ln Zgt
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Infinite randomness at the MIT?

• 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) .)

a-1 variance of log(Z)
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Size of the rare events?
Rare event!
Typical sample
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
U0.96UcW2.5D
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Size of the rare events a movie
Killing the Mott droplet
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Spectrosopic signatures disorder screening

• The effective disorder at the Fermi level is
  given by the distribution of

This quantity is strongly renormalized close to
the Mott MIT
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Energy-resolved inhomogeneity!

• However, the effect is lost even slightly away
  from the Fermi energy

EEF - 0.05D
The strong disorder effects reflect the wide
fluctuations of Zi
EEF
Similar to high-Tc materials, as seen by STM,
tunneling (e.g. K. McElroy et al. Science 309,
1048 (2005))
Generic to the strongly correlated materials?
U0.96UcW0.375D
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Mottness-induced contrast
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Conclusions and...

• 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 D2???

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Localization-induced electronic Griffiths
phase (Miranda Dobrosavljevic, 2001)
The physical picture
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Electronic Griffiths Phase in Kondo
Alloys (Tanaskovic, Dobrosavljevic, Miranda also
Grempel et al.)
EGP sets in for W gt W (pt2ravJK)1/2
EGP always comes BEFORE the MIT
MIT at W Wc EF
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EGP RKKY interactions beyond semi-classical
spins! (Tanaskovic, Dobrosavljevic, Miranda)!

• Similar non-Ohmic (strong) dissipation
• Quantum (S1/2) spin dynamics (Berry phase)
• Local action Bose-Fermi (BF) Kondo model
• (E-DMFT A. Sengupta, Q. Si, Grempel,...)

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Destruction of the Kondo effect and two-fluid
behavior

• BF model has a (local) phase transition for a
  sub-Ohmic dissipative bath (e gt 0 )

• 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 JK flows to zero they
• form a spin fluid (Sachdev-Ye)
• (frustrated insulating magnet)

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Spin-glass (SG) instability of the EGP

• ?(T) ln(To/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.)

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Conclusions

• In metals dissipation destroys QGP at lowest T
• ? (quantum) glassy ordering
• Magnetic (QCP) QGP ? semi-classical dynamics
  at T gt TG
• Fluctuationdriven first-order QCP of the
  cluster glass
• Spin liquid in EGP at T gt TG