Strongly Correlated Electron Systems a Dynamical Mean Field Perspective

1
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
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Strongly Correlated Electron Systems Display
remarkable 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.
Takagis work on Vanadates), temperature
dependence of the integrated optical weight up
to high frequency (e.g. Vandermarels work on
Silicides).
THE WHY
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.
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THE HOW
Need non perturbative tool.
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.

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What did we learn ? Schematic DMFT phase diagram
and DOS of a partially frustrated integer filled
Hubbard model and pressure driven Mott transition.
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Pressure driven Mott transition.
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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.

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Ising critical endpoint found! In V2O3 P.
Limelette et.al. (Science 2003)
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Anomalous transfer of optical spectral weight,
NiSeS. Miyasaka and Takagi 2000
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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
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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.

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• Mott transition across the 5fs, 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 ?

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DMFT Phonons in fcc d-Pu connect bonding to
energy and length scales.
( Dai, Savrasov, Kotliar,Ledbetter, Migliori,
Abrahams, Science, 9 May 2003)
(experiments from Wong et.al, Science, 22 August
2003)
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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 )?
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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)
A rapidly convergent algorithm ?
U/t4.
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Links, Ti2O3 Coulomb and Pauling

• LTS 250 K, HTS 750 K.

C.E.Rice et all, Acta Cryst B33, 1342 (1977)
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Evolution of the k resolved Spectral Function at
zero frequency. (Parcollet Biroli and GK PRL,
92, 226402. (2004)) )
U/D2.25
U/D2
Uc2.35-.05, Tc/D1/44
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U/t16,t 0.9
U/t8, 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), T0 ED solution of the
C-DMFT equations. M. Civelli, M. Capone, O.
Parcollet and GK
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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

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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 ,
  115s(Nakatsuji, Pines and Fisk )

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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. AmAt
room pressure a localised 5f6 systemj5/2. S
-L 3 J 0 apply pressure ?
S
S
.g T
Log2J1
???
Uc
S0
U
U
g 1/(Uc-U)
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Americium under pressure J.C. Griveaux J.
Rebizant G. Lander
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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)
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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.

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Total Energy as a function of volume for Pu W
(ev) vs (a.u. 27.2 ev)
(Savrasov, Kotliar, Abrahams, Nature ( 2001) Non
magnetic correlated state of fcc Pu.
Zein Savrasov and Kotliar (2004)
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DMFT Phonons in fcc d-Pu
( Dai, Savrasov, Kotliar,Ledbetter, Migliori,
Abrahams, Science, 9 May 2003)
(experiments from Wong et.al, Science, 22 August
2003)
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Epsilon Plutonium.
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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 T0. 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).
  TC 600 K.

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Transverse Phonon along (0,1,1) in epsilon Pu in
self consistent Born approximation.
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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. AmAt
room pressure a localised 5f6 systemj5/2. S
-L 3 J 0 apply pressure ?
S
S
.g T
Log2J1
???
Uc
S0
U
U
g 1/(Uc-U)
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Americium under pressure J.C. Griveaux J.
Rebizant G. Lander
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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.

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DMFT study in the fcc structure. S. Murthy and G.
Kotliar
fcc
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LDADMFT spectra. Notice the rapid occupation of
the f7/2 band.
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One electron spectra. Experiments (Negele) and
LDADFT theory (S. Murthy and GK )
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Conclusion Am

• Crude LDADMFT 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 ?

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H.Q. Yuan et. al. CeCu2(Si2-x Gex). Am under
pressure Griveau et. al.
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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!

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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).

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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.
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Dynamical Mean Field Theory (DMFT) Cavity
Construction A. Georges and G. Kotliar PRB 45,
6479 (1992).
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Site? Cell. Cellular DMFT. C-DMFT. G.
Kotliar,S.. Savrasov, G. Palsson and G. Biroli,
Phys. Rev. Lett. 87, 186401 (2001)
t(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)

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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.
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LDADMFT 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.

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Functional formulation. Chitra and Kotliar
(2001), Savrasov and Kotliarcond- matt0308053
(2003).
IrgtR, rgt
Double loop in Gloc and Wloc
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Impurity model representability of spectral
density functional.
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RVB phase diagram of the Cuprate Superconductors

• P.W. Anderson. Baskaran Zou and Anderson.
  Connection between high Tc and Mott physics.
• ltbgt coherence order parameter.
• K, D singlet formation order paramters.

G. Kotliar and J. Liu Phys.Rev. B 38,5412 (1988)
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• 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

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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
• rTx-Ta x2 , Exp rT x-T a.
• Theory has uniform Z on the Fermi surface, in
  contradiction with ARPES.

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Evolution of the spectral function at low
frequency.
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. Frustration.
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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.

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Spectral Evolution at T0 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 spectral
  weight.

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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)
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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)
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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)

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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).
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Anomalous Resistivity and Mott transition Ni
Se2-x Sx
Crossover from Fermi liquid to bad metal to
semiconductor to paramagnetic insulator.
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k-(ET)2X are across Mott transition
ET
Insulating anion layer
X-1
conducting ET layer
(ET)21
Prof. Kanoda U. Tokyo
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Mott transition in layered organic conductors
S Lefebvre et al. cond-mat/0004455, Phys. Rev.
Lett. 85, 5420 (2000)
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• Theoretical issue is there a Mott transition
• in the integer filled Hubbard model, and is it
• well described by the single site DMFT ?

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Evolution of the spectral function at low
frequency.
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.
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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

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Mechanism for hot spot formation nn self energy
! General phenomena.
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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.

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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.

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Pauling and Coulomb Ti2O3S. Poteryaev S.
Lichtenstein and GK PRL (2004)
Dynamical Goodenough-Honig Pauling picture
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2site-Cluster DMFT with intersite Coulomb
U 2, J 0.5, W 0.5 ß 20 eV-1, LT structure
U 2, J 0.5, W 0.5 ß 10 eV-1, HT structure
A. Poteryaev
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U/t16,t 0.9
Underlying normal state of the Hubbard model near
the Mott transition, (force the Weiss field to
its paramagnetic value), T0 ED solution of the
C-DMFT equations. M. Civelli, M. Capone, O.
Parcollet and GK
U/t8, 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
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U/t16 t-.3 n.95 and t.9 n.95
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Insights into the differences between electron
and hole doped cuprates ?

• t lt0 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.
• tgto has the quasiparticles around (pi,0), does
  not connect smoothly with the SC.

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What did we learn ? Schematic DMFT phase diagram
and DOS of a partially frustrated integer filled
Hubbard model and pressure driven Mott transition.