Transcript Strongly Correlated Electron Systems a Dynamical Mean Field Perspective G. Kotliar

Strongly Correlated Electron
Systems a Dynamical Mean
Field Perspective
G. Kotliar
Physics Department and Center for
Materials Theory
Rutgers
ICAM meeting: Frontiers in Correlated Matter
Snowmass September 2004
Strongly Correlated Electron Systems Display remarkable
THE WHY
phenomena, that cannot be understood within the standard model of
solids. Resistivities that rise without sign of saturation beyond the
Mott limit, (e.g. H. Takagi’s work on Vanadates), temperature
dependence of the integrated optical weight up to high frequency
(e.g. Vandermarel’s work on Silicides).
Correlated electrons do “big things”, large volume collapses, colossal
magnetoresitance, high temperature superconductivity . Properties are
very sensitive to structure chemistry and stoichiometry, and control
parameters large non linear susceptibilites,etc……….
Need non perturbative tool.
THE HOW
How to think about their electronic
states ?
How to compute their properties ?
Mapping onto connecting their
properties, a simpler “reference
system”. A self consistent impurity
model
living on SITES, LINKS and
PLAQUETTES......
DYNAMICAL MEAN FIELD THEORY.
"Optimal Gaussian Medium " + " Local Quantum Degrees of Freedom " + "their interaction "
is a good reference frame for understanding, and predicting physical properties
of correlated materials. Focus on local quantities, construct functionals of those quantities, similarities with DFT.
What did we learn ? Schematic DMFT phase diagram
and DOS of a partially frustrated integer filled
Hubbard model and pressure driven Mott transition.
Pressure driven Mott transition.
How do we know there is some truth in this
picture ? Qualitative Predictions Verified
• Two different features in spectra. Quasiparticles
bands and Hubbard bands.
• Transfer of spectral weight which is non local in
frequency. Optics and Photoemission.
• Two crossovers, associated with gap closure
and loss of coherence. Transport.
• Mott transition endpoint, is Ising like, couples to
all electronic properties.
• An “exact numerical approach PRG “ recently
found the first order line(M. Imada), C-DMFT
offers a consistency check.
Ising critical endpoint found! In V2O3 P.
Limelette et.al. (Science 2003)
Anomalous transfer of optical spectral
weight, NiSeS. [Miyasaka and Takagi
2000]
Why does it work: Energy Landscape of
a Correlated Material and a top to bottom
approach to correlated materials.
Single site DMFT. High temperature
universality vs low temperature
sensitivity to detail for materials
near a temperature-pressure driven
Mott transition
Energy
T
Configurational Coordinate in the space of Hamiltonians
What did we gain?
• Conceptual understanding of how the electronic
structure evolves when the electron goes from
localized to itinerant.
• Uc1 Uc2, transfer of spectral weight, ….
• A general methodology which was extended to
clusters (non trivial!) and integrated into an
electronic structure method, which allows us to
incorporate structure and chemistry. Both are
needed away from the high temperature
universal region.
•
Mott transition across the 5f’s, a very
interesting playground for studying
correlated electron phenomena.
•
DMFT ideas have been extended into
a framework capable of making first
principles first principles studies of
correlated materials. Pu Phonons.
Combining theory and experiments to
separate the contributions of different
energy scales, and length scales to
the bonding
In single site DMFT , superconductivity
is an unavoidable consequence when
we try to go move from a metallic state
to a Mott insulator where the atoms
have a closed shell (no entropy).
Realization in Am under pressure ?
•
DMFT Phonons in fcc d-Pu: connect bonding to
energy and length scales.
C11 (GPa)
C44 (GPa)
C12 (GPa)
C'(GPa)
Theory
34.56
33.03
26.81
3.88
Experiment
36.28
33.59
26.73
4.78
( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003)
(experiments from Wong et.al, Science, 22 August 2003)
Big question: will we be nearly as successful
in our attemps to understand and predict
(some ) physical properties of correlated
materials, with DMFT, as we have been for
weakly correlated materials using (
approximate DFT and perturbation theory in
screened Coulomb interactions eg.GW )?
A rapidly convergent algorithm ?
One dimensional Hubbard model 2 site (LINK) CDMFT compare with Bethe Anzats, [V.
Kancharla C. Bolech and GK PRB 67, 075110 (2003)][[M.CaponeM.Civelli V Kancharla
C.Castellani and GK P. R B 69,195105 (2004) ]
U/t=4.
Links, Ti2O3 : Coulomb and
Pauling
C.E.Rice et all, Acta Cryst B33, 1342 (1977)

LTS 250 K, HTS 750 K.
Evolution of the k resolved Spectral
Function at zero frequency. (Parcollet Biroli and
GK PRL, 92, 226402. (2004)) )
A(  0, k )vs k
U/D=2
U/D=2.25
Uc=2.35+-.05, Tc/D=1/44
U/t=16,t’= +0.9
U/t=8, t’= -0.3
Density= 0.88, 0.89, 0.9, 0.91, 0.922,
0.96, 0.986, 0.988, 0.989, 0.991,
0.993
Underlying normal state
of the Hubbard model
near the Mott transition,
(force the Weiss field to
its paramagnetic value),
T=0 ED solution of the
C-DMFT equations. M.
Civelli, M. Capone, O.
Parcollet and GK
Approaching the Mott transition:
plaquette Cdmft.
• Qualitative effect, momentum space
differentiation. Formation of hot –cold regions is
an unavoidable consequence of the approach to
the Mott insulating state!
• D wave gapping of the single particle spectra as
the Mott transition is approached.
• Study the “normal state” of the Hubbard model.
General phenomena, but the location of the cold
regions depends on parameters. [Civelli Capone
Parcollet and Kotliar ]
Where do we go now ?
• One can study a large number of experimentally
relevant problems within the single site
framework.
• Continue the methodological development, we
need tools!
• Solve the CDMFT Mott transition problem on
the plaquette problem, hard, but it is a significant
improvement, the early mean field theories while
keeping its physical appeal.
• Study material trends, make contact with
phenomenological approaches, doped
semiconductors (Bhatt and Sachdev), heavy
fermions , 115’s(Nakatsuji, Pines and Fisk )……
Mott transition into an open (right) and closed (left) shell
systems. In single site DMFT, superconductivity must intervene
before reaching the Mott insulating state.[Capone et. al. ] Am
At room pressure a localised 5f6 system;j=5/2. S = -L = 3: J
= 0 apply pressure ?
S
.g T
Log[2J+1]
S
???
Uc
U
S=0
g ~1/(Uc-U)
U
Americium under pressure [J.C.
Griveaux J. Rebizant G. Lander]
Evolution of the Spectral Function with
Temperature
Anomalous transfer of spectral weight connected to the
proximity to the Ising Mott endpoint (Kotliar Lange nd
Rozenberg Phys. Rev. Lett. 84, 5180 (2000)
Answer: cautiously optimistic yes,
but it needs a lot of work.
• Focus on short distance intermediate
energy scale properties. [Method is
designed for that]
• Need analytic +numerical work.
Connection with other approaches/DMRG
• Need adaptive k space.
• One can already do a lot with single site
DMFT in many many many materials.
• Plaquette equations are one order of
magnitude harder to solve.
Total Energy as a function of volume for Pu
W
(ev) vs iw (a.u. 27.2 ev)
(Savrasov, Kotliar, Abrahams, Nature ( 2001)
Non magnetic correlated state of fcc Pu.
Zein Savrasov and Kotliar (2004)
DMFT Phonons in fcc d-Pu
C11 (GPa)
C44 (GPa)
C12 (GPa)
C'(GPa)
Theory
34.56
33.03
26.81
3.88
Experiment
36.28
33.59
26.73
4.78
( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003)
(experiments from Wong et.al, Science, 22 August 2003)
Epsilon Plutonium.
Phonon entropy drives the
epsilon delta phase transition
• Epsilon is slightly more delocalized than delta,
has SMALLER volume and lies at HIGHER
energy than delta at T=0. But it has a much
larger phonon entropy than delta.
• At the phase transition the volume shrinks but
the phonon entropy increases.
• Estimates of the phase transition following
Drumont and G. Ackland et. al. PRB.65, 184104
(2002); (and neglecting electronic entropy).
Transverse Phonon along
(0,1,1) in epsilon Pu in self
consistent Born approximation.
Mott transition into an open (right) and closed (left) shell
systems. In single site DMFT, superconductivity must intervene
before reaching the Mott insulating state.[Capone et. al. ] Am
At room pressure a localised 5f6 system;j=5/2. S = -L = 3: J
= 0 apply pressure ?
S
.g T
Log[2J+1]
S
???
Uc
U
S=0
g ~1/(Uc-U)
U
Americium under pressure [J.C.
Griveaux J. Rebizant G. Lander]
Overview of rho (p, T)
of Am
• Note strongly
increasing
resistivity as f(p)
at all T. Shows
that more
electrons are
entering the
conduction band
• Superconducting
at all pressure
• IVariation of rho
vs. T for
increasing p.
DMFT study in the fcc structure. S.
Murthy and G. Kotliar
fcc
LDA+DMFT spectra. Notice the
rapid occupation of the f7/2 band.
One electron spectra. Experiments (Negele) and LDA+DFT
theory (S. Murthy and GK )
Conclusion Am
• Crude LDA+DMFT calculations describe the crude
energetics of the material, eq. volume, even p vs V .
• Superconductivity near the Mott transition.
Tc increases first and the decreases as we approach the
Mott boundary.
Dramatic effect in the f bulk module.
 What is going on at the Am I- Am II boundary ??? Subtle
effect (bulk moduli do not change much ), but crucial
modifications at low energy.
 Mott transition of the f7/2 band ? Quantum critical point
?:
H.Q. Yuan et. al. CeCu2(Si2-x Gex).
Am under pressure Griveau et. al.
Electronic states in weakly and
strongly correlated materials
• Simple metals, semiconductors. Fermi Liquid
Description: Quasiparticles and quasiholes, (and their
bound states ). Computational tool: Density functional
theory + perturbation theory in W, GW method.
• Correlated electrons. Atomic states. Hubbard bands.
Narrow bands. Many anomalies.
• Need tool that treats Hubbard bands, and quasiparticle
bands, real and momentum space on the same footing.
DMFT!
Weakly correlated electrons. FLT and DFT, and what goes wrong in
correlated materials.
• Fermi Liquid . . Correspondence between a
system of non interacting particles and the full
Hamiltonian.
• A band structure is generated (Kohn Sham
system).and in many systems this is a good
starting point for perturbative computations of the
spectra (GW).
 [  (r) ,  (r) ]
DMFT Cavity Construction: A. Georges and G. Kotliar
PRB 45, 6479 (1992). Figure from : G. Kotliar and D.
Vollhardt Physics Today 57,(2004)
http://www.physics.rutgers.edu/~kotliar/RI_gen.html
The self consistent impurity model is a new
reference system, to describe strongly
correlated materials.
H  H cluster  H cluster exterior  H exterior
H
H cluster  H cluster exterior  H exterior
Simpler "medium" Hamiltonian
Dynamical Mean Field
Theory (DMFT) Cavity
Construction: A. Georges
and G. Kotliar PRB 45, 6479
(1992).
Site Cell. Cellular DMFT. C-DMFT.
G. Kotliar,S.. Savrasov,
G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001)
tˆ(K) hopping expressed in the superlattice notations.
•Other cluster extensions (DCA Jarrell Krishnamurthy,
Katsnelson and Lichtenstein periodized scheme, Nested
Cluster Schemes Schiller Ingersent ), causality issues, O.
Parcollet, G. Biroli and GK cond-matt 0307587 (2003)
Two paths for ab-initio
calculation of electronic
structure of strongly
correlated materials
Crystal structure +Atomic
positions
Model Hamiltonian
Correlation Functions Total
Energies etc.
DMFT ideas can be used in both cases.
LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin
and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). A Lichtenstein and M.
Katsnelson PRB 57, 6884 (1988).
• The light, SP (or SPD) electrons are extended,
well described by LDA .The heavy, D (or F)
electrons are localized treat by DMFT.
• LDA Kohn Sham Hamiltonian already contains an
average interaction of the heavy electrons, subtract
this out by shifting the heavy level (double counting
term)
Kinetic energy is provided by the Kohn Sham
Hamiltonian (sometimes after downfolding ). The U
matrix can be estimated from first principles of
viewed as parameters. Solve resulting model using
DMFT.
Functional formulation. Chitra and Kotliar (2001), Savrasov and Kotliarcondmatt0308053 (2003).
+
Ir>=|R, >
G = - < y ( Rr ') y † ( Rr ') >
1
f ( x)VC - 1 ( x, x ')f ( x ') +
ò
ò
2
ò if ( x)y
†
( x )y ( x )
< f ( Rr ')f ( Rr ) > - < f ( Rr ') > < f ( Rr ) > = W
1
1
[G,W ]  TrLnG  Tr[G01  G 1 ]G  TrLnW  Tr[VC1  W 1 ]W  Ehartree  [G,W ]
2
2
[G,W ]  EDMFT [Gloc ,Wloc , Gnonloc  0,Wnonloc  0]
Double loop in Gloc and Wloc
Impurity model representability of
spectral density functional.
RVB phase diagram of the Cuprate
Superconductors
• P.W. Anderson.
Baskaran Zou and
Anderson.
Connection between
high Tc and Mott
physics.
• <b> coherence order
parameter.
• K, D singlet formation
order paramters.
G. Kotliar and J. Liu Phys.Rev. B
38,5412 (1988)
• High temperature superconductivity is an
unavoidable consequence of the need to
connect with Mott insulator that does not break
any symmetries to a metallic state.
• Tc decreases as the quasiparticle residue goes
to zero at half filling and as the Fermi liquid
theory is approached.
• Early on, accounted for the most salient features
of the phase diagram. [d-wave
superconductivity, anomalous metallic state,
pseudo-gap state ]
Problems with the approach.
• Numerous other competing states. Dimer phase,
box phase , staggered flux phase , Neel order,
• Stability of the pseudogap state at finite
temperature.
• Missing finite temperature . [ fluctuations of slave
bosons , ]
• Temperature dependence of the penetration
depth [Wen and Lee , Ioffe and Millis ] Theory:
 [T]=x-Ta x2 , Exp: [T]= x-T a.
• Theory has uniform Z on the Fermi surface, in
contradiction with ARPES.
Evolution of the spectral
function at low frequency.
A(  0, k )vs k
Ek=t(k)+Re( k ,   0)  
g k = Im( k ,   0)
A( k ,   0) 
gk
g k 2  Ek 2
If the k dependence of the self energy is
weak, we expect to see contour
lines corresponding to Ek = const
and a height increasing as we
approach the Fermi surface.
Study a model of kappa organics.
Keeps all the goodies of the slave boson mean
field and make many of the results more solid
but also removes the main difficulties.
• Can treat coherent and incoherent spectra.
• Not only superconductivity, but also the
phenomena of momentum space differentiation
(formation of hot and cold regions on the Fermi
surface) are unavoidable consequence of the
approach to the Mott insulator.
• Can treat dynamical fluctuations between
different singlet order parameters.
• Surprising role of the off diagonal self energy
which renormalizes t’.
Spectral Evolution at T=0 half filling full
frustration figure from X.Zhang M. Rozenberg G.
Kotliar (PRL 70,16661993)
• Spectra of the
strongly correlated
metallic regime
contains both
quasiparticle-like and
Hubbard band-like
features.
• Mott transition is
driven by transfer of
Evolution of the Spectral Function with
Temperature
Anomalous transfer of spectral weight connected to the
proximity to the Ising Mott endpoint (Kotliar Lange nd
Rozenberg Phys. Rev. Lett. 84, 5180 (2000)
Consequences for the optical conductivity Evidence
for QP peak in V2O3 from optics.
M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkine J Honig and P
Metcalf Phys. Rev. Lett. 75, 105 (1995)
Anomalous transfer of optical
spectral weight V2O3
:M Rozenberg G. Kotliar and H. Kajuter Phys. Rev. B 54, 8452 (1996).
M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig
and P Metcalf Phys. Rev. Lett. 75, 105 (1995)
Optical transfer of spectral weight , kappa organics.
Eldridge, J., Kornelsen, K.,Wang, H.,Williams, J.,
Crouch, A., and Watkins, D., Sol. State. Comm., 79, 583
(1991).
Anomalous Resistivity and Mott
transition Ni Se2-x Sx
Crossover from Fermi liquid to bad metal to
semiconductor to paramagnetic insulator.
ET =
k-(ET)2X are across Mott transition
Insulating
anion layer
X-
Ground
State
U/t
t’/t
Cu2(CN)3
Mott
insulator
8.2
1.06
Cu[N(CN)2]Cl
Mott
insulator
7.5
0.75
Cu[N(CN)2]Br
SC
7.2
0.68
Cu(NCS)2
SC
6.8
0.84
Cu(CN)[N(CN)2 SC
]
Ag(CN)2 H2O
SC
6.8
0.68
6.6
0.60
I3
6.5
0.58
X-1
conducting
ET layer
[(ET)2]+1
modeled to triangular lattice
t’
t
SC
Prof. Kanoda U. Tokyo
Mott transition in layered organic conductors
cond-mat/0004455, Phys. Rev. Lett. 85, 5420 (2000)
S Lefebvre et al.
• Theoretical issue: is there a Mott transition
in the integer filled Hubbard model, and is it
well described by the single site DMFT ?
Evolution of the spectral
function at low frequency.
A(  0, k )vs k
Ek=t(k)+Re( k ,   0)  
g k = Im( k ,   0)
A( k ,   0) 
gk
g k 2  Ek 2
If the k dependence of the self energy is
weak, we expect to see contour
lines corresponding to Ek = const
and a height increasing as we
approach the Fermi surface.
Approaching the Mott transition:
plaquette Cdmft.
• Qualitative effect, momentum space
differentiation. Formation of hot –cold
regions is an unavoidable consequence of
the approach to the Mott insulating state!
• D wave gapping of the single particle
spectra as the Mott transition is
approached..
• Square symmetry is restored as we
approched the insulator
Mechanism for hot spot formation: nn
self energy ! General phenomena.
Conclusion.
• Mott transition survives in the cluster
setting. Role of magnetic frustration.
• Surprising result: formation of hot and cold
regions as a result of an approach to the
Mott transition. General result ?
• Unexpected role of the next nearest
neighbor self energy. CDMFT a new
window to extend DMFT to lower
temperatures.
Conclusion
• DMFT mapping onto “self consistent impurity
models” offer a new “reference frame”, to think
about correlated materials and compute their
physical properties.Formal parallel with DFT.
• .Plaquettes-Kappa organics-Hot and cold
regions.
• Titanium sesquioxides. Dynamical Pauling
Goodenough mechanism.
• Sites. Phonons in Plutonium. Mott transition
across the actinide series.
Pauling and Coulomb Ti2O3[S.
Poteryaev S. Lichtenstein and GK PRL (2004)
Dynamical GoodenoughHonig Pauling
picture
2site-Cluster DMFT with intersite Coulomb
U = 2, J = 0.5, W = 0.5
β = 20 eV-1, LT structure
U = 2, J = 0.5, W = 0.5
β = 10 eV-1, HT
structure
A. Poteryaev
U/t=16,t’= +0.9
U/t=8, t’= -0.3
Density= 0.88, 0.89, 0.9, 0.91, 0.922,
0.96, 0.986, 0.988, 0.989, 0.991,
0.993
Underlying normal state
of the Hubbard model
near the Mott transition,
(force the Weiss field to
its paramagnetic value),
T=0 ED solution of the
C-DMFT equations. M.
Civelli, M. Capone, O.
Parcollet and GK
U/t=16 t’=-.3 n=.95 and t’=.9 n=.95
Insights into the differences between
electron and hole doped cuprates ?
• t’ <0 has an underlying normal state with
QP around (pi/2, pi/2). This is a state
which can naturally evolve into the d-wave
superconductor.
• t’>o has the quasiparticles around (pi,0),
does not connect smoothly with the SC.
What did we learn ? Schematic DMFT phase diagram
and DOS of a partially frustrated integer filled
Hubbard model and pressure driven Mott transition.