Correlated Materials: A Dynamical Mean Field Theory (DMFT) Perspective.

1
Correlated Materials A Dynamical Mean Field
Theory (DMFT) Perspective.

• Gabriel Kotliar
• Center for Materials Theory Rutgers University
• CPhT Ecole Polytechnique Palaiseau, and SPhT
  CEA Saclay , France

Toulouse April 18th 2006
Collaborators M. Civelli, K. Haule (Haule), M.
Capone (U. Rome), O. Parcollet(SPhT Saclay), T.
D. Stanescu, (Rutgers) V. Kancharla
(RutgersSherbrooke) A. M Tremblay, D. Senechal
B. Kyung (Sherbrooke)
Support NSF DMR . Blaise Pascal Chair
Fondation de lEcole Normale.
2
Outline

• Strongly Correlated Electrons. Basic
  Dynamical Mean Field Ideas and Cluster
  Extensions.
• High Temperature Superconductivity and
  Proximity to the Mott Transition. Early Ideas.
  Slave Boson Implementation.
• High Temperature Superconductors. What can we
  learn from the study of the doped Mott
  insulator within plaquette Cellular DMFT ?
• CDMFT results for the 2x2 plaquette.
• a) Normal State Photoemission. Civelli et. al.
  PRL (2005) Stanescu and Kotliar cond-mat
• b) Superconducting State Tunnelling Density
  of States. Kancharla et.al. Capone et.al
• c) Optical Conductivity near optimal doping
  and near Tc K. Haule and G. Kotliar

3
High Temperature Superconductors. What can we
learn from the study of the doped Mott
insulator within plaquette Cellular DMFT ?

• We can learn a lot, but there is still a lot of
  work to be done until we reach the same level of
  understanding that we have of the single site
  DMFT solution. This work is definitely in
  progress.
• a) Either that we can account semiquantitatively
  for the large body of experimental data once we
  study more realistic models of the material.
• Or b) we do not, in which case other degrees of
  freedom, or inhomgeneities or long wavelength non
  Gaussian modes are essential as many authors
  have surmised.
• It is still too early to tell, but some evidence
  in favor of a) was presented in this seminar.

Collaborators M. Civelli, K. Haule (Haule), M.
Capone (U. Rome), O. Parcollet(SPhT Saclay), T.
D. Stanescu, (Rutgers) V. Kancharla
(RutgersSherbrooke) A. M Tremblay, D. Senechal
B. Kyung (Sherbrooke)
Support NSF DMR . Blaise Pascal Chair
Fondation de lEcole Normale.
4
Correlated Electron Materials

• Are not well described by either the itinerant
  or the localized framework . Do not fit in the
  Standard Model Solid State Physics. Reference
  System QP. Fermi Liquid Theory and Kohn Sham
  DFTGW
• Compounds with partially filled f and d shells.
• Have consistently produce spectacular big
  effects thru the years. High temperature
  superconductivity, colossal magneto-resistance,
  huge volume collapses..
• Need new starting point for their description.
  Non perturbative problem. DMFT New reference
  frame for thinking about correlated materials
  and computing their physical properties.

5
Breakdown of the Standard Model Large Metallic
Resistivities (Takagi)
6
Transfer of optical spectral weight non local in
frequency Schlesinger et. al. (1994), Van der
Marel (2005) Takagi (2003 ) Neff depends on T
7
DMFT Cavity Construction. A. Georges and G.
Kotliar PRB 45, 6479 (1992). First happy marriage
of atomic and band physics.
Reviews A. Georges G. Kotliar W. Krauth and M.
Rozenberg RMP68 , 13, 1996 Gabriel Kotliar and
Dieter Vollhardt Physics Today 57,(2004)
8
Mean-Field Classical vs Quantum
Classical case
Quantum case
A. Georges, G. Kotliar (1992)
Phys. Rev. B 45, 6497
9
Cluster Extensions of Single Site DMFT
Many Techniques for solving the impurity model
QMC, (Fye-Hirsch), NCA, ED(Krauth Caffarel),
IPT, For a review see Kotliar et. Al to
appear in RMP (2006)
10
For reviews of cluster methods see Georges
et.al. RMP (1996) Maier et.al RMP (2005), Kotliar
et.al cond-mat 0511085. to appear in RMP (2006)
Kyung et.al cond-mat 0511085
Parametrizes the physics in terms of a few
functions .
D , Weiss Field
Alternative (T. Stanescu and G. K. ) periodize
the cumulants rather than the self energies.
11
Testing CDMFT (G.. Kotliar,S. Savrasov, G.
Palsson and G. Biroli, Phys. Rev. Lett. 87,
186401 (2001) ) with two sites in the Hubbard
model in one dimension V. Kancharla C. Bolech and
GK PRB 67, 075110 (2003)M.Capone M.Civelli V
Kancharla C.Castellani and GK PR B 69,195105
(2004)
U/t4.
12
Finite T, DMFT and the Energy Landscape of
Correlated Materials
T
13
Pressure Driven Mott transition
How does the electron go from the localized to
the itinerant limit ?
14
M. Rozenberg et. al. Phys. Rev. Lett. 75, 105
(1995)
T/W
Phase diagram of a Hubbard model with partial
frustration at integer filling. Thinking about
the Mott transition in single site DMFT. High
temperature universality
15
Single site DMFT and kappa organics. Qualitative
phase diagram Coherence incoherence crosover.
16
Three peak structure, predicted Georges and
Kotliar (1992) Transfer of spectral weight near
the Mott transtion. Predicted Zhang Rozenberg and
GK (1993) . ARPES measurements on
NiS2-xSexMatsuura et. Al Phys. Rev B 58 (1998)
3690. Doniaach and Watanabe Phys. Rev. B 57, 3829
(1998) Mo et al., Phys. Rev.Lett. 90, 186403
(2003).
.
17
Conclusions.

• Three peak structure, quasiparticles and Hubbard
  bands.
• Non local transfer of spectral weight.
• Large metallic resistivities.
• The Mott transition is driven by transfer of
  spectral weight from low to high energy as we
  approach the localized phase.
• Coherent and incoherence crossover. Real and
  momentum space.
• Theory and experiments begin to agree on the
  broad picture.

18
Some References

• Reviews A. Georges G. Kotliar W. Krauth and M.
  Rozenberg RMP68 , 13, (1996).
• Reviews G. Kotliar S. Savrasov K. Haule V.
  Oudovenko O. Parcollet and C. Marianetti.
  Submitted to RMP (2006).
• Gabriel Kotliar and Dieter Vollhardt Physics
  Today 57,(2004)

19
Cuprate superconductors and the Hubbard Model .
PW Anderson 1987 . Schematic Phase Diagram (Hole
Doped Case)
20
Methodological Remarks

• Leave out inhomogeneous states and ignore
  disorder.
• What can we understand about the evolution of
  the electronic structure from a minimal model of
  a doped Mott insulator, using Dynamical Mean
  Field Theory ?
• Approach the problem directly from finite
  temperatures,not from zero temperature. Address
  issues of finite frequency temperature
  crossovers. As we increase the temperature DMFT
  becomes more and more accurate.
• DMFT provides a reference frame capable of
  describing coherent and incoherent regimes within
  the same scheme.

21
RVB physics and Cuprate Superconductors

• P.W. Anderson. Connection between high Tc and
  Mott physics. Science 235, 1196 (1987)
• Connection between the anomalous normal state of
  a doped Mott insulator and high Tc. t-J limit.
• Slave boson approach. ltbgt
  coherence order parameter. k, D singlet formation
  order parameters.Baskaran Zhou Anderson ,
  Ruckenstein et.al (1987) .

Other states flux phase or sid ( G. Kotliar
(1988) Affleck and Marston (1988) have point
zeors.
22
RVB phase diagram of the Cuprate Superconductors.
Superexchange.

• The approach to the Mott insulator renormalizes
  the kinetic energy Trvb increases.
• The proximity to the Mott insulator reduce the
  charge stiffness , TBE goes to zero.
• Superconducting dome. Pseudogap evolves
  continously into the superconducting state.

G. Kotliar and J. Liu Phys.Rev. B 38,5412 (1988)
Related approach using wave functionsT. M. Rice
group. Zhang et. al. Supercond Scie Tech 1, 36
(1998, Gross Joynt and Rice (1986) M. Randeria
N. Trivedi , A. Paramenkanti PRL 87, 217002
(2001)
23
Problems with the approach.

• Neel order. How to continue a Neel insulating
  state ? Need to treat properly finite T.
• Temperature dependence of the penetration depth
  Wen and Lee , Ioffe and Millis .
  TheoryrTx-Ta x2 , Exp rT x-T a.
• Mean field is too uniform on the Fermi surface,
  in contradiction with ARPES.
• No quantitative computations in the regime
  where there is a coherent-incoherent
  crossover,compare well with experiments. e.g.
  Ioffe Kotliar 1989

CDMFT may solve some of these problems.!!
24
Photoemission spectra near the antinodal
direction in a Bi2212 underdoped sample.
Campuzano et.al
EDC along different parts of the zone, from Zhou
et.al.
25
M. Rozenberg et. al. Phys. Rev. Lett. 75, 105
(1995)
T/W
Phase diagram of a Hubbard model with partial
frustration at integer filling. Thinking about
the Mott transition in single site DMFT. High
temperature universality
26
.

• Functional of the cluster Greens function. Allows
  the investigation of the normal state underlying
  the superconducting state, by forcing a symmetric
  Weiss function, we can follow the normal state
  near the Mott transition.
• Earlier studies use QMC (Katsnelson and
  Lichtenstein, (1998) M Hettler et. T. Maier
  et. al. (2000) . ) used QMC as an impurity
  solver and DCA as cluster scheme. (Limits U to
  less than 8t )
• Use exact diag ( Krauth Caffarel 1995 ) as a
  solver to reach larger Us and smaller
  Temperature and CDMFT as the mean field
  scheme.
• Recently (K. Haule and GK ) the region near the
  superconducting normal state transition
  temperature near optimal doping was studied
  using NCA DCA .
• DYNAMICAL GENERALIZATION OF SLAVE BOSON ANZATS
• w-S(k,w)m w/b2 -(Db2 t) (cos kx cos ky)/b2
  l
• b--------gt b(k), D -----? D(w), l -----?
  l (k )
• Extends the functional form of the self energy
  to finite T and higher frequency.

CDMFT study of cuprates
27

• Can we continue the superconducting state towards
  the Mott insulating state ?

28
Competition of AF and SC
or
SC
AF
SC
AF
AFSC
d
d
29
Competition of AF and SC M. Capone and GK (2006)
30

• Can we continue the superconducting state towards
  the Mott insulating state ?

For U gt 8t YES. For U lt 8t NO,
magnetism really gets in the way.
31
Superconducting State t0

• Does the Hubbard model superconduct ?
• Is there a superconducting dome ?
• Does the superconductivity scale with J ?
• Is it BCS like ?

32
Superconductivity in the Hubbard model role of
the Mott transition and influence of the
super-exchange. ( work with M. Capone V.
Kancharla. CDMFTED, 4 8 sites t0) .
33
Order Parameter and Superconducting Gap do not
always scale! ED study in the SC state Capone
Civelli Parcollet and GK (2006)
34
How is the Mott insulatorapproached from the
superconducting state ?
Work in collaboration with M. Capone M Civelli O
Parcollet
35
Superconducting DOS
Superconductivity is destroyed by transfer of
spectral weight. M. Capone et. al. Similar to
slave bosons d wave RVB.
36

• In BCS theory the order parameter is tied to the
  superconducting gap. This is seen at U4t, but
  not at large U.
• How is superconductivity destroyed as one
• approaches half filling ?

37
Superconducting State t0

• Does it superconduct ?
• Yes. Unless there is a competing phase.
• Is there a superconducting dome ?
• Yes. Provided U /W is above the Mott transition .
• Does the superconductivity scale with J ?
• Yes. Provided U /W is above the Mott transition .
• Is superconductivity BCS like?
• Yes for small U/W. No for large U, it is RVB
  like!

38

• The superconductivity scales
• with J, as in the RVB approach.
• Qualitative difference between large and small U.
  The superconductivity goes to zero at half
  filling ONLY above the Mott transition.

39
Anomalous Self Energy. (from Capone et.al.)
Notice the remarkable increase with decreasing
doping! True superconducting pairing!! U8t
Significant Difference with Migdal-Eliashberg.
40
Can we connect the superconducting state with the
underlying normal state ? What does the
underlying normal state look like ?
41
Follow the normal state with doping. Civelli
et.al. PRL 95, 106402 (2005) Spectral Function
A(k,??0) -1/p G(k, ? ?0) vs k U16 t, t-.3
K.M. Shen et.al. 2004
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.
2X2 CDMFT
42
Dependence on periodization scheme.
43
Comparison of 2 and 4 sites
44
Spectral shapes. Large Doping Stanescu and GK
cond-matt 0508302
45
Small Doping. T. Stanescu and GK cond-matt 0508302
46
Interpretation in terms of lines of zeros and
lines of poles of G T.D. Stanescu and G.K
cond-matt 0508302
47
Lines of Zeros and Spectral Shapes. Stanescu and
GK cond-matt 0508302
48
Connection between superconducting and normal
state.

• Transfer of spectral weight in optics. Elucidate
  how the spin superexchange energy and the
  kinetic energy of holes changes upon entering the
  superconducting state!
• Mechanism of superconductivity.

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RESTRICTED SUM RULES
Below energy
Low energy sum rule can have T and doping
dependence . For nearest neighbor it gives the
kinetic energy.
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Treatement needs refinement

• The kinetic energy of the Hubbard model contains
  both the kinetic energy of the holes, and the
  superexchange energy of the spins.
• Physically they are very different.
• Experimentally only measures the kinetic energy
  of the holes.

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Validation of the superexchange Mechanism.

• J drives superconductivity.
• Underdoped gains hole kinetic energy
• Overdoped loses kinetic energy. BCS like.

54
Connection between superconducting and normal
state.

• Transfer of spectral weight in optics. Elucidate
  how the spin superexchange energy and the
  kinetic energy of holes changes upon entering the
  superconducting state!
• Origin of the powerlaws discovered in the groups
  of N. Bontemps and D. VarDerMarel.
• K. Haule and GK development of an EDDCANCA
  approach to the problem. New tool for addressing
  the neighborhood of the dome in Tc vs doping.

55
Optical conductivity t-J . K. Haule
56
Behavior of the optical mass and the plasma
frequency.
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Optical conductivity at optimal doping
59
Conclusions

• DMFT is a useful mean field tool to study
  correlated electrons. Provide a zeroth order
  picture of a physical phenomena.
• Provide a link between a simple system (mean
  field reference frame) and the physical system
  of interest. Sites, Links, and Plaquettes
• Formulate the problem in terms of local
  quantities (which we can usually compute better).
• Allows to perform quantitative studies and
  predictions . Focus on the discrepancies between
  experiments and mean field predictions.Substantia
  tes and improves over early slave boson studies
  of the phenomena
• Generate useful language and concepts. Follow
  mean field states as a function of parameters.
• K dependence gets strong as we approach the Mott
  transition. Psedogap. Fermi surfaces and lines
  of zeros of Tsvelik (quasi-one dimensional
  systems ) T. Stanescu and GK (proximity to a
  Mott transition in 2 d).

60
High Temperature Superconductors. What can we
learn from the study of the doped Mott
insulator within plaquette Cellular DMFT ?

• We can learn a lot, but there is still a lot of
  work to be done until we reach the same level of
  understanding that we have of the single site
  DMFT solution. This work is definitely in
  progress.
• a) Either that we can account semiquantitatively
  for the large body of experimental data once we
  study more realistic models of the material.
• Or b) we do not, in which case other degrees of
  freedom, or inhomgeneities or long wavelength non
  Gaussian modes are essential as many authors
  have surmised.
• It is still too early to tell, but some evidence
  in favor of a) was presented in this seminar.

Collaborators M. Civelli, K. Haule (Haule), M.
Capone (U. Rome), O. Parcollet(SPhT Saclay), T.
D. Stanescu, (Rutgers) V. Kancharla
(RutgersSherbrooke) A. M Tremblay, D. Senechal
B. Kyung (Sherbrooke)
Support NSF DMR . Blaise Pascal Chair
Fondation de lEcole Normale.