Collaborators: G.Kotliar, S. Savrasov, V. Oudovenko

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Electronic Structure of Strongly Correlated
Electron Materials A Dynamical Mean Field
Perspective.
Kristjan Haule, Physics Department and Center
for Materials Theory Rutgers University Rutgers
University

• Collaborators G.Kotliar, S. Savrasov, V.
  Oudovenko

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Overview

• Application of DMFT to real materials (Spectral
  density functional approach). Examples
• alpha to gamma transition in Ce, optics near the
  temperature driven Mott transition.
• Mott transition in Americium under pressure
• Extensions of DMFT to clusters.
• Examples
• Superconducting state in t-J the model
• Optical conductivity of the t-J model

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Universality of the Mott transition
Crossover bad insulator to bad metal
Critical point
First order MIT
Ni2-xSex
V2O3
k organics
1B HB model (DMFT)
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Coherence incoherence crossover in the 1B HB
model (DMFT)
Phase diagram of the HM with partial frustration
at half-filling M. Rozenberg et.al., Phys. Rev.
Lett. 75, 105 (1995).
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Dynamical Mean Field Theory
Basic idea of DMFT reduce the quantum many body
problem to a one site or a cluster of sites, in a
medium of non interacting electrons obeying a
self consistency condition. Basic idea of
Spectral density functional approach instead of
using functionals of the density, use more
sensitive functionals of the one electron
spectral function. density of states for adding
or removing particles in a solid, measured in
photoemission
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DFT DMFT from the unifying point of view
Density functional theory
observable of interest is the electron density
Dynamical mean field theory
observable of interest is the local Green's
function (on the lattice uniquely defined)
DMFT approximation
exact BK functional
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Spectral density functional theory
G. Kotliar et.al., cond-mat/0511085
observable of interest is the "local Green's
functions (spectral function)
Currently feasible approximations LDADMFT and
GWDMFT
basic idea sum-up all local diagrams for
electrons in correlated orbitals
LDAU corresponds to LDADMFT when impurity is
solved in the Hartree Fock approximation
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Periodic table
f1 L3,S1/2 J5/2
f6 L3,S3 J0
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Cerium
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Ce overview
? isostructural phase transition ends in a
critical point at (T600K, P2GPa) ? ? (fcc)
phase magnetic moment (Curie-Wiess law), large
volume, stable high-T, low-p ? ? (fcc) phase
loss of magnetic moment (Pauli-para), smaller
volume, stable low-T, high-p with large
volume collapse ?v/v ? 15?
volumes exp. LDA LDAU
a 28Å3 24.7Å3
g 34.4Å3 35.2Å3

• Transition is 1.order
• ends with CP

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LDA and LDAU
ferromagnetic
volumes exp. LDA LDAU
a 28Å3 24.7Å3
g 34.4Å3 35.2Å3
f DOS
total DOS
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LDADMFT alpha DOS
TK(exp)1000-2000K
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LDADMFT gamma DOS
TK(exp)60-80K
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Photoemissionexperiment

• A. Mc Mahan K Held and R. Scalettar (2002)
• K. Haule V. Udovenko and GK. (2003)

Fenomenological Landau approach
Kondo volume colapse (J.W. Allen, R.M. Martin,
1982)
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Optical conductivity


J.W. van der Eb, A.B. Kuzmenko, and D. van der
Marel, Phys. Rev. Lett. 86, 3407 (2001)
K. Haule, et.al., Phys. Rev. Lett. 94, 036401
(2005)
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Americium
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Americium
f6 -gt L3, S3, J0
Mott Transition?
"soft" phase f localized
"hard" phase f bonding
A.Lindbaum, S. Heathman, K. Litfin, and Y.
Méresse, Phys. Rev. B 63, 214101 (2001)
J.-C. Griveau, J. Rebizant, G. H. Lander, and
G.KotliarPhys. Rev. Lett. 94, 097002 (2005)
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Am within LDADMFT
F(0)4.5 eV F(2)8.0 eV F(4)5.4 eV F(6)4.0 eV
Large multiple effects
S. Y. Savrasov, K. Haule, and G. KotliarPhys.
Rev. Lett. 96, 036404 (2006)
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Am within LDADMFT
from J0 to J7/2
Comparisson with experiment
VV0 Am I
V0.76V0 Am III
V0.63V0 Am IV
nf6
nf6.2
Exp J. R. Naegele, L. Manes, J. C. Spirlet, and
W. MüllerPhys. Rev. Lett. 52, 1834-1837 (1984)

• Soft phase very different from g Ce
• not in local moment regime since J0 (no entropy)

Theory S. Y. Savrasov, K. Haule, and G.
KotliarPhys. Rev. Lett. 96, 036404 (2006)

• "Hard" phase similar to a Ce,
• Kondo physics due to hybridization, however,
• nf still far from Kondo regime

Different from Sm!
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Beyond single site DMFT
What is missing in DMFT?

• Momentum dependence of the self-energy m/m1/Z
• Various orders d-waveSC,
• Variation of Z, m,t on the Fermi surface
• Non trivial insulator (frustrated magnets)
• Non-local interactions (spin-spin, long range
  Columb,correlated hopping..)

• Present in cluster DMFT
• Quantum time fluctuations
• Spatially short range quantum fluctuations

• Present in DMFT
• Quantum time fluctuations

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The simplest model of high Tcs
t-J, PW Anderson
Hubbard-Stratonovich -gt(to keep some
out-of-cluster quantum fluctuations)
BK Functional, Exact
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What can we learn from small Cluster-DMFT?
Phase diagram
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Insights into superconducting state
(BCS/non-BCS)?
BCS upon pairing potential energy of electrons
decreases, kinetic energy increases (cooper pairs
propagate slower) Condensation energy is the
difference
non-BCS kinetic energy decreases upon
pairing (holes propagate easier in superconductor)
J. E. Hirsch, Science, 295, 5563 (2226)
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Optical conductivity
optimally doped
overdoped
cond-mat/0601478
D van der Marel, Nature 425, 271-274 (2003)
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Optical weight, plasma frequency
Weight bigger in SC, K decreases (non-BCS)
Weight smaller in SC, K increases (BCS-like)
D. van der Marel et.al., in preparation
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Hubbard versus t-J model

• Kinetic energy in Hubbard model
• Moving of holes
• Excitations between Hubbard bands

Hubbard model
U
Drude
t2/U
Excitations into upper Hubbard band

• Kinetic energy in t-J model
• Only moving of holes

Drude
t-J model
J
no-U
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Kinetic energy change
Kinetic energy increases
cluster-DMFT, cond-mat/0601478
Kinetic energy decreases
Kinetic energy increases
cond-mat/0503073
Exchange energy decreases and gives largest
contribution to condensation energy
Phys Rev. B 72, 092504 (2005)
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Kinetic energy upon condensation
underdoped
overdoped
electrons gain energy due to exchange
energy holes gain kinetic energy (move faster)
electrons gain energy due to exchange energy hole
loose kinetic energy (move slower)
BCS like
same as RVB (see P.W. Anderson Physica C, 341, 9
(2000), or slave boson mean field (P. Lee,
Physica C, 317, 194 (1999)
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Optics mass and plasma frequency
Extended Drude model

• Within DMFT, optics mass is m/m1/Z and diverges
  at the Mott transition
• Plasma frequency vanishes as 1/Z (Drude shrinks
  as Kondo peak shrinks)

• In cluster-DMFT optics mass constant at low
  doping doping
• Plasma frequency vanishes because the active
  (coherent) part of the Fermi surface shrinks

line cluster DMFT (cond-mat 0601478), symbols
Bi2212, Van der Marel (in preparation)
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Optimal doping Powerlaws
cond-mat/0605149
D. van der Marel et. al., Nature 425, 271 (2003).
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Optimal doping Coherence scale seems to vanish
underdoped
scattering at Tc
optimally
overdoped
Tc
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Local density of states of SC
Cluster DMFT
STM study
1 Larger doping 6 Smaller doping

• V shaped gap (d-wave)
• size of gap decreases with doping
• CDMFT-optimall doping PH symmetric

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41meV resonance

• Resonance at 0.16t48meV
• Most pronounced at optimal doping
• Second peak shifts with doping (at 0.38120meV
  opt.d.) and changes below Tc contribution to
  condensation energy

local susceptibility
YBa2Cu3O6.6 (Tc62.7K)
Pengcheng et.al., Science 284, (1999)
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Pseudoparticle insight
PH symmetry, Large t
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Conclusions

• LDADMFT can describe interplay of lattice and
  electronic structure near Mott transition. Gives
  physical connection between spectra, lattice
  structure, optics,....
• Allows to study the Mott transition in open and
  closed shell cases.
• In both Ce and Am single site LDADMFT gives the
  zeroth order picture
• Am Rich physics, mixed valence under pressure.
• 2D models of high-Tc require cluster of sites.
  Some aspects of optimally doped, overdoped and
  slightly underdoped regime can be described with
  cluster DMFT on plaquette
• Evolution from kinetic energy saving to BCS
  kinetic energy cost mechanism
• Optical mass approaches a constant at the Mott
  transition and plasma frequency vanishes
• At optimal doping Physical observables like
  optical conductivity and spin susceptibility show
  powerlaw behavior at intermediate frequencies,
  very large scattering rate vanishing of
  coherence scale, PH symmetry is dynamically
  restored, 41meV resonance appears in spin response

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LDADMFT implementation
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Comparison of spectral weight cluster DMFT /
Bi2212
Spectral weight (kinetic energy) changes faster
with T in overdoped system larger coherence
scale
Carbone et.al., in preparation
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Partial DOS
4f
Z0.33
5d
6s
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More complicated f systems

• Hunds coupling is important when more than one
  electron in the correlated (f) orbital
• Spin orbit coupling is very small in Ce, while it
  become important in heavier elements

The complicated atom embedded into fermionic
bath (with crystal fileds) is a serious chalange
so solve!
Coulomb interaction is diagonal in the base of
total LSJ -gt LS base while the SO coupling is
diagonal in the j-base -gt jj base Eigenbase of
the atom depends on the strength of the Hund's
couling and strength of the spin-orbit
interaction
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Classical theories
Mott transition (B. Johansson, 1974)
Hubbard model
f electrons insulating
changes and causes Mott tr.
spd electrons pure spectators
Anderson (impurity) model
Kondo volume colapse (J.W. Allen, R.M. Martin,
1982)
hybridization with spd electrons is crucial
(Lavagna, Lacroix and Cyrot, 1982)
changes ? chnange of TK
bath
f electrons in local moment regime
either constant or taken from LDA and rescaled
Fenomenological Landau approach
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LDADMFT
ab initio calculation
is self-consistently determined
bath for AIM
contains tff and Vfd hopping
Kondo volume colapse model resembles DMFT
picture Solution of the Anderson impurity model
? Kondo physics Difference with DMFT the
lattice problem is solved (and therefore ? must
self-consistently determined) while in KVC ? is
calculated for a fictious impurity (and needs to
be rescaled to fit exp.) In KVC scheme there is
no feedback on spd bans, hence optics is not much
affected.
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An example
Atomic physics of selected Actinides
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optics mass and plasma w
Basov, cond-mat/0509307
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Two Site CDMFT in the 1D Hubbard model
M.Capone M.Civelli V. Kancharla C.Castellani and
Kotliar, PRB 69,195105 (2004)
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Slave particle diagrammatic impurity solvers
every atomic state represented with a unique
pseudoparticle atomic eigenbase - full (atomic)
base
, where
general AIM
OCA
TCA
( )
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SUNCA vs QMC
three band Hubbard model, Bethe lattice, U5D,
T0.0625D
two band Hubbard model, Bethe lattice, U4D
three band Hubbard model, Bethe lattice, U5D,
T0.0625D