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The Mott transition in f electron systems, Pu, a dynamical mean field perspective Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University Collaborators: S. Savrasov (NJIT) ASR2002 Tokai Japan November 12-24 2002

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the mott transition in f electron systems pu a dynamical mean field perspective

The Mott transition in f electron systems, Pu, a dynamical mean field perspective

Gabriel Kotliar

Physics Department and

Center for Materials Theory

Rutgers University

Collaborators: S. Savrasov (NJIT)

ASR2002

Tokai Japan November 12-24 2002

mott transition in the actinide series smith kmetko phase diagram
Mott transition in the actinide series (Smith Kmetko phase diagram)

THE STATE UNIVERSITY OF NEW JERSEY

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small amounts of ga stabilize the d phase a lawson lanl
Small amounts of Ga stabilize the d phase (A. Lawson LANL)

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outline
Introduction: some Pu puzzles.

Introduction to DMFT.

Some qualitative insights from model Hamiltonian studies.

Towards and understanding of elemental Pu.

Conclusions

Outline

THE STATE UNIVERSITY OF NEW JERSEY

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plutonium puzzles
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 phaseIs 35% lower than experiment

This is the largest discrepancy ever known in DFT based calculations.

Plutonium Puzzles

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dft studies
LSDA predicts magnetic long range (Solovyev et.al.)

Experimentally d Pu is not magnetic.

If one treats the f electrons as part of the core LDA overestimates the volume by 30%

DFT in GGA predicts correctly the volume of the a phase of Pu, when full potential LMTO (Soderlind Eriksson and Wills) is used. This is usually taken as an indication that a Pu is a weakly correlated system

U/W is not so different in alpha and delta

DFT Studies

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pu specific heat
Pu Specific Heat

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anomalous resistivity
Anomalous Resistivity

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pu is not magnetic
Pu is NOT MAGNETIC

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plutonium puzzles10
How to think about the alpha and delta phases and compute their physical properties?

Why does delta have a negative thermal expansion?

Why do minute amount of impurities stablize delta?

Where does epsilon fit? Why is it smaller than delta?

Plutonium puzzles.

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outline11
Introduction: some Pu puzzles.

Introduction to DMFT.

Some qualitative insights from model Hamiltonian studies.

Towards and understanding of elemental Pu.

Conclusions

Outline

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mean field classical vs quantum
Mean-Field : Classical vs Quantum

Classical case

Quantum case

A. Georges, G. Kotliar (1992)

Phys. Rev. B 45, 6497

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extensions of dmft
Renormalizing the quartic term in the local impurity action.

EDMFT.

Taking several sites (clusters) as local entity.

CDMFT

Combining DMFT with other methods.

LDA+DMFT, GWU.

Extensions of DMFT.

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dmft effective action point of view r chitra and g kotliar phys rev b 62 12715 2000 b63 115110 2001
Identify observable, A. Construct an exact functional of <A>=a, G [a] which is stationary at the physical value of a.

Construct approximations to the functional G to perform practical calculations.

Example: Density functional theory (Fukuda et. al.),density, LDA, GGA.

Example: model DMFT. Observable: Local Greens function Gii (w). Exact functional G [Gii (w) ]. DMFT Approximation the functional keeping 2PI graphs

DMFT: effective action point of view. R. Chitra and G. KotliarPhys. Rev. B 62, 12715 (2000), B63, 115110 (2001)

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slide15
Effective action construction.

Introduce local orbitals, caR(r-R), 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, G[r(r),G(R,R)(iw)]

Spectral Density Functionals for Electronic Structure Calculations, Sergej Savrasov and Gabriel Kotliar,  cond-mat/0106308

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slide16
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 (Gunnarson and Anisimov, McMahan et.al. Hybertsen et.al) of viewed as parameters

Construct approximate functional which gives the LDA+DMFT equations.V. Anisimov, A. Poteryaev, M. Korotin, A.  Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359-7367 (1997).

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outer loop relax
Outer loop relax

Edc

G0

Impurity Solver

Imp. Solver: Hartree-Fock

G,S

U

SCC

DMFT

LDA+U

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review
A. Georges G. Kotliar W. Krauth M. Rozenberg. Rev Mod Phys 68,1 (1996)

Kotliar, Savrasov, in New Theoretical approaches to strongly correlated systems, Edited by A. Tsvelik, Kluwer Publishers, (2001).

Held Nekrasov Blumer Anisimov and Vollhardt et.al. Int. Jour. of Mod PhysB15, 2611 (2001).

A. Lichtenstein M. Katsnelson and G. Kotliar condmat 0211076(2002)

Review

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outline19
Introduction: some Pu puzzles.

Introduction to DMFT.

Some qualitative insights from model Hamiltonian studies.

Towards and understanding of elemental Pu.

Conclusions

Outline

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slide20
Schematic DMFT phase diagram one band Hubbard model (half filling, semicircular DOS, partial frustration) Rozenberg et.al PRL (1995)

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cerium
Cerium

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slide22
Spectral Evolution at T=0 , half filling full frustration, Hubbard bands in the metal, transfer of spectral weight.

X.Zhang M. Rozenberg G. Kotliar (PRL 1993) A. Georges, G. Kotliar (1992) Phys. Rev. B 45, 6497

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qualitative phase diagram in the u t m plane two band kotliar murthy rozenberg prl 2002
Coexistence regions between localized and delocalized spectral functions.

k diverges at generic Mott endpoints

Qualitative phase diagram in the U, T , m plane (two band Kotliar Murthy Rozenberg PRL (2002).

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minimum in melting curve and divergence of the compressibility at the mott endpoint
Minimum in melting curve and divergence of the compressibility at the Mott endpoint

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cerium25
Cerium

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generalized phase diagram
Generalized phase diagram

T

U/W

Structure, bands, orbitals

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slide27
Case study in f electrons, Mott transition in the actinide series. B. Johanssen 1974 Smith and Kmetko Phase Diagram 1984.

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outline28
Introduction: some Pu puzzles.

DMFT , qualitative aspects of the Mott transition in model Hamiltonians.

DMFT as an electronic structure method.

DMFT results for delta Pu, and some qualitative insights.

Conclusions

Outline

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what is the dominant atomic configuration local moment
Snapshots of the f electron

Dominant configuration:(5f)5

Naïve view Lz=-3,-2,-1,0,1

ML=-5 mB

S=5/2 Ms=5 mB

Mtot=0

What is the dominant atomic configuration? Local moment?

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gga u bands savrasov kotliar phys rev lett 84 3670 3673 2000
GGA+U bands. Savrasov Kotliar ,Phys. Rev. Lett. 84, 3670-3673, (2000)

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how is the magnetic moment quenched
L=5, S=5/2, J=5/2, Mtot=Ms=mB gJ =.7 mB

Crystal fields G7 +G8

GGA+U estimate (Savrasov and Kotliar 2000) ML=-3.9 Mtot=1.1

This bit is quenched by Kondo effect of spd electrons [ DMFT treatment]

Experimental consequence: neutrons large magnetic field induced form factor (G. Lander).

How is the Magnetic moment quenched.

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pu dmft total energy vs volume s savrasov g kotliar and e abrahams nature 410 793 2001
Pu: DMFT total energy vs VolumeS. Savrasov, G. Kotliar, and E. Abrahams, Nature 410, 793 (2001),

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double well structure and d pu
Qualitative explanation of negative thermal expansionDouble well structure and d Pu

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dynamical 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.

Dynamical Mean Field View of Pu(Savrasov Kotliar and Abrahams, Nature 2001)

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comments on the hf static limit for pu
Describes only the Hubbard bands.

No QP states.

Single well structure in the E vs V curve.

(Savrasov and Kotliar PRL). Same if one uses a Hubbard one impurity solver.

Comments on the HF static limit for Pu

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lda vs exp spectra
Lda vs Exp Spectra

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pu spectra dmft savrasov exp arko joyce morales wills jashley prb 62 1773 2000
Pu Spectra DMFT(Savrasov) EXP (Arko Joyce Morales Wills Jashley PRB 62, 1773 (2000)

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summary

Spectra

Method

E vs V

Summary

LDA

LDA+U

DMFT

the delta epsilon transition
The high temperature phase, (epsilon) is body centered cubic, and has a smaller volume than the (fcc) delta phase.

What drives this phase transition?

A functional, that computes total energies opens the way to the computation of phonon frequencies in correlated materials (S. Savrasov and G. Kotliar 2002)

The delta –epsilon transition

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effects of structure gga dmft hubbard1 imp solver
Effects of structure. GGA+DMFT_Hubbard1 imp.solver

Ee-Ed=350 K

GGA gives

Ee-Ed=

-6000 K

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phonon freq thz vs q in delta pu s savrasov
Phonon freq (THz) vs q in delta Pu (S. Savrasov)

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phonon frequency thz vs q in epsilon pu
Phonon frequency (Thz ) vs q in epsilon Pu.

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epsilon plutonium
Compute the energy of the most unstable frozen mode. Transverse mode at ( 0,pi, pi) with polarization (0,1,-1).

Extrapolate the form of the quartic interaction to the whole Brillouin zone.

Carry out a self consistent Born approximation to obtain the restabilize phones. Recompute the entropy difference between delta and epsilon.

Estimate the critical temperatures: 500-700 K , depending on the detials of the extrapolation.

Epsilon plutonium.

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phonon entropy drives the epsilon delta phase transition
Epsilon is slightly more metallic than delta, but it has a much larger phonon entropy than delta.

At the phase transition the volume shrinks but the phonon entropy increases.

Phonon entropy drives the epsilon delta phase transition

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phonons epsilon
Phonons epsilon

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outline46
Introduction: some Pu puzzles.

DMFT , qualitative aspects of the Mott transition in model Hamiltonians.

DMFT as an electronic structure method.

DMFT results for delta Pu, and some qualitative insights into other phases.

Conclusions

Outline

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conclusions
DMFT produces non magnetic state, around a fluctuating (5f)^5 configuraton with correct volume the qualitative features of the photoemission spectra, and a double minima structure in the E vs V curve.

Correlated view of the alpha and delta phases of Pu. Interplay of correlations and electron phonon interactions (delta-epsilon).

Calculations can be refined.

Conclusions

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conclusions48
Phonons matter. Role of electronic entropy.

In the making, new generation of DMFT programs, QMC with multiplets, full potential DMFT, frequency dependent U’s, multiplet effects , combination of DMFT with GW

Other materials, Cerium and Yterbium compounds…………

Conclusions

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acknowledgements development of dmft
Acknowledgements: Development of DMFT

Collaborators: V. Anisimov, R. Chitra, V. Dobrosavlevic, D. Fisher, A. Georges, H. Kajueter, W.Krauth, E. Lange, A. Lichtenstein, G. Moeller, Y. Motome, G. Palsson, S. Pankov, M. Rozenberg,S. Murthy , S. Savrasov, Q. Si, V. Udovenko, X.Y. Zhang

Funding: National Science Foundation.

Department of Energy and LANL.

Office of Naval Research.

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technical details
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)/2]^3 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)

Technical details

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technical details54
Atomic sphere approximation.

Ignore crystal field splittings in the self energies.

Fully relativistic non perturbative treatment of the spin orbit interactions.

Technical details

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solving the dmft equations
Solving the DMFT equations
  • Wide variety of computational tools (QMC,ED….)Analytical Methods
  • Extension to ordered states.
  • Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]

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temperature stabilizes a very anharmonic phonon mode
Temperature stabilizes a very anharmonic phonon mode

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lsda dmft functional
LSDA+DMFT functional

F Sum of local 2PI graphs with local U matrix and local G

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e dmft references
H. Kajueter and G. Kotliar (unpublished and Kajuter’s Ph.D thesis (1995)).

Q. Si and J L Smith PRL 77 (1996)3391 .

R. Chitra and G.Kotliar Phys. Rev. Lett 84, 3678-3681 (2000 )

Y. Motome and G. Kotliar. PRB 62, 12800 (2000)

R. Chitra and G. Kotliar Phys. Rev. B 63, 115110 (2001)

S. Pankov and G. Kotliar PRB 66, 045117 (2002)

E-DMFT references

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dmft impurity cavity construction
DMFT Impurity cavity construction

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cluster extensions of dmft
Two impurity method. [A. Georges and G. Kotliar (1995 unpublished ) and RMP 68,13 (1996) , A. Schiller PRL75, 113 (1995)]

M. Jarrell et al Dynamical Cluster Approximation [Phys. Rev. B 7475 1998]

Periodic cluster] M. Katsenelson and A. Lichtenstein PRB 62, 9283 (2000).

G. Kotliar S. Savrasov G. Palsson and G. Biroli Cellular DMFT [PRL87, 186401 2001]

Cluster extensions of DMFT

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c dmft
C-DMFT

C:DMFT The lattice self energy is inferred

from the cluster self energy.

Alternative approaches DCA (Jarrell et.al.) Periodic clusters (Lichtenstein and Katsnelson)

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c dmft test in one dimension bolech kancharla gk cond mat 2002
C-DMFT: test in one dimension. (Bolech, Kancharla GK cond-mat 2002)

Gap vs U, Exact solution Lieb and Wu, Ovshinikov

Nc=2 CDMFT

vs Nc=1

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dmft models
DMFT MODELS.

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mean field classical vs quantum67
Mean-Field : Classical vs Quantum

Classical case

Quantum case

A. Georges, G. Kotliar (1992)

Phys. Rev. B 45, 6497

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example single site dmft functional formulation
Express in terms of Weiss field (G. Kotliar EPJB 99)Example: Single site DMFT, functional formulation

Local self energy (Muller Hartman 89)

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dmft impurity cavity construction69
DMFT Impurity cavity construction

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dmft review a georges g kotliar w krauth and m rozenberg rev mod phys 68 13 1996
DMFT Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]

Weissfield

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case study ipt half filled hubbard one band
(Uc1)exact = 2.2+_.2 (Exact diag, Rozenberg, Kajueter, Kotliar PRB 1996) , confirmed by Noack and Gebhardt (1999) (Uc1)IPT =2.6

(Uc2)exact =2.97+_.05(Projective self consistent method, Moeller Si Rozenberg Kotliar Fisher PRL 1995 ), (Confirmed by R. Bulla 1999) (Uc2)IPT =3.3

(TMIT ) exact =.026+_ .004 (QMC Rozenberg Chitra and Kotliar PRL 1999), (TMIT )IPT =.045

(UMIT )exact =2.38 +- .03 (QMC Rozenberg Chitra and Kotliar PRL 1999), (UMIT )IPT =2.5 (Confirmed by Bulla 2001)

For realistic studies errors due to other sources (for example the value of U, are at least of the same order of magnitude).

Case study: IPT half filled Hubbard one band

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spectral 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 GDFT[r(r)] exist, e.g. LDA, GGA

A useful approximation to the exact functional can be constructed, the DMFT +LDA functional.

Spectral Density Functional

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interfacing dmft in calculations of the electronic structure of correlated materials
Interfacing DMFT in calculations of the electronic structure of correlated materials

Crystal Structure

+atomic positions

Model

Hamiltonian

Correlation functions

Total energies etc.

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e dmft gw effective action
E-DMFT+GW effective action

G=

D=

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e dmft gw p sun and g kotliar phys rev b 2002
E-DMFT +GW P. Sun and G. Kotliar Phys. Rev. B 2002

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lda dmft and lda u
Static limit of the LDA+DMFT functional ,

with FatomFHF reduces to the LDA+U functional

of Anisimov Andersen and Zaanen.

Crude approximation. Reasonable in ordered Mott insulators. Short time picture of the systems.

Total energy in DMFT can be approximated by LDA+U with an effective U . Extra screening processes in DMFT produce smaller Ueff.

ULDA+U < UDMFT

LDA+DMFT and LDA+U

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minimum in melting curve and divergence of the compressibility at the mott endpoint77
Minimum in melting curve and divergence of the compressibility at the Mott endpoint

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interface dmft with electronic structure
Derive model Hamiltonians, solve by DMFT

(or cluster extensions). Total energy?

Full many body aproach, treat light electrons by GW or screened HF, heavy electrons by DMFT [E-DMFT frequency dependent interactionsGK and S. Savrasov, P.Sun and GK cond-matt 0205522]

Treat correlated electrons with DMFT and light electrons with DFT (LDA, GGA +DMFT)

Interface DMFT with electronic structure.

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lda dmft outer loop relax
LDA+DMFT-outer loop relax

Edc

U

DMFT

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outer loop relax80
Outer loop relax

Edc

G0

Impurity Solver

Imp. Solver: Hartree-Fock

G,S

U

SCC

DMFT

LDA+U

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lda dmft and lda u81
Static limit of the LDA+DMFT functional ,

with FatomFHF reduces to the LDA+U functional

of Anisimov Andersen and Zaanen.

Crude approximation. Reasonable in ordered Mott insulators. Short time picture of the systems.

Total energy in DMFT can be approximated by LDA+U with an effective U .

LDA+DMFT and LDA+U

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very partial list of application of realistic dmft to materials
QP bands in ruthenides: A. Liebsch et al (PRL 2000)

N phase of Pu: S. Savrasov G. Kotliar and E. Abrahams (Nature 2001)

MIT in V2O3: K. Held et al (PRL 2001)

Magnetism of Fe, Ni: A. Lichtenstein M. Katsenelson and G. Kotliar et al PRL (2001)

J-G transition in Ce: K. Held A. Mc Mahan R. Scalettar (PRL 2000); M. Zolfl T. et al PRL (2000).

3d doped Mott insulator La1-xSrxTiO3 Anisimov et.al 1997, Nekrasov et.al. 1999, Udovenko et.al 2002)

………………..

Very Partial list of application of realistic DMFT to materials

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lda dmft references
Anisimov Poteryaev Korotin Anhokin and Kotliar J. Phys. Cond. Mat. 35, 7359 (1997).

Lichtenstein and Katsenelson PRB (1998).

Reviews:Kotliar, Savrasov, in New Theoretical approaches to strongly correlated systems, Edited by A. Tsvelik, Kluwer Publishers, (2001).

Held Nekrasov Blumer Anisimov and Vollhardt et.al. Int. Jour. of Mod PhysB15, 2611 (2001).

A. Lichtenstein M. Katsnelson and G. Kotliar (2002)

LDA+DMFT References

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lda dmft self consistency loop
LDA+DMFT Self-Consistency loop

E

U

DMFT

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lda dmft functional
LDA+DMFT functional

F Sum of local 2PI graphs with local U matrix and local G

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comments on lda dmft
Static limit of the LDA+DMFT functional , with F= FHF reduces to LDA+U

Gives the local spectra and the total energy simultaneously, treating QP and H bands on the same footing.

Luttinger theorem is obeyed.

Functional formulation is essential for computations of total energies, opens the way to phonon calculations.

Comments on LDA+DMFT

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references
LDA+DMFT:

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 G.Kotliar funcional formulation for full self consistent implementation of a spectral density functional.

Application to Pu S. Savrasov G. Kotliar and E. Abrahams (Nature 2001).

References

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debye temperatures
Debye temperatures

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references89
Long range Coulomb interactios, E-DMFT. R. Chitra and G. Kotliar

Combining E-DMFT and GW, GW-U , G. Kotliar and S. Savrasov

Implementation of E-DMFT , GW at the model level. P Sun and G. Kotliar.

Also S. Biermann et. al.

References

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wilson and kadowaki woods ratio
Wilson and Kadowaki Woods Ratio

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dependence on structure
Dependence on structure

Expt: Ve-Vd=.54 A

Theory: Ve-Vd=.35 A

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slide92
Local approximation (Treglia and Ducastelle PRB 21,3729), local self energy, as in CPA.

Exact the limit defined by Metzner and Vollhardt prl 62,324(1989) inifinite.

Mean field approach to many body systems, maps lattice model onto a quantum impurity model (e.g. Anderson impurity model )in a self consistent medium for which powerful theoretical methods exist. (A. Georges and G. Kotliar prb45,6479 (1992)). See also M. Jarrell (PRL 1992) .Connect local spectra and lattice total energy.

Dynamical Mean Field Theory(DMFT)Review: A. Georges G. Kotliar W. Krauth M. Rozenberg. Rev Mod Phys 68,1 (1996)

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correlation betwee the minimum of the melting point and the mott transition endpoint
Divergence of the compressibility at the Mott transition endpoint.

Rapid variation of the density of the solid as a function of pressure, in the localization delocalization crossover region.

Slow variation of the volume as a function of pressure in the liquid phase

Elastic anomalies, more pronounced with orbital degeneracy.

Correlation betwee the Minimum of the melting point and the Mott transition endpoint.

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localization delocalization transition and f electrons
Mott phenomena. Evolution of the electronic structure between the atomic limit and the band limit in an open shell situation.

The “”in between regime” is ubiquitous central them in strongly correlated systems, gives rise to interesting physics. Example Mott transition across the actinide series [ B. Johansson Phil Mag. 30,469 (1974)]

Connection between local spectra and cohesive energy using Anderson impurity models foreshadowed by J. Allen and R. Martin PRL 49, 1106 (1982) in the context of KVC for cerium.

Localization delocalization transition and f electrons.

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dmft and f electrons
These views of the localization delocalization transition are not orthogonal and were incorporated into a more complete Dynamical Mean Field picture of the Mott transition.

G. Kotliar, EPJ-B, 11, (1999), 27 . A. Georges G. Kotliar W. Krauth M. Rozenberg. Rev Mod Phys 68,1 (1996) . Moeller Q. Si G. Kotliar M. Rozenberg and D. S Fisher, PRL 74 (1995) 2082.

DMFT: Powerful new tool for studying f electrons.

Qualitative insights into complex materials.

Turn technology developed to solve models into practical quantitative electronic structure method , to study eg. PU.

DMFT and f electrons.

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evolution of the spectral function with temperature near mott endpoint
Evolution of the Spectral Function with Temperature near Mott endpoint.

Anomalous transfer of spectral weight connected to the proximity to the Ising Mott endpoint (Kotliar Lange and Rozenberg Phys. Rev. Lett. 84, 5180 (2000)

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dmft effective action point of view r chitra and g kotliar phys rev b 2000 2001
DMFT: Effective Action point of view.R. Chitra and G. Kotliar Phys Rev. B.(2000), (2001).
  • Identify observable, A. Construct an exact functional of <A>=a, G [a] which is stationary at the physical value of a.
  • Example, density in DFT theory. (Fukuda et. al.)
  • When a is local, it gives an exact mapping onto a local problem, defines a Weiss field.
  • The method is useful when practical and accurate approximations to the exact functional exist. Example: LDA, GGA, in DFT.
  • It is useful to introduce a Lagrange multiplier l conjugate to a, G [a, l ].
  • It gives as a byproduct a additional lattice information.

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slide98
Schematic DMFT phase diagram one band Hubbard model (half filling, semicircular DOS, partial frustration) Rozenberg et.al PRL (1995)

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slide99
Mott transition in layered organic conductors S Lefebvre et al. Ito et.al, Kanoda’s talk Bourbonnais talk

Magnetic Frustration

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ultrasound study of
Ultrasound study of

Fournier et. al. (2002)

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comparaison with lda u
Comparaison with LDA+U

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slide102
DMFT: Effective Action point of view.R. Chitra and G. Kotliar Phys Rev. B.(2000), Phys. Rev. B 63, 115110 (2001)
  • Identify observable, A. Construct an exact functional of <A>=a, G [a] which is stationary at the physical value of a.
  • Example, density in DFT theory. (Fukuda et. al.)
  • When a is local, it gives an exact mapping onto a local problem, defines a Weiss field.
  • The method is useful when practical and accurate approximations to the exact functional exist. Example: LDA, GGA, in DFT.

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example dmft for lattice model e g single band hubbard
Observable: Local Greens function Gii (w).

Exact functional G [Gii (w) ].

DMFT Approximation to the functional.

Example: DMFT for lattice model (e.g. single band Hubbard).

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wilson and kadowaki woods ratio104
Wilson and Kadowaki Woods Ratio

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comparaison with the hartree fock static limit gga u
Comparaison with the Hartree Fock static limit: GGA+U.

Volume, total energies are OK much better than LDA, but no double minima

Ee-Ed=350 K

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