Title: High Temperature Superconductors. What can we learn from the study of the doped Mott insulator within plaquette Cellular DMFT.
1High Temperature Superconductors. What can we
learn from the study of the doped Mott
insulator within plaquette Cellular DMFT.
- Gabriel Kotliar
- Center for Materials Theory Rutgers University
- CPhT Ecole Polytechnique Palaiseau, and SPhT
CEA Saclay , France
Geneve February 10th 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.
2Outline
- 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. - 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
3Correlated 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.
4 Breakdown of the Standard Model Large Metallic
Resistivities (Takagi)
5Transfer of optical spectral weight non local in
frequency Schlesinger et. al. (1994), Van der
Marel (2005) Takagi (2003 ) Neff depends on T
6DMFT 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)
7Mean-Field Classical vs Quantum
Classical case
Quantum case
A. Georges, G. Kotliar (1992)
Phys. Rev. B 45, 6497
8Cluster 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)
9 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.
10Testing 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.
11Effective Action point of view.
- Identify observable, A. Construct a free
energy functional of ltAgta, G a which is
stationary at the physical value of a. - Example, density in DFT theory. (Fukuda et. al.).
- DMFT Local Spectral Function. (R. Chitra and G.K
(2000) (2001). - HH0l H1. G a,J0F0J0 a J0 _ Ghxc
a - Functional of two variables, a ,J0.
- H0 A J0 Reference system to think about H.
- J0 a Is the functional of a with the
property ltAgt0 a lt gt0 computed with H0
A J0 - Many choices for H0 and for A
- Extremize a to get G J0 exta G a,J0
12 Finite T, DMFT and the Energy Landscape of
Correlated Materials
T
13Pressure Driven Mott transition
How does the electron go from the localized to
the itinerant limit ?
14M. 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
15Single site DMFT and kappa organics. Qualitative
phase diagram Coherence incoherence crosover.
16Ising critical endpoint prediction Kotliar Lange
Rozenberg Phys. Rev. Lett. 84, 5180
(2000)Observed! In V2O3 P. Limelette et.al.
Science 302, 89 (2003)
17Three 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).
.
18Conclusions.
- 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.
19Some 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)
20Cuprate superconductors and the Hubbard Model .
PW Anderson 1987 . Schematic Phase Diagram (Hole
Doped Case)
21Methodological 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.
22RVB 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.
23RVB 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)
24Problems 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.!!
25Photoemission spectra near the antinodal
direction in a Bi2212 underdoped sample.
Campuzano et.al
EDC along different parts of the zone, from Zhou
et.al.
26M. 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
27.
- 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
28- Can we continue the superconducting state towards
the Mott insulating state ?
29Competition of AF and SC
or
SC
AF
SC
AF
AFSC
d
d
30Competition of AF and SC M. Capone M. Civelli and
GK (2006)
31- 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.
32Superconducting State t0
- Does the Hubbard model superconduct ?
- Is there a superconducting dome ?
- Does the superconductivity scale with J ?
- Is it BCS like ?
33Superconductivity 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) .
34Order Parameter and Superconducting Gap do not
always scale! ED study in the SC state Capone
Civelli Parcollet and GK (2006)
35How is the Mott insulatorapproached from the
superconducting state ?
Work in collaboration with M. Capone.
36Evolution of DOS with doping U12t. Capone et.al.
Superconductivity is driven by transfer of
spectral weight , slave boson b2 !
37Superconductivity is destroyed by transfer of
spectral weight. M. Capone et. al. Similar to
slave bosons d wave RVB.
38- 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 ?
39Superconducting 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!
40- 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. -
41Anomalous Self Energy. (from Capone et.al.)
Notice the remarkable increase with decreasing
doping! True superconducting pairing!! U8t
Significant Difference with Migdal-Eliashberg.
42Can we connect the superconducting state with the
underlying normal state ? What does the
underlying normal state look like ?
43Follow 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
44Dependence on periodization scheme.
45Comparison of 2 and 4 sites
46Spectral shapes. Large Doping Stanescu and GK
cond-matt 0508302
47Small Doping. T. Stanescu and GK cond-matt 0508302
48Interpretation in terms of lines of zeros and
lines of poles of G T.D. Stanescu and G.K
cond-matt 0508302
49Lines of Zeros and Spectral Shapes. Stanescu and
GK cond-matt 0508302
50Connection 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.
51Optical Conductivity near optimal doping.
Theory DCA EDNCA study, K. Haule and GK
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53Kristjan Haule there is an avoided quantum
critical point near optimal doping.
54What is happening near optimal doping ?? Avoided
Quantum Criticality K. Haule and GK
55Optical conductivity t-J . K. Haule
56Behavior of the optical mass and the plasma
frequency.
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58RESTRICTED 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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60Treatement 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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62Conclusions
- 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).
63 Conclusions Superconductivity
- Reproduced the basic general features of the
early slave boson treatment. - The approach naturally introduces a strong
anisotropy and particle hole asymmetry in the
problem. - It reveals the frequency dependence of the self
energy which is growing with doping J and U - Establishes the clear differences between
superconductivity above and below the Mott
transition.
64Conclusions
- Qualitative effect, momentum space
differentiation. Formation of hot cold regions
is an unavoidable consequence of the approach to
the Mott insulating state! - Truncation of the Fermi surface as a STRONG
COUPLING instability (compare weak coupling RG
e.g. Honerkamp, Metzner, Rice ) - General phenomena, but the location of the cold
regions depends on parameters. - Fundamental difference between electron and hole
doped cuprates.
65- Qualitative Difference between the hole doped and
the electron doped phase diagram is due to the
underlying normal state. In the hole doped, it
has nodal quasiparticles near (p/2,p/2) which
are ready to become the superconducting
quasiparticles. Therefore the superconducing
state can evolve continuously to the normal
state. The superconductivity can appear at very
small doping. - Electron doped case, has in the underlying normal
state quasiparticles leave in the (p, 0) region,
there is no direct road to the superconducting
state (or at least the road is tortuous) since
the latter has QP at (p/2, p/2).
66Approaching the Mott transition from high T
CDMFT Picture
- Fermi Surface Breakup. Qualitative effect,
momentum space differentiation. Formation of hot
cold regions is an unavoidable consequence of
the approach to the Mott insulating state! It can
be seen starting from high temperatures. - D wave gapping of the single particle spectra as
the Mott transition is approached. Real and
Imaginary part of the self energies grow
approaching half filling. Unlike weak coupling! - Scenario was first encountered in previous study
of the kappa organics. O Parcollet G. Biroli and
G. Kotliar PRL, 92, 226402. (2004) .
67High 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.
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70What is the origin of the asymmetry ? Comparison
with normal state near Tc. K. Haule
Early slave boson work, predicted the asymmetry,
and some features of the spectra. Notice that
the superconducting gap is smaller than
pseudogap!!
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74Magnetic Susceptibility
75Outline
- Theoretical Point of View, and Methodological
Developments. - Local vs Global observables.
- Reference Frames. Functionals. Adiabatic
Continuity. - The basic RVB pictures.
- CDMFT as a numerical method, or as a boundary
condition.Tests. - The superconducting state.
- The underdoped region.
- The optimally doped region.
- Materials Design. Chemical Trends. Space of
Materials.
76Connection with large N studies.
77References
- Dynamical Mean Field Theory and a cluster
extension, CDMFT G.. Kotliar,S. Savrasov, G.
Palsson and G. Biroli, Phys. Rev. Lett. 87,
186401 (2001) - Cluser Dynamical Mean Field Theories Causality
and Classical Limit. - G. Biroli O. Parcollet G.Kotliar Phys. Rev. B 69
205908 - Cluster Dynamical Mean Field Theories a Strong
Coupling Perspective. T. Stanescu and G. Kotliar
( 2005)
78Evolution of the normal state Questions.
- Origin of electron hole asymmetry in electron and
doped cuprates. - Detection of lines of zeros and the Luttinger
theorem.
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83ED and QMC
84(No Transcript)
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86Testing 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.
87Electron Hole Asymmetry Puzzle
88What about the electron doped semiconductors ?
89 Spectral Function A(k,??0) -1/p G(k, ? ?0) vs
k
electron doped
P. Armitage et.al. 2001
Momentum space differentiation a we approach the
Mott transition is a generic phenomena.
Location of cold and hot regions depend on
parameters.
Civelli et.al. 2004
90Approaching the Mott transition CDMFT Picture
- 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. - Similar scenario was encountered in previous
study of the kappa organics. O Parcollet G.
Biroli and G. Kotliar PRL, 92, 226402. (2004) .
91High 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. - Either that we can account semiquantitatively
for the large body of experimental data once we
study more realistic models of the material. - Or 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.
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.
92Conclusion
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94OPTICS
95 CDMFT and NCS as truncations of the Baym
Kadanoff functional
96Ex Baym Kadanoff functional,a G, H0 free
electrons.
Viewing it as a functional of J0, Self Energy
functional(Potthoff)
97Weiss Field Functional
98Example single site DMFT semicircular density of
states. GKotliar EPJB (1999)
Extremize Potthoffs self energy functional. It
is hard to find saddles using conjugate
gradients. Extremize the Weiss field
functional.Analytic for saddle point equations
are available Minimize some distance
99Approaching the Mott transition CDMFT Picture
- Fermi Surface Breakup. 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. - Similar scenario was encountered in previous
study of the kappa organics. O Parcollet G.
Biroli and G. Kotliar PRL, 92, 226402. (2004) .
100Dynamical RVB brings in strong anistropy in the
underdoped regime.
101What about the electron doped semiconductors ?
102 Spectral Function A(k,??0) -1/p G(k, ? ?0) vs
k
electron doped
P. Armitage et.al. 2001
Momentum space differentiation a we approach the
Mott transition is a generic phenomena.
Location of cold and hot regions depend on
parameters.
Civelli et.al. 2004
103- Qualitative Difference between the hole doped and
the electron doped phase diagram is due to the
underlying normal state. In the hole doped, it
has nodal quasiparticles near (p/2,p/2) which
are ready to become the superconducting
quasiparticles. Therefore the superconducing
state can evolve continuously to the normal
state. The superconductivity can appear at very
small doping. - Electron doped case, has in the underlying normal
state quasiparticles leave in the (p, 0) region,
there is no direct road to the superconducting
state (or at least the road is tortuous) since
the latter has QP at (p/2, p/2).
104- Can we connect the superconducting state with
the underlying normal state ? - Yes, within our resolution in the hole doped
case. - No in the electron doped case.
- What does the underlying normal state look
like ? - Unusual distribution of spectra (Fermi arcs) in
the normal state.
105To test if the formation of the hot and cold
regions is the result of the proximity to
Antiferromagnetism, we studied various values of
t/t, U16.
106Introduce much larger frustration t.9t
U16tn.69 .92 .96
107Approaching the Mott transition
- Qualitative effect, momentum space
differentiation. Formation of hot cold regions
is an unavoidable consequence of the approach to
the Mott insulating state! - General phenomena, but the location of the cold
regions depends on parameters. - With the present resolution, t .9 and .3 are
similar. However it is perfectly possible that
at lower energies further refinements and
differentiation will result from the proximity to
different ordered states.
108Fermi Surface Shape Renormalization (
teff)ijtij Re(Sij(0))
109Fermi Surface Shape Renormalization
- Photoemission measured the low energy
renormalized Fermi surface. - If the high energy (bare ) parameters are doping
independent, then the low energy hopping
parameters are doping dependent. Another failure
of the rigid band picture. - Electron doped case, the Fermi surface
renormalizes TOWARDS nesting, the hole doped case
the Fermi surface renormalizes AWAY from nesting.
Enhanced magnetism in the electron doped side.
110Understanding the location of the hot and cold
regions. Interplay of lifetime and fermi surface.
111 Superconductivity as the cure for a sick
normal state.