Transcript Document 7350442

The Mott Transition: a CDMFT
study
G. Kotliar
Physics Department and Center for
Materials Theory
Rutgers
Sherbrook July 2005
References
o Model for kappa organics. [O. Parcollet, G. Biroli
and G. Kotliar PRL, 92, 226402. (2004)) ]
o Model for cuprates [O. Parcollet (Saclay), M.
Capone (U. Rome) M. Civelli (Rutgers) V.
Kancharla (Sherbrooke) GK(2005).
• Cluster Dynamical Mean Field Theories a Strong
Coupling Perspective. T. Stanescu and G. Kotliar
(in preparation 2005)
• Talk by B. Kyung et. al. Tomorrow. condmat/0502565 Short-Range Correlation Induced
Pseudogap in Doped Mott Insulators
Talk by V. Kancharla Sarma (this morning)
Outline
• Motivation and Objectives.Schematic Phase
Diagram(s) of the Mott Transition.
• Finite temperature study of very frustrated
anisotropic model. [O. Parcollet ]
• Low temperature study of the normal state of
the isotropic Hubbard model. [M. Civelli, T.
Stanescu ] [See also B. Kyung’s talk]
• Superconducting state near the Mott transition. [
M. Capone. V. Kancharla Sarma ]
• Conclusions.
RVB phase diagram of the 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.
• Slave boson approach.
<b>
coherence order parameter. k, D singlet formation
order parameters.
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 functions:T. 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)
Problems with the approach.
• Neel order
• Stability of the pseudogap state at finite
temperature. [Ubbens and Lee]
• Missing incoherent spectra . [ fluctuations
of slave bosons ]
• Dynamical Mean Field Methods are ideal
to remove address these difficulties.
COHERENCE INCOHERENCE CROSSOVER
T/W
Phase diagram of a Hubbard model with partial frustration at integer
filling. M. Rozenberg et.al., Phys. Rev. Lett. 75, 105-108 (1995). .
Focus of this work
• Generalize and extend these early mean field
approaches to systems near the Mott transition.
• Obtain the solution of the 2X 2 plaquette and
gain physical understanding of the different
CDMFT states.
• Even if the results are changed by going
to larger clusters, the short range physics is
general and will teach us important lessons.
Follow states as a function of parameters.
Adiabatic continuity. Furthermore the results can
be stabilized by adding further interactions.
Finite T Mott tranisiton in CDMFT
Parcollet Biroli and GK PRL, 92, 226402. (2004))
Evolution of the spectral
function at low frequency.
A(  0, k )vs k
Ek=t(k)+Re( k ,   0)  
 k = Im( k ,   0)
A( k ,   0) 
k
 k 2  Ek 2
If the k dependence of the self energy is
weak, we expect to see contour lines
corresponding to t(k) = const and a
height increasing as we approach the
Fermi surface.
Evolution of the k resolved Spectral
Function at zero frequency. (QMC
study 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. Tmott~.01 W
Momentum Space
Differentiation the high
temperature story
T/W=1/88
Physical Interpretation
• Momentum space differentiation. The Fermi liquid –Bad
Metal, and the Bad Insulator - Mott Insulator regime are
realized in two different regions of momentum space.
• Cluster of impurities can have different characteristic
temperatures. Coherence along the diagonal
incoherence along x and y directions.
• Connection with slave
Boson theory divergence of
Sigma13 . Connections with
RVB (Schmalian and Trivedi)
Cuprate superconductors and the Hubbard Model . PW
Anderson 1987

 (t
i , j  ,
ij
  ij )(c c j  c c )  U  nini
†
i
†
j i
i
CDMFT study of cuprates
.
• 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 (Katsnelson and Lichtenstein, M. Jarrell, M
Hettler et. al. Phys. Rev. B 58, 7475 (1998). T. Maier et. al. Phys. Rev. Lett
85, 1524 (2000) . ) used QMC as an impurity solver and DCA as
cluster scheme.
• We use exact diag ( Krauth Caffarel 1995 with effective
temperature 32/t=124/D ) as a solver and Cellular DMFT as the
mean field scheme. Connect the solution of the 2X2 plaquette
to simpler mean field theories.
Follow the “normal state” with
doping. Evolution of the spectral
function at low frequency.
A(  0, k )vs k
Ek=t(k)+Re( k ,   0)  
 k = Im( k ,   0)
A( k ,   0) 
k
 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.
Hole doped case t’=-.3t, U=16 t
n=.71 .93 .97
K.M . Shen et. al. Science (2005).
For a review Damascelli et. al. RMP (2003)
Approaching 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) and
Senechal and tremblay for cuprates with VCPT.
Results of many numerical studies of electron hole asymmetry in t-t’
Hubbard models Tohyama Maekawa Phys. Rev. B 67, 092509 (2003)
Senechal and Tremblay. PRL 92 126401 (2004) Kusko et. al. Phys. Rev 66,
Experiments. Armitage et. al. PRL (2001).
Momentum dependence of the low-energy
Photoemission spectra of NCCO
Approaching 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!
• General phenomena, BUT the location of the cold
regions depends on parameters.
• Quasiparticles are now generated from the Mott insulator
at (p, 0).
• Results of many l studies of electron hole asymmetry in
t-t’ Hubbard models Tohyama Maekawa Phys. Rev. B 67,
092509 (2003) Senechal and Tremblay. PRL 92 126401 (2004)
Kusko et. al. Phys. Rev 66, 140513 (2002). Kusunose and Rice PRL
91, 186407 (2003).
Comparison with Experiments in Cuprates:
Spectral Function A(k,ω→0)= -1/π G(k, ω →0) vs k
hole doped
K.M. Shen et.al. 2004
electron doped
P. Armitage et.al. 2001
To 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, U=16.
Introduce much larger frustration:
t’=.9t U=16t
n=.69 .92 .96
Approaching 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.
Understanding the result in terms of
cluster self energies (eigenvalues)
A
(0,p )
 B ~ (p ,p )
A
(0,0)
Cluster Eigenvalues
Evolution of the real part of the self
energies.
Fermi Surface Shape
Renormalization ( teff)ij=tij+ Re(ij(0))
Fermi 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.
Understanding the location of the
hot and cold regions. Interplay of
lifetime and fermi surface.
Comparison of periodization
methods for A(=0,k)

M

M

M
Pseudogap: insights from cumulant
periodization.
• Qualitative Difference between the
periodization methods. The cumulant
periodization, is a non linear interpolation
of the self energies (linear interpolation of
the cumulants). When the off diagonal
elements of the self energy get large, it
gives rise to lines of poles of the self
energy, in addition to the Fermi lines.
• Quasi-one d, Essler and Tsvelik.
• Subtle topological phase transition at intermediate
doping ?
• Is there a quantum critical point related to the change of
topology of the Fermi surface ?
• Is there a quantum critical point associated to the
emergence of lines of zeros.
• This is NOT a Fermi surface instability, invisible to weak
coupling analysis.
• We checked with PCMDFT that the results survive,
unlike the corresponding quasi-1d case (Arrigoni et. al.
Giamarchi Georges and Biermann).
Large Doping
Small Doping
How is the Mott insulator
approached from the
superconducting state ?
Work in collaboration with M. Capone, see also V.
Kancharla’s talk.
Superconductivity in the Hubbard model role of
the Mott transition and influence of the superexchange. (M. Capone, V. Kancharla.
CDMFT+ED, 4+ 8 sites t’=0) .
• 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.
Order Parameter and
Superconducting Gap .
• In BCS theory the order parameter is tied
to the superconducting gap. This is seen
at U=4t, but not at large U.
• How is superconductivity destroyed as one
approaches half filling ?
Evolution of the low energy tunneling density of
state with doping. Decrease of spectral weight
as the insulator is approached. Low energy
particle hole symmetry.
• Superconductivity is destroyed at half
filling due to a reduction of the one
electron weight. Just like in the slave
boson.
• High energy ph asymmetry.
• Low energy ph symmetry.
Alternative view
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.
• Generate useful language and concepts. Follow mean
field states as a function of parameters.
• K dependence gets strong as we approach the Mott
transition. Fermi surfaces and lines of zeros of G.
Conclusions
• 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. Study the
“normal state” of the Hubbard model is useful.
• Character of the superconductivity is different for
small and large U.
the Hubbard model does not capture the trend of supra
with t’. Need augmentation.(Venky Kancharla
Sarma).Same trend observed in DCA Maier and Jarrell.
D wave Superconductivity and
Antiferromagnetism t’=0 M. Capone
V. Kancharla (see also VCPT Senechal and
Tremblay ).
Antiferromagnetic (left) and d wave superconductor (right) Order Parameters
Estimates of upper bound for Tc
exact diag. M. Capone. U=16t, t’=0, (
t~.35 ev, Tc ~140 K~.005W)
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, M Hettler
et. al. Phys. Rev. B 58, 7475 (1998)Katsnelson and Lichtenstein
periodized scheme, Nested Cluster Schemes , causality
issues, O. Parcollet, G. Biroli and GK Phys. Rev. B 69, 205108 (2004)
Understanding in terms of cluster
self-energies. Civelli et. al.
k-(ET)2X are across Mott
transition
ET =
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
Electron doped case t’=.9t U=16t
n=.69 .92 .96
Color scale x=.9,.32,.22
Two paths for 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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INTUITIVE NOTIONS OF DMFT AND WEISS FIELD CAVITY CONSTRUCTION.
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EFFECTIVE ACTION CONSTRUCTION.
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Examples.
Outline
Mapping of lattice onto a cluster in a medium. With
a prescription for building the medium from the computation of
the cluster quantities.
Prescription for reconstructing lattice quantities.
Weiss field describe the medium.
Cavity Construction is highly desireable. Delta is non zero
on the boundary.
------------------------------------------------------------------------------------------------------------
Show how it is done. < chitra >
It is not perturbative.
It is general.
It includes everything you want to know.
It gives you a reference system.
APPROXIMATE WEISS FIELD.
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SUBTLETIES WITH THE WEISS FIELD, WHAT ABOUT GETTING
NOT ONLY Gii but also Gij, and admiting Delta_ij,
is the unique solution t_ij ?
-----------------------------------------------------------------------------------------------------------------------------------------How to go from lattice to impurity model.
CDMFT construction.
Picture giving simga_c, M_cluster chi_cluster Gamma_cluster
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a) Baym Kadanoff Functional
b) Self energy functional.
c) DFT.
Want to generate good approximations.
and their hybrids.
----------------------------------------------------------------------------------------------WEIS FIELD.
I NEED TO GET THIS RIGHT HOW I DO THE SEPARATION,
INTO PIECES WHAT IS THE EXACT AND WHA IS TEH
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Nested Cluster Schemes.
Explicit Cavity constructions.
-------------------------------------------------------------------------------------------------CAUSALITY PROBLEMS.
Parcollet Biorli KOtliar clausing.
---------------------------------------------------------------------------------CONVERGENCE. 1/L vs e-L.
DCA converges as 1/L2 < put reference >
Classical limits. See what olivier ahs done
on w-cdmft.
--------------------------------------------------------------------------Impurity Solvers.
OCA-QMC-ED.
Lessons form the past. Combinations
of methods.
----------------------------------------------------------------------------ED discretization
Role of distance.
Role of functionals.
Why not variational is usesless.
Few sites.
H  H cluster  H cluster exterior  H exterior
H
H cluster  H cluster exterior  H exterior
Simpler "medium" Hamiltonian
Medium of free
electrons :
impurity model.
Solve for the
medium using
Self Consistency
G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87,
186401 (2001)
Mott transition in layered organic conductors
cond-mat/0004455, Phys. Rev. Lett. 85, 5420 (2000)
S Lefebvre et al.
k-(ET)2X are across Mott
transition
ET =
Insulating
anion layer
X-1
conducting
ET layer
modeled to triangular lattice
]+1
[(ET)2
t’
modeled to triangular lattice
t’
t
t
References
CDMFT one electron spectra
n=.96 t’/t=.-.3 U=16 t
• i
Other cluster extensions (DCA Jarrell Krishnamurthy, M
Hettler et. al. Phys. Rev. B 58, 7475 (1998)Katsnelson and
Lichtenstein periodized scheme. Causality issues O. Parcollet,
G. Biroli and GK Phys. Rev. B 69, 205108 (2004)
H  H cluster  H cluster exterior  H exterior
H
H cluster 
Medium of free
electrons :
impurity model.
Solve for the
medium using
Self Consistency
G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87,
186401 (2001)
Difference in Periodizations
Advantages of the Weiss field
functional. Simpler analytic
structure near the Mott transition.
How to determine the parameters
of the bath ?
•Extremize Potthoff’s 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.
Convergence of Cluster Schemes
as a function of cluster size.
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Aryamanpour et. al. DCA observables converge as 1/L^2.
cond-mat/0205460 [abs, ps, pdf, other] :
–
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Title: Two Quantum Cluster Approximations
Authors: Th. A. Maier (1), O. Gonzalez (1 and 2), M. Jarrell (1), Th. Schulthess (2) ((1)
University of Cincinnati, Cincinnati, USA, (2) Oak Ridge National Laboratory, Oak Ridge,
USA)
Aryamampour et. al. The Weiss field in CDMFT converges as 1/L. Title: The
Dynamical Cluster Approximation (DCA) versus the Cellular Dynamical
Mean Field Theory (CDMFT) in strongly correlated electrons systems
Authors: K. Aryanpour, Th. A. Maier, M. Jarrell
Comments: Comment on Phys. Rev. B 65, 155112 (2002). 3 pages, 2
figures
Subj-class: Strongly Correlated Electrons
Journal-ref: Phys. Rev. B 71, 037101 (2005)
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Biroli and Kotliar. Phys. Rev. B 71, 037102 (2005);cond-mat/0404537. Local
observables (i.e. observables contained in the cluster ) converge EXPONENTIALLY
at finite temperatures away from critical points.
Hole doped case t’=-.3t, U=16 t
n=.71 .93 .97
Color scale x= .37 .15 .13
o 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.
o 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).
ED and QMC
Systematic Evolution