Infinite Randomness Fixed Point of the
disordered Mott transition?
Eric Castro Andrade1,2
E. Miranda1
Vlad Dobrosavljević2
1Campinas,
Brazil
2Florida State University FL, U.S.A.
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}
E.Miranda, V. Dobrosavljevic,
Reports on Progress in Physics 68, 2337 (2005)
How generic is IRFP?
Quantum insulating magnets + disorder (D. Fisher et al.)
• Ising (and other discrete symm.) models – YES
• Heisenberg (and other continuous symm.) – NO
More recently (2003-2008)
Dissipative (itinerant?) magnets + disorder
• Ising model: transition “rounding” – NO (T. Vojta, 2003)
• Heisenberg (and other continuous symm.) – YES!!!
(T. Vojta, J. Schmalian, J. Hoyos, 2005-2008)
• Heisenberg + RKKY interactions – cluster glass – NO
(V.D. and E. Miranda, 2005)
Metal-Insulator Transition?
• Example: Doped semiconductors Si:P,B
Paalanen et al, Physica B 169, 223 (1991)
Uncompensated Si:P
Compensated Si:P, B
2-Fluid Phenomenology
• Strongly localized states dominate the spin response even in
the metallic state. Spin and charge seem to decouple.
insulator
M. J. Hirsch et al.,
PRL 68, 1418 (1992)
metal
Two transitions?
• Possible phase diagram
Fermi-liquid metal
(T=0)=o
Non-Fermi-liquid
metal
(T=0)=
Insulator
MIT
Disorder
• Previous studies:
– (Milovanović, Sachdev, and Bhatt, PRL 63, 82 (1989)) Mean-field theory of the
disordered Hubbard model
• Shows localized magnetic states in the presence of delocalized singleparticle states.
• The mean-field theory (Anderson, 1962) is known to break down at low T
due to the Kondo effect: need to describe the scales at which this occurs?
– R. N. Bhatt and D. S. Fisher, PRL 68, 3072 (1992): Kondo vs. RKKY; non-Fermi
liquid behavior, divergent thermodynamics slower than any power-law: need
quantitative calculation (fluctuations of the density of states).
Incorporating the Kondo effect
• Statistical Dynamical Mean Field Theory (Kotliar and
Dobrosavljević, PRL 78, 3943 (1997)): assume local (though spatially
non-uniform)
self-energiesEach local site is governed by an impurity action:
Local
renormalizations
: hybridization to the neighboring sites
Physics content of statDMFT
• Treat each impurity action within the Gutzwiller approximation
(PRL 10, 159 (1963)); equivalent to the Kotliar-Ruckenstein slave
boson formalism (PRL 57, 1362 (1986)).
• Clean case (W=0): Mott metal-insulator transition at U=Uc,
where Z (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 vanish (Mott transition) like in the clean case
(Brinkman and Rice, PRB 2, 4302 (1970)).
• If we re-scale all Zi by Z0 ~ Uc(W)-U, we can look at P(Zi /Z0)
• In the limit of infinite coordination (dynamical mean field theory), P(Z/Z0)
tends to a universal form at Uc.
Z0
gap
Results in d=2
(E. C. Andrade, Eduardo Miranda, V. D., PRL, 2009)
• In low dimensions, the environment of each site (“bath”) has
strong spatial fluctuations
• New physics: rare evens due to fluctuations and spatial
correlations
DMFT
However, in d=2, the
distribution P(Z/Z0) acquires a
low-Z tail:
Results: thermodynamics
• 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))
Phase diagram (W<U)
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
-1 – variance of log(Z)
Quantum Griffiths phases and IRFP
• 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}
E.Miranda, V. D.,
Reports on Progress in Physics 68, 2337 (2005))
“Size” of the rare events?
Typical sample
Rare event!
Replace the environment of
given site outside square by
uniform (DMFT-CPA) effective
medium.
Reduce square size down to
DMFT limit.
Rare events 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???