Transcript Document

Quantum Griffiths Phases of Correlated Electrons
Vladimir Dobrosavljevic
Department of Physics and National High Magnetic Field Laboratory
Florida State University,USA
Collaborators:
Eric Adrade (Campinas)
Matthew Case (FSU)
Eduardo Miranda (Campinas)
Funding:
NSF grants:
DMR-9974311
DMR-0234215
DMR-0542026
REVIEW: Reports on Progress in Physics 68, 2337–2408 (2005)
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
Long wavelength modes rule!
Ken Wilson’s RG
GRIFFITHS singularities, Harris criterion:
“Weak” disorder corrections
Trouble Starts (circa 1990)
Dissidents run away
over the Berlin Wall
Weak coupling RG finds
run away flows
for QCPs with disorder
(Sachdev,...,Vojta,...)
Quantum Griffiths phases and IRFP (1990s)
• D. Fisher (1991): new scenario for (insulating) QCPs with disorder (Ising)
T
clean phase
boundary
Griffiths phase (Till + Huse):
ordered
phase
g
Griffiths
phase
P(L) ~ exp{-ρLd}
P(Δ) ~ Δα-1 ; χ ~ Tα-1
α → 0 at QCP (IRFP)
Rare, dilute magnetically
ordered cluster tunnels
with rate Δ(L) ~ exp{-ALd}
General classification for single-droplet dynamics (T. Vojta)
• Large droplets: SEMICLASSICAL!
τ
• LGW action (L > ξ)
L
Droplet dynamics
(all symmetry classes)
Classical spin chain
(Kosrerlitz, 1976)
2
Symmetry and dissipation (SINGLE DROPLET)
1.
Insulating magnets (z=1) – short-range interaction (in time)
•
•
2.
Ising: at “LCD” – tunneling rate Δ(L) ~ τξ -1~ exp{-π|r|Ld}
Heisenberg: below LCD – powerlaw only no QGP!
Metallic magnets (z=2) – long-range 1/τ2 interaction (dissipation)
•
Ising: above “LCD” – dissipative phase transition
Large droplets (L > Lc) freeze!! (Caldeira-Leggett, i.e. K-T)
“ROUNDING” of QCP (T. Vojta, Hoyos)
•
Heisenberg: at “LCD”
Δ(L) ~ τξ -1 ~ exp{-π|r|Ld}
QGP ??? (Vojta-Schmalian)
* SDRG theory (Vojta+Hoyos, 2008)
RKKY-interacting droplets?
(Dobrosavljevic, Miranda, PRL 2005)
• How RKKY affect the droplet dynamics??
2
random sign
3
R12
JRKKY
1
• NOTE: Droplet-QGP – all dimensions!
• Strategy: integrate-out “other” droplets
δSRKKY=J2 ∑ ∫dτ ∫dτ’ φ(τ) χav(τ-τ’) φ(τ’)
4
J14
L1
additional dissipation due
to spin fluctuations
χav(ωn) = ∫dΔ P(Δ) χ(Δ; ωn) =
~ ∫dΔ Δα-1 [iωn+ Δ]-1 = χ(0) - |ω|α-1
non-Ohmic (strong) dissipation
for α < 2!!
Cluster-glass phase (“foot”): generic case of QGP in metals
(Dobrosavljevic, Miranda, PRL 2005)
T
clean phase
boundary
cluster
glass
ordered
phase
a=0
Griffiths
a=1
a=2
a=
g
eo>0
(non-Ohmic dissipation)
Fluctuation-driven
first-order glass transition
Matthew Case & V.D.
(PRL, 2007)
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a(ri-r)/u]
full self-consistency:
av()  0 - ||a’-1
• Novel fluctuation-driven first order transition!!
Experimental candidates?
Double-Layered Ruthenate alloys (Sr1-xCax)3Ru2O7
[Z. Mao (Tulane) + V.D., Phys. Rev. B (Rapid Comm.) 78, 180407 (2008)]
Electronic Griffiths Phases?
• Disorder near Mott transitions: generic phase diagram
Fermi-liquid metal
(T=0)=o
Non-Fermi-liquid
metal
(T=0)= 
Insulator
MIT
Disorder
• 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.!!)
statDMFT in D=2
Eric Andrade, E. Miranda, V.D., cond-mat/0808.0913
• statDMFT: local (though spatially non-uniform) self-energies
Local renormalizations
Each local site is governed by an impurity action:
: hybridization to the neighboring sites
Physical content of statDMFT
• Clean case (W=0): Mott metal-insulator transition at U=Uc, where
Z 0 (Brinkman and Rice, 1970).
• Fermi liquid approach in which each fermion acquires a quasiparticle renormalization and each site-energy is renormalized:
Results in D = ∞
(D. Tanasković 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 =∞ (DMFT), P(Z/Z0) - universal form at Uc.
Z0
gap
Results in D=2
(E. C. Andrade, Eduardo Miranda, V. D., arXiv:0808.0913v1)
• In D=2, the environment of each site (“bath”) has strong spatial
fluctuations
• New physics: rare evens due to fluctuations and spatial
correlations
DMFT
Distribution P(Z/Z0) acquires
a low-Z tail:
Results: Thermodynamics
• Remembering that the local Kondo temperature
Singular thermodynamic response
The exponent a is a function of
disorder and interaction strength.
a=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))
and
Phase Diagram (U > W)
Ztyp = exp{ < ln Z> }
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); ….)
Compatible with infinite
randomness
fixed point scenario
a-1 – variance of log(Z)
“Size” of the rare events?
Replace the environment of
given site outside square by
uniform (DMFT-CPA) effective
medium.
Typical sample
Rare event!
Reduce square size down to
DMFT limit.
Rare evrents due to rare regions
with weaker disorder
The rare event is preserved for a
box of size l > 9: rather smooth
profile with a characteristic size.
U=0.96Uc;W=2.5D
“Size” of the rare events: a movie
Killing the Mott droplet
Spectrosopic signatures: disorder screening
• The effective disorder at the Fermi level is given by the
distribution of:
Width of the vi distribution
This quantity is strongly
renormalized close to the
Mott MIT
is pinned to Fermi level
(Kondo resonance)
Energy-resolved inhomogeneity!
• However, the effect is lost even slightly away from the Fermi
energy:
Aspen
E=EF - 0.05D
E=EF
The strong disorder effects reflect the
wide fluctuations of Zi
Similar to high-Tc materials, as seen by STM,
tunneling (e.g. K. McElroy et al.
Science 309, 1048 (2005))
Tallahassee
U=0.96Uc;W=0.375D
Generic to the strongly
correlated materials?
Mottness-induced contrast
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 D=2???
Localization-induced electronic Griffiths phase
(Miranda & Dobrosavljevic, 2001)
The physical picture
Electronic Griffiths Phase in Kondo Alloys
(Tanaskovic, Dobrosavljevic, Miranda; also Grempel et al.)
EGP sets in for W > W* = (t2ravJK)1/2
EGP always comes BEFORE the MIT
MIT at W = Wc ~ EF
W
W*
Wc
EGP +RKKY interactions: beyond semi-classical spins!
(Tanaskovic, Dobrosavljevic, Miranda)!
• 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,...)
Destruction of the Kondo effect and two-fluid behavior
• BF model has a (local) phase transition for a sub-Ohmic dissipative bath (e > 0 )
• EGP model: distribution of Kondo
couplings all the way to zero!
Kondo screened
• A finite fraction of spins fall on each
side of the critical line
• Kondo effect destroyed by dissipation
on a finite fraction of spins
spin fluid
~e
• Decoupled spins JK flows to zero; they
form a spin fluid (Sachdev-Ye)
(frustrated insulating magnet)
g (RKKY coupling)
Fermi liquid
NFL - spin fluid
e>0
W1
W* “bare EGP” MIT
insulator
disorder W
Spin-glass (SG) instability of the EGP
•
χ(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.)
- disorder
Conclusions:
• In metals dissipation destroys QGP at lowest T
→ (quantum) glassy ordering
• Magnetic (QCP) QGP: → semi-classical dynamics at T > TG
• Fluctuation–driven first-order QCP of the “cluster glass”
• Spin liquid in EGP at T > TG