Disorder screening near theMott-Anderson transition
M. C. O. Aguiar a,b,∗, V. Dobrosavljević c, E. Abrahams b, G. Kotliar b
aDepartamento de F́ısica, Universidade Federal de Minas Gerais, Av. Antônio Carlos, 6627, Belo Horizonte, MG, Brazil
bCenter for Materials Theory, Serin Physics Laboratory, Rutgers University, 136 Frelinghuysen Road, Piscataway, New Jersey 08854, USA
cDepartment of Physics and National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32306, USA
Abstract
Correlation-driven screening of disorder is studied within the typical-medium dynamical mean-field theory (TMT-DMFT) of the
Mott-Anderson transition. In the strongly correlated regime, the site energies εiR characterizing the effective disorder potential are
strongly renormalized due to the phenomenon of Kondo pinning. This effect produces very strong screening when the interaction
U is stronger then disorder W , but applies only to a fraction of the sites in the opposite limit (U < W ).
Key words: Strong correlation; disorder; metal-insulator transition; Hubbard model
PACS: 71.27.+a, 72.15.Rn, 71.30.+h
Introduction - Theories that are able to capture both the
Mott [1] and the Anderson [2] mechanisms for electron lo-
calization have remained elusive despite many years of ef-
fort. An attractive approach to this difficult problem has
recently been proposed by combining the dynamical mean-
field theory (DMFT) [3] of the Mott transition, and the
typical medium theory (TMT) [4] of Anderson localiza-
tion. This new formulation of the Mott-Anderson problem
has been explored in recent work by Vollhardt and collabo-
rators [5] using numerical renormalization-group methods,
but the precise mechanism for the critical behavior of this
model remains to be elucidated. Here we examine the mech-
anism for disorder screening within this theory, which ex-
plains several aspects of the results found in Ref. [5]
Within TMT-DMFT, a lattice problem is mapped onto
an ensemble of single-impurities problems, which are em-
bedded in a self-consistently determined bath. Recent work
of Ref. [6] examined the behavior of a collection of single-
impurity models in the situation where the bath seen by the
impurities was chosen to mimic the approach to the Mott-
Anderson transition. In this work, the impurity quasipar-
ticle weight Zi was shown to present a scaling behavior as
a function of the on-site energy εi and the distance t to
the transition. These findings, however, are not sufficient
to address the disorder screening behavior of the model,
which requires the description of the renormalized disor-
∗ Corresponding author. Tel: +55 31 3499-5671 fax: +55 31 3499-
Email address: aguiar@fisica.ufmg.br (M. C. O. Aguiar).
-0.5 -0.2 0.0 0.2 0.5
t = 0.10
t = 0.08
t = 0.06
t = 0.04
t = 0.02
t = 0.01
t = 0.004
t = 0.001
Fig. 1. Renormalized energy εR as a function of the on-site energy ε
for a collection of single-impurity problems close to the Mott-An-
derson transition (t → 0). The parameters used were U = 1.75 and
W = 2.8.
der potential. In this paper, we demonstrate that a scaling
procedure similar to that presented in Ref. [6] can also be
carried on for the renormalized energy εiR.
Renormalization of the disorder potential - We consider a
collection of Anderson impurity models [6] with on-site re-
pulsion U , on-site energies εi, and the total spectral weight
t of the cavity field. Without loss of generality [6], we con-
sider a featureless model bath with non-vanishing spectral
weight for −t/2 < ω < t/2 and zero otherwise. Our goal
is to describe the statistics of the renormalized site ener-
gies as the metal-insulator transition is approached, corre-
sponding to t → 0 within the TMT-DMFT scheme.
Preprint submitted to Elsevier 1 November 2018
http://arxiv.org/abs/0704.0450v1
δε’=|ε/W-0.3125|
ln(t*) = 0.58 + 2 ln(δε’)
ln(t*) = 3.4 + 2 ln(δε’)
Fig. 2. Scaled renormalized energy εR/t
1/2 as a function of t/t∗(δε)
showing that the results for different (and positive) ε can be collapsed
onto a single scaling function with two branches. Different symbols
correspond to different ε; the upper (bottom) branch contains results
for ε > U/2 (ε < U/2). The inset shows the scaling parameter t∗
as a function of |ε/W − 0.3125| for the upper (squares) and bottom
(circles) branches. The parameters used were U = 1.75 and W = 2.8.
The impurity models were solved at zero temperature us-
ing the SB4 method [7,6], which provides a parametrization
of the low-energy (quasiparticle) part of the local Green’s
function, given by
Gi(ωn) =
iωn − ε
R − Zi∆(ωn)
. (1)
Here Zi is the local quasiparticle weight and ε
R is the renor-
malized site energy. The details of the calculations mirror
those of Ref. [6].
The results for the renormalized energy εiR as a func-
tion of −W/2 < εi < W/2, in the vicinity of the Mott-
Anderson transition (t → 0), are shown in Fig. 1. As in
Ref. [6], we find two-fluid behavior, where sites with |εi| <
U/2 turn into local magnetic moments, corresponding to
“Kondo pinning” [8] εiR → 0. For the remaining sites, ε
εi + U/2 or ε
R → εi − U/2, as they become, respectively,
doubly occupied (those with εi < −U/2) or empty (those
with εi > U/2). We should emphasize that such two-fluid
behavior thus emerges only for sufficiently strong disorder,
such that U < W . Otherwise all sites turn into local mag-
netic moments, and the Mott transition for moderate dis-
order retains a character similar to that found within the
standard DMFT approach [8].
Scaling analysis - These results can alternatively be pre-
sented in a scaling form, as shown in Fig. 2. Here, we
show that it is possible to collapse the family of curves
εR(t, δε)/t
0.5, where δε = (εi − ε
∗) /ε∗ and ε∗ = U/2,
onto a single universal scaling function εR(t, δε)/t
0.5 =
f [t/t∗(δε)] with two branches, one for εi < ε
∗ and other
for εi > ε
∗. In agreement with Ref. [6] (inset of Fig. 2),
the crossover scale t∗(δε) ∼ |δε|φ, with exponent φ = 2. In
the limit t → 0, we find that the branch corresponding to
εi < ε
∗ has the asymptotic form f(x) ∼ x3/2 (here x =
t/t∗(δε)), corresponding to εR(t) ∼ t
2. Similarly, for εi >
ε∗, f(x) ∼ x−1/2 corresponding to εR(t) ∼ constant. For
0.0 0.5 1.0 1.5 2.0 2.5
clean value
Fig. 3. Arithmetic and geometric density-of-states (ADOS and
TDOS, respectively) at the Fermi level as a function of U , for
W = 1.5, when the TMT-DMFT self-consistent loop is performed.
x ≫ 1 the two branches merge, viz. f(x) ∼ A±B±x−0.5.
Disorder screening - Within TMT-DMFT, the Anderson
localization effects are manifested by the reduction of the
typical density of states (TDOS), since the (algebraic) aver-
age (ADOS) remains finite even in an Anderson insulator.
When disorder is strongly screened due to the correlation
effects, the two quantities should not differ much, as illus-
trated by the results of Fig. 3. Here we present the results of
the fully self-consistent solution, as the Mott-like transition
is approached by increasing U for W = 1.5. Close to the
transition, both averages approach the clean limit (dashed
line), indicating a strong screening effect. These results are
consistent with those found in the numerical renormaliza-
tion group solution of the TMT-DMFT equation of Ref. [5].
As discussed above, strong disorder screening is expected
near the Mott-like transition (U > W ), which indeed cor-
responds to the mechanism responsible for the results in
Fig. 3.When the transition is approached at strong disorder
(U < W ) (not shown), strong screening effects are found
only for a fractions of the sites (i.e of the volume of the
sample), indicating different critical behavior at the Mott-
Anderson transition. The details of the critical behavior in
this case will be discussed elsewhere.
Acknowledgements - This work was partially supported
by NSF grants DMR-0312495 (M.C.O.A.), DMR-0234215
and DMR-0542026 (V.D.) and DMR-0096462 (G.K.).
References
[1] N.F. Mott, Metal-insulator Transitions (Taylor and Francis,
London, 1974).
[2] P.W. Anderson, Phys. Rev. 109 (1958) 1498.
[3] A. Georges et al., Rev. Mod. Phys. 68 (1996) 13.
[4] V. Dobrosavljević et al., Europhys. Lett. 62 (2003) 76.
[5] K. Byczuk, W. Hofstetter, and D. Vollhardt, Phys. Rev. Lett. 94
(2005) 056404.
[6] M.C.O. Aguiar et al., Phys. Rev. B 73 (2006) 115117.
[7] G. Kotliar and A.E. Ruckenstein, Phys. Rev. Lett. 57 (1986)
1362.
[8] D. Tanasković et al., Phys. Rev. Lett. 91 (2003) 066603.
	References

  Correlation-driven screening of disorder is studied within the typical-medium
dynamical mean-field theory (TMT-DMFT) of the Mott-Anderson transition. In the
strongly correlated regime, the site energies epsilon_R^i characterizing the
effective disorder potential are strongly renormalized due to the phenomenon of
Kondo pinning. This effect produces very strong screening when the interaction
U is stronger then disorder W, but applies only to a fraction of the sites in
the opposite limit (U<W).

Introduction - Theories that are able to capture both the
Mott [1] and the Anderson [2] mechanisms for electron lo-
calization have remained elusive despite many years of ef-
fort. An attractive approach to this difficult problem has
recently been proposed by combining the dynamical mean-
field theory (DMFT) [3] of the Mott transition, and the
typical medium theory (TMT) [4] of Anderson localiza-
tion. This new formulation of the Mott-Anderson problem
has been explored in recent work by Vollhardt and collabo-
rators [5] using numerical renormalization-group methods,
but the precise mechanism for the critical behavior of this
model remains to be elucidated. Here we examine the mech-
anism for disorder screening within this theory, which ex-
plains several aspects of the results found in Ref. [5]
Within TMT-DMFT, a lattice problem is mapped onto
an ensemble of single-impurities problems, which are em-
bedded in a self-consistently determined bath. Recent work
of Ref. [6] examined the behavior of a collection of single-
impurity models in the situation where the bath seen by the
impurities was chosen to mimic the approach to the Mott-
Anderson transition. In this work, the impurity quasipar-
ticle weight Zi was shown to present a scaling behavior as
a function of the on-site energy εi and the distance t to
the transition. These findings, however, are not sufficient
to address the disorder screening behavior of the model,
which requires the description of the renormalized disor-
∗ Corresponding author. Tel: +55 31 3499-5671 fax: +55 31 3499-
Email address: aguiar@fisica.ufmg.br (M. C. O. Aguiar).
-0.5 -0.2 0.0 0.2 0.5
t = 0.10
t = 0.08
t = 0.06
t = 0.04
t = 0.02
t = 0.01
t = 0.004
t = 0.001
Fig. 1. Renormalized energy εR as a function of the on-site energy ε
for a collection of single-impurity problems close to the Mott-An-
derson transition (t → 0). The parameters used were U = 1.75 and
W = 2.8.
der potential. In this paper, we demonstrate that a scaling
procedure similar to that presented in Ref. [6] can also be
carried on for the renormalized energy εiR.
Renormalization of the disorder potential - We consider a
collection of Anderson impurity models [6] with on-site re-
pulsion U , on-site energies εi, and the total spectral weight
t of the cavity field. Without loss of generality [6], we con-
sider a featureless model bath with non-vanishing spectral
weight for −t/2 < ω < t/2 and zero otherwise. Our goal
is to describe the statistics of the renormalized site ener-
gies as the metal-insulator transition is approached, corre-
sponding to t → 0 within the TMT-DMFT scheme.
Preprint submitted to Elsevier 1 November 2018
http://arxiv.org/abs/0704.0450v1
δε’=|ε/W-0.3125|
ln(t*) = 0.58 + 2 ln(δε’)
ln(t*) = 3.4 + 2 ln(δε’)
Fig. 2. Scaled renormalized energy εR/t
1/2 as a function of t/t∗(δε)
showing that the results for different (and positive) ε can be collapsed
onto a single scaling function with two branches. Different symbols
correspond to different ε; the upper (bottom) branch contains results
for ε > U/2 (ε < U/2). The inset shows the scaling parameter t∗
as a function of |ε/W − 0.3125| for the upper (squares) and bottom
(circles) branches. The parameters used were U = 1.75 and W = 2.8.
The impurity models were solved at zero temperature us-
ing the SB4 method [7,6], which provides a parametrization
of the low-energy (quasiparticle) part of the local Green’s
function, given by
Gi(ωn) =
iωn − ε
R − Zi∆(ωn)
. (1)
Here Zi is the local quasiparticle weight and ε
R is the renor-
malized site energy. The details of the calculations mirror
those of Ref. [6].
The results for the renormalized energy εiR as a func-
tion of −W/2 < εi < W/2, in the vicinity of the Mott-
Anderson transition (t → 0), are shown in Fig. 1. As in
Ref. [6], we find two-fluid behavior, where sites with |εi| <
U/2 turn into local magnetic moments, corresponding to
“Kondo pinning” [8] εiR → 0. For the remaining sites, ε
εi + U/2 or ε
R → εi − U/2, as they become, respectively,
doubly occupied (those with εi < −U/2) or empty (those
with εi > U/2). We should emphasize that such two-fluid
behavior thus emerges only for sufficiently strong disorder,
such that U < W . Otherwise all sites turn into local mag-
netic moments, and the Mott transition for moderate dis-
order retains a character similar to that found within the
standard DMFT approach [8].
Scaling analysis - These results can alternatively be pre-
sented in a scaling form, as shown in Fig. 2. Here, we
show that it is possible to collapse the family of curves
εR(t, δε)/t
0.5, where δε = (εi − ε
∗) /ε∗ and ε∗ = U/2,
onto a single universal scaling function εR(t, δε)/t
0.5 =
f [t/t∗(δε)] with two branches, one for εi < ε
∗ and other
for εi > ε
∗. In agreement with Ref. [6] (inset of Fig. 2),
the crossover scale t∗(δε) ∼ |δε|φ, with exponent φ = 2. In
the limit t → 0, we find that the branch corresponding to
εi < ε
∗ has the asymptotic form f(x) ∼ x3/2 (here x =
t/t∗(δε)), corresponding to εR(t) ∼ t
2. Similarly, for εi >
ε∗, f(x) ∼ x−1/2 corresponding to εR(t) ∼ constant. For
0.0 0.5 1.0 1.5 2.0 2.5
clean value
Fig. 3. Arithmetic and geometric density-of-states (ADOS and
TDOS, respectively) at the Fermi level as a function of U , for
W = 1.5, when the TMT-DMFT self-consistent loop is performed.
x ≫ 1 the two branches merge, viz. f(x) ∼ A±B±x−0.5.
Disorder screening - Within TMT-DMFT, the Anderson
localization effects are manifested by the reduction of the
typical density of states (TDOS), since the (algebraic) aver-
age (ADOS) remains finite even in an Anderson insulator.
When disorder is strongly screened due to the correlation
effects, the two quantities should not differ much, as illus-
trated by the results of Fig. 3. Here we present the results of
the fully self-consistent solution, as the Mott-like transition
is approached by increasing U for W = 1.5. Close to the
transition, both averages approach the clean limit (dashed
line), indicating a strong screening effect. These results are
consistent with those found in the numerical renormaliza-
tion group solution of the TMT-DMFT equation of Ref. [5].
As discussed above, strong disorder screening is expected
near the Mott-like transition (U > W ), which indeed cor-
responds to the mechanism responsible for the results in
Fig. 3.When the transition is approached at strong disorder
(U < W ) (not shown), strong screening effects are found
only for a fractions of the sites (i.e of the volume of the
sample), indicating different critical behavior at the Mott-
Anderson transition. The details of the critical behavior in
this case will be discussed elsewhere.
Acknowledgements - This work was partially supported
by NSF grants DMR-0312495 (M.C.O.A.), DMR-0234215
and DMR-0542026 (V.D.) and DMR-0096462 (G.K.).
References
[1] N.F. Mott, Metal-insulator Transitions (Taylor and Francis,
London, 1974).
[2] P.W. Anderson, Phys. Rev. 109 (1958) 1498.
[3] A. Georges et al., Rev. Mod. Phys. 68 (1996) 13.
[4] V. Dobrosavljević et al., Europhys. Lett. 62 (2003) 76.
[5] K. Byczuk, W. Hofstetter, and D. Vollhardt, Phys. Rev. Lett. 94
(2005) 056404.
[6] M.C.O. Aguiar et al., Phys. Rev. B 73 (2006) 115117.
[7] G. Kotliar and A.E. Ruckenstein, Phys. Rev. Lett. 57 (1986)
1362.
[8] D. Tanasković et al., Phys. Rev. Lett. 91 (2003) 066603.
	References
