The disordered Mott metal-insulator
transition
Eduardo Miranda
University of Campinas, Brazil
Daniel C. B. Miranda
Campinas, Brazil
Martha Y. Suárez Villagrán
Eric C. Andrade
Marcelo Rozenberg (Université Paris-Sud, Orsay, France)
EPSRC Symposium Workshop on Quantum Simulations
University of Warwick, August 25, 2009
The Mott-Hubbard transition: beating Bloch’s
theorem
Neville Mott
Localization by interaction: the price paid in kinetic energy by
localizing a quantum particle is compensated by the having the
electrons avoid each other and interact less.
one electron per site
Coulomb repulsion
Kinetic energy
CuSO4.5H2O
Mott insulators:
NiO, MnO, V2O3
“Mott-Hubbard
turtles”
The Mott transition
 (V0.989Cr0.011)2O3 under pressure (P. Limelette et al., Science 302, 89 (2003)).
(V0.989Cr0.011)2O3
Hysteresis: metal-insulator coexistence, first-order transition (like liquid-gas)
The Mott transition: other examples
k-(ET)2Cu[N(CN)2]Cl
NiS2
Ni(S1-xSex)2
Takagi’s group
Kanoda, J. Phys. Soc. Jpn. 75, 051007 (2006)
Hysteresis: metal-insulator coexistence, first-order transition (like liquid-gas)
Theoretical approach: DMFT
Focus on a particular site
Inspired by the infinite-D limit:
Mean-field function: how can we self-consistently
determine it?
(from Kotliar and Vollhardt, Phys. Today, March 2004)
First, find
From it, get the self-energy
Note that, as in D , the self-energy is local
Now impose that the above G (t ) coincides with
the local Green´s function obtained from the
lattice Green´s function
DMFT phase diagram of the Hubbard model
mean-field critical behavior
paramagnetic solution
critical end-point
(V0.989Cr0.011)2O3
H. Park, K. Haule and G. Kotliar, arXiv:0803.1324v2
coexistence region, 1st order transition
(P. Limelette et al., Science 302, 89 (2003)).
What about disorder?
NiS2
Ni(S1-xSex)2
(V1-xMx)2O3
Takagi’s group
How does disorder
affect the transition?
McWhan et al., PRB (1971)
DMFT or not DMFT?
Let us focus on a diagonally disordered Hubbard model:
uniform distribution, width W
What is the DMFT description? (M. C. O. Aguiar et al., PRB 71, 205115 (2005))
 D= limit: each site “sees” an infinite number of neighbors
corresponding bath is self-averaging.
 Ok, but poor description of fluctuations of the local density of states, no Friedel
oscillations, no Anderson transition.
One can keep the local correlations while allowing for full spatial fluctuations of the
density of states:
Statistical DMFT: the method
(V. Dobrosavljevic and G. Kotliar, Phys. Rev. Lett. 78, 3943 (1997))
Time- and space-dependent
mean field
Compute the local Green’s
function at each site (hard part)
Get the local self-energy for each site:
The shift
defines the lattice Green’s function/resolvent
Self-consistency:
Note that the theory is exact at U=0.
non-interacting Hamilt.
Statistical DMFT: the altgorithm
Algorithm:
1. Start with an initial guess for
2. Solve the effective action and find
(hardest part)
3. Find new
from the self-consistency condition
4. Check for convergence. If not there yet, go back to 1.
Step 2 – Impurity solver
a) Choose from: QMC, NRG, DMRG, slave bosons...
b) Usually the slowest step, but “embarassingly” parallel (each site is an
independent problem).
Step 3 – Non-Hermitian matrix inversion for each frequency (need only diagonal terms)
a) Costly for large lattices, but also parallelizable.
Whole algorithm
a) Can be accelerated by the modified Broyden’s method (see recent Rok Zitko,
arXiv:0908.0613v1).
b) Other suggestions?
Results for the disordered Mott transition
(Martha Y. S. Villagrán, Daniel B. C. Miranda, Eric C. Andrade, M. Rozenberg, E.M., in preparation)
uniform distribution, width W
Impurity solver: Hirsch-Fye QMC (PRL 56, 2521 (1986)).
100,000 lattice sweeps of the auxiliary (64 or 128) Ising spin variables were necessary to
get the hysteresis loop.
We have solved for the paramagnetic phase of 2D L x L lattices up to L=20 at T0
clean phase diagram
H. Park, K. Haule and G. Kotliar, arXiv:0803.1324v2
Telling the metal from the insulator at T0
How does one characterize the conducting character locally, through the local
Metal
Value of
Insulator
at the first Matsubara frequency can serve as indicator.
?
Conducting landscape above Tc
droplet of bad metal
droplet of bad insulator
Diagonal disorder as local doping
Spatial fluctuations of the site energy
locally dope away from half-filling.
Effect is fairly localized for weak
disorder.
more metallic
Sites organized in
decreasing order of
more insulating
corresponding site
energies
Above Tc: density of states fluctuations
The relative spatial fluctuations of the density of states
close to the Fermi energy peaks around Tc
Below Tc: from metal to insulator
Behavior of
as a function of iteration number
for the 10x10 sites
Metal
Metal
Converged solution for one value
of U is used as starting point for
the next one: hysteresis
Average local Green’s function
Uc2: line of instability (spinodal)
of the metastable metallic phase
Insulator
Insulator
Below Tc: from insulator to metal
Uc1: line of instability (spinodal) of
the metastable insulating phase
Insulator
Insulator
Metal
Metal
Hysteresis below Tc:
metal-insulator coexistence
Metal-insulator coexistence:
the landscape
Insulator
Metal
Tc suppression by disorder
Tc is suppressed and the Uc lines are
pushed to larger interactions, both in
DMFT and statDMFT
DMFT
statDMFT
(M. C. O. Aguiar et al., PRB 71, 205115 (2005))
Conclusions and perspectives
• StatDMFT: powerful tool for stongly correlated
disordered systems.
• Coexisting droplets of bad metals and bad insulators
above Tc.
• Relative fluctuations of DOS peak close to the critical
end-point.
• Below Tc , the metal-insulator coexistence region
survives the introduction of weak disorder (L?).
• Tc is suppressed and Uc is enhanced by disorder.
• Fate of the metallic phase as T0 ???
DMFT phase diagram of the Hubbard model
paramagnetic solution
antiferromagnetic solution
critical end-point
H. Park, K. Haule and G. Kotliar, arXiv:0803.1324v2
coexistence region, 1st order transition
M. Rozenberg, G. Kotliar, X. Y. Zhang, PRB 1994
The convergence process