Title: THE STATE UNIVERSITY OF NEW JERSEY
1Outline
- Model Hamiltonians and qualitative
considerations in the physics of materials. Or
what do we want to know? An example from the
physics of the Mott transition. - Merging band structure methods with many body
theory, where to improve? A) basis set? B)
parameter estimates of your model Hamiltonian C)
DMFT impurity solver? D) Improvements of DMFT ?
An intro to Cellular DMFT G. Kotliar S.
Savrasov G. Palsson and G. Biroli PRL87, 186401
2001
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3 Extensions of DMFT
- Spin Orbital Ordered States
- Longer range interactions Coulomb, interactions,
Random Exchange (Sachdev and Ye, Parcollet and
Georges, Kajueter and Kotliar, Si and Smith,
Chitra and Kotliar,) - Short range magnetic correlations. Cluster
Schemes. (Ingersent and Schiller, Georges and
Kotliar, cluster expansion in real space,
momentum space cluster DCA Jarrell et.al., C-DMFT
Kotliar et. al ).
4Extensions of DMFT
- Formulation as an electronic structure method
(Chitra and Kotliar) - Density vs Local Spectral Function
- Extensions to treat strong spatial
inhomogeneities. Anderson Localization
(Dobrosavlevic and Kotliar),Surfaces
(Nolting),Stripes (Fleck Lichtenstein and Oles) - Practical Implementation (Anisimov and Kotliar,
Savrasov, Katsenelson and Lichtenstein)
5Cuprates Photoemission
Transfer of Spectral Weight with a) temperature
and b) doping
6Challenges
- The photoemission in cuprates has a strong
momentum dependence - Strong Magnetic Correlations (no orbital
degeneracy) - Single Site DMFT does not capture these effects
7Cuprates Photoemission
Transfer of Spectral Weight with a) temperature
and b) doping
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9 Mott transition in the actinide series. B.
Johanssen 1974 Smith and Kmetko Phase Diagram
1984.
10Schematic DMFT phase diagram one band Hubbard
model (half filling, semicircular DOS, partial
frustration) Rozenberg et.al PRL (1995)
11Robustness of the finite T results
- Underlying Landau Free energy which is
responsible of all the qualitative features of
the phase diagram. Of the frustrated Hubbard
model in large d G. Kotliar EPJB 99 - Around the finite temperature Mott endpoint, the
Free energy has a simple Ising like form as in a
liquid gas transition R. Chitra, G. Kotliar
E.Lange M. Rozenberg - Changing the model (DOS, degeneracy, etc) just
changes the coefficients of the Landau theory.
12Robustness of the finite T results and Functional
Approach
- Different impurity solvers, different values of
the Landau coefficients, as long as they preserve
the essential (non) analytic properties of the
free energy functional. - The functional approach can be generalized to
combine DFT and DMFT R. Chitra G. Kotliar , S.
Savrasov and G. Kotliar - Justification for applying simple models to some
aspects of the crossover in Ni(SeS)2And V2O3.
13Qualitative phase diagram in the U, T , m plane
(two band Kotliar and Rozenberg (2001) cond-matt
0110625)
- Coexistence regions between localized and
delocalized spectral functions.
14QMC calculationof n vs m (Murthy Rozenberg and
Kotliar 2001, 2 band, U3.0, cond-matt 0110625)
k diverges at generic Mott endpoints
15Compressibilty divergence One band case
(Kotliar Murthy and Rozenberg 2001, cond-matt
0110625)
16Minimum in melting curve and divergence of the
compressibility at the Mott endpoint
17Minimum in melting curve and divergence of the
compressibility at the Mott endpoint
18A (non comprehensive )list of extensions of DMFT
- Two impurity method. A. Georges and G. Kotliar,
A. Schiller PRL75, 113 (1995) - M. Jarrell Dynamical Cluster Approximation Phys.
Rev. B 7475 1998 - Continuous version periodic cluster M.
Katsenelson and A. Lichtenstein PRB 62, 9283
(2000). - Extended DMFT H. Kajueter and G. Kotliar
- Rutgers Ph.D thesis 2001, Q. Si and J L Smith PRL
77 (1996)3391 Coulomb interactions R . Chitra - Cellular DMFT PRL87, 186401 2001
19DMFT cavity construction
Weiss field
20Elements of the Dynamical Mean Field Construction
and Cellular DMFT, G. Kotliar S. Savrasov G.
Palsson and G. Biroli PRL 2001
- Definition of the local degrees of freedom
- Expression of the Weiss field in terms of the
local variables (I.e. the self consistency
condition) - Expression of the lattice self energy in terms of
the cluster self energy.
21Cellular DMFT Basis selection
22Lattice action
23Elimination of the medium variables
24Determination of the effective medium.
25Connection between cluster and lattice self
energy.
The estimation of the lattice self energy in
terms of the cluster energy has to be done using
additional information Ex. Translation invariance
- C-DMFT is manifestly causal causal impurity
solvers result in causal self energies and Green
functions (GK S. Savrasov G. Palsson and G.
Biroli PRL 2001) - In simple cases C-DMFT converges faster than
other causal cluster schemes.
26Improved estimators
- Improved estimators for the lattice self energy
are available (Biroli and Kotliar)
27Real Space Formulation of the DCA approximation
of Jarrell et.al.
28Affleck Marston model.
29Convergence test in the Affleck Marston
30Convergence of the self energy
31Recent application to high Tc
- A. Perali et.al. cond-mat 2001, two patch model,
phenomenological fit of the functional form of
the vertex function of C-DMFT to experiments in
optimally doped and overdoped cuprates - Flexibility in the choice of basis seems
important.
32Extended DMFT electron phonon
33Extended DMFT e.ph. Problem
34E-DMFT classical case, soft spins
35E-DMFT classical case Ising limit
36E-DMFT test in the classical caseBethe Lattice,
S. Pankov 2001
37Advantage and Difficulties of E-DMFT
- The transition is first order at finite
temperatures for dlt 4 - No finite temperature transition for d less than
2 (like spherical approximation) - Improved values of the critical temperature
38Conclusion
- For first principles work there are several
many body tools waiting to be used, once the one
electron aspects of the problem are clarified. - E-DMFT or C-DMFT for Ni, and Fe ?
- Promising problem Qualitative aspects of the
Mott transition within C-DMFT ?? Cuprates?
39Realistic Theories of Correlated Materials
- ITP, Santa-Barbara
- July 20 December 20 (2002)
- O.K. Andesen, A. Georges,
- G. Kotliar, and A. Lichtenstein
- http//www.itp.ucsb.edu/activities/future/
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41 Functional Approach
G. Kotliar EPJB (1999)
42Recent phase diagram of the frustrated Half
filled Hubbard model with semicircular DOS (QMC
Joo and Udovenko PRB2001).
43Case study IPT half filled Hubbard one band
- (Uc1)exact 2.1 (Exact diag, Rozenberg,
Kajueter, Kotliar 1995) , (Uc1)IPT 2.4 - (Uc2)exact 2.95 (Projective self consistent
method, Moeller Si Rozenberg Kotliar PRL 1995 )
(Uc2)IPT 3.3 - (TMIT ) exact .026_ .004 (QMC Rozenberg Chitra
and Kotliar PRL 1999), (TMIT )IPT .5 - (UMIT )exact 2.38 - .03 (QMC Rozenberg Chitra
and Kotliar PRL 1991), (UMIT )IPT 2.5 For
realistic studies errors due to other sources
(for example the value of U, are at least of the
same order of magnitude).
44The Mott transition as a bifurcation in effective
action
Zero mode with S0 and p0, couples generically
Divergent compressibility (R. Chitra and
G.Kotliar
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46Realistic implementation of the self consistency
condition
- H and S, do not commute
- Need to do k sum for each frequency
- DMFT implementation of Lambin Vigneron
tetrahedron integration (Poteryaev et.al 1987)
47Solving the impurity
- Multiorbital situation and several atoms per unit
cell considerably increase the size of the space
H (of heavy electrons). - QMC scales as N(N-1)/23 N dimension of H
- Fast interpolation schemes (Slave Boson at low
frequency, Roth method at high frequency, 1st
mode coupling correction), match at intermediate
frequencies. (Savrasov et.al 2001)
48Good method to study the Mott phenomena
- Evolution of the electronic structure between
the atomic limit and the band limit. Basic solid
state problem. Solved by band theory when the
atoms have a closed shell. Motts problem Open
shell situation. - The in between regime is ubiquitous central
them in strongly correlated systems. Some
unorthodox examples - Fe, Ni, Pu .
-
49Two Roads for calculations of the electronic
structure of correlated materials
Crystal Structure atomic positions
Model Hamiltonian
Correlation functions Total energies etc.
50LDA functional
Conjugate field, VKS(r)
51Minimize LDA functional
52LDAU functional
53LDADMFT
- The light, SP (or SPD) electrons are extended,
well described by LDA - The heavy, D (or F) electrons are localized,treat
by DMFT. - LDA already contains an average interaction of
the heavy electrons, substract this out by
shifting the heavy level (double counting term) - The U matrix can be estimated from first
principles of viewed as parameters
54Spectral Density Functional effective action
construction (Fukuda, Valiev and Fernando ,
Chitra and GK).
- DFT, consider the exact free energy as a
functional of an external potential. Express the
free energy as a functional of the density by
Legendre transformation. GDFTr(r) - Introduce local orbitals, caR(r-R)orbitals, and
local GF - G(R,R)(i w)
- The exact free energy can be expressed as a
functional of the local Greens function and of
the density by introducing sources for r(r) and G
and performing a Legendre transformation,
Gr(r),G(R,R)(iw)
55Spectral Density Functional
- The exact functional can be built in perturbation
theory in the interaction (well defined
diagrammatic rules )The functional can also be
constructed from the atomic limit, but no
explicit expression exists. - DFT is useful because good approximations to the
exact density functional GDFTr(r) exist, e.g.
LDA, GGA - A useful approximation to the exact functional
can be constructed, the DMFT LDA functional.
56LDADMFT functional
F Sum of local 2PI graphs with local U matrix and
local G
57Comments on LDADMFT
- Static limit of the LDADMFT functional , with F
FHF reduces to LDAU - Removes inconsistencies of this approach,
- Only in the orbitally ordered Hartree Fock limit,
the Greens function of the heavy electrons is
fully coherent - Gives the local spectra and the total energy
simultaneously, treating QP and H bands on the
same footing.
58LDADMFTConnection with atomic limit
Weiss field
59LDADMFT Self-Consistency loop
E
U
DMFT
60Realistic DMFT loop
61LDADMFT References
- V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin
and G. Kotliar, J. Phys. Cond. Mat. 35,
7359-7367 (1997). - A Lichtenstein and M. Katsenelson Phys. Rev. B
57, 6884 (1988). - S. Savrasov and G.Kotliar, funcional
formulation for full self consistent
implementation (2001)
62Functional Approach
- The functional approach offers a direct
connection to the atomic energies. One is free to
add terms which vanish quadratically at the
saddle point. - Allows us to study states away from the saddle
points, - All the qualitative features of the phase
diagram, are simple consequences of the non
analytic nature of the functional. - Mott transitions and bifurcations of the
functional .
63 Functional Approach
G. Kotliar EPJB (1999)
64 Case study in f electrons, Mott transition in
the actinide series
65Pu Anomalous thermal expansion (J. Smith LANL)
66Small amounts of Ga stabilize the d phase
67Delocalization-Localization across the actinide
series
- f electrons in Th Pr U Np are itinerant . From
Am on they are localized. Pu is at the
boundary. - Pu has a simple cubic fcc structure,the d phase
which is easily stabilized over a wide region in
the T,p phase diagram. - The d phase is non magnetic.
- Many LDA , GGA studies ( Soderlind et. Al 1990,
Kollar et.al 1997, Boettger et.al 1998, Wills
et.al. 1999) give an equilibrium volume of the d
phase Is 35 lower than experiment - This is one of the largest discrepancy ever known
in DFT based calculations.
68Problems with LDA
- DFT in the LDA or GGA is a well established tool
for the calculation of ground state properties. - Many studies (Freeman, Koelling 1972)APW methods
- ASA and FP-LMTO Soderlind et. Al 1990, Kollar
et.al 1997, Boettger et.al 1998, Wills et.al.
1999) give - an equilibrium volume of the d phase Is 35
lower than experiment - This is the largest discrepancy ever known in DFT
based calculations.
69Problems with LDA
- LSDA predicts magnetic long range order which is
not observed experimentally (Solovyev et.al.) - If one treats the f electrons as part of the core
LDA overestimates the volume by 30 - LDA predicts correctly the volume of the a phase
of Pu, when full potential LMTO (Soderlind and
Wills). This is usually taken as an indication
that a Pu is a weakly correlated system
70Conventional viewpoint
- Alpha Pu is a simple metal, it can be described
with LDA correction. In contrast delta Pu is
strongly correlated. - Constrained LDA approach (Erickson, Wills,
Balatzki, Becker). In Alpha Pu, all the 5f
electrons are treated as band like, while in
Delta Pu, 4 5f electrons are band-like while one
5f electron is deloclized. - Same situation in LDA U (Savrasov and Kotliar,
Bouchet et. Al. ) Delta Pu has U4, - Alpha Pu has U 0.
71Problems with the conventional viewpoint of Pu
- The specific heat of delta Pu, is only twice as
big as that of alpha Pu. - The susceptibility of alpha Pu is in fact larger
than that of delta Pu. - The resistivity of alpha Pu is comparable to that
of delta Pu. - Only the structural and elastic properties are
completely different.
72Pu Specific Heat
73Anomalous ResistivityJ. Smith LANL
74MAGNETIC SUSCEPTIBILITY
75Dynamical Mean Field View of Pu(Savrasov Kotliar
and Abrahams, Nature 2001)
- Delta and Alpha Pu are both strongly correlated,
the DMFT mean field free energy has a double
well structure, for the same value of U. One
where the f electron is a bit more localized
(delta) than in the other (alpha). - Is the natural consequence of the model
hamiltonian phase diagram once electronic
structure is about to vary. - This result resolves one of the basic paradoxes
in the physics of Pu.
76Pu DMFT total energy vs Volume
77Lda vs Exp Spectra
78Pu Spectra DMFT(Savrasov) EXP (Arko et. Al)
79Earlier Studies of Magnetic Anisotropy
80Case study Fe and Ni
- Archetypical itinerant ferromagnets
- LSDA predicts correct low T moment
- Band picture holds at low T
- Main challenge, finite T properties
(Lichtensteins talk). - Magnetic anisotropy puzzle. LDA predicts the
incorrect easy axis for Nickel . - LDA Fermi surface has features which are not seen
in DeHaas Van Alphen ( Lonzarich)
81Iron and Nickel crossover to a real space
picture at high T
82Photoemission Spectra and Spin Autocorrelation
Fe (U2, J.9ev,T/Tc.8) (Lichtenstein,
Katsenelson,GK prl 2001)
83Photoemission and T/Tc.8 Spin Autocorrelation
Ni (U3, J.9 ev)
84Iron and Nickelmagnetic properties
(Lichtenstein, Katsenelson,GK cond-mat 0102297)
85Ni and Fe theory vs exp
- m( T.9 Tc)/ mB ordered moment
- Fe 1.5 ( theory) 1.55 (expt)
- Ni .3 (theory) .35 (expt)
- meff / mB high T moment
- Fe 3.1 (theory) 3.12 (expt)
- Ni 1.5 (theory) 1.62 (expt)
- Curie Temperature Tc
- Fe 1900 ( theory) 1043(expt)
- Ni 700 (theory) 631 (expt)
86Fe and Ni
- Satellite in minority band at 6 ev, 30
reduction of bandwidth, exchange splitting
reduction .3 ev - Spin wave stiffness controls the effects of
spatial flucuations, it is about twice as large
in Ni and in Fe - Mean field calculations using measured exchange
constants(Kudrnovski Drachl PRB 2001) right Tc
for Ni but overestimates Fe , RPA corrections
reduce Tc of Ni by 10 and Tc of Fe by 50.
87Ni moment
88Fe moment\
89Magnetic anisotropy Ni
90Magnetic anisotropy Fe
91Magnetic anisotropy
92Conclusion
- The character of the localization delocalization
in simple( Hubbard) models within DMFT is now
fully understood, nice qualitative insights. - This has lead to extensions to more realistic
models, and a beginning of a first principles
approach interpolating between atoms and band,
encouraging results for simple elements
93DMFT Review A. Georges, G. Kotliar, W. Krauth
and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)
Weiss field
94Outlook
- Systematic improvements, short range
correlations. - Take a cluster of sites, include the effect of
the rest in a G0 (renormalization of the
quadratic part of the effective action). What
to take for G0 - DCA (M. Jarrell et.al) , CDMFT ( Savrasov and GK
) - include the effects of the electrons to
renormalize the quartic part of the action (spin
spin , charge charge correlations) E. DMFT
(Kajueter and GK, Si et.al)
95Outlook
- Extensions of DMFT implemented on model systems,
(e.g. Motome and GK ) carry over to more
realistic framework. Better determination of Tcs. - First principles approach determination of the
Hubbard parameters, and the double counting
corrections long range coulomb interactions
E-DMFT - Improvement in the treatement of multiplet
effects in the impurity solvers, phonon
entropies,
96 Functional Approach
G. Kotliar EPJB (1999)