Dynamical Mean Field Theory from Model Hamiltonian Studies of the Mott Transition to Electronic Structure Calculations

1
Dynamical Mean Field Theory from Model
Hamiltonian Studies of the Mott Transition to
Electronic Structure Calculations

• Gabriel Kotliar
• Physics Department and
• Center for Materials Theory
• Rutgers University

11 Conference on Recent Progress in Many Body
Physics UMIST July 9-15th 2001
2
Outline

• What is DMFT, when is it useful and how is it
  done.
• What has been accomplished. Ex. model
  Hamiltonian studies of the finite temperature
  Mott transition.
• How to combine DMFT and band structure, formal
  aspects.
• Results for some real materials.

3
References, Collaborators

• Review A. Georges, G. Kotliar, W. Krauth and M.
  Rozenberg Rev. Mod. Phys. 68,13 (1996)
• Finite T Mott endpoint Kotliar Lange and
  Rozenberg PRL 84, 5180 (2000))
• Realistic CalculationsS. Savrasov and GK
  cond-mat 0106308. Application to Pu, S.Savrasov
  GK and E. Abrahams Nature 410, 793 (2001). Fe and
  Ni A. Lichtenstein M. Katsnelson and GK (PRL in
  press).

4
DMFT Review A. Georges, G. Kotliar, W. Krauth
and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)
Weiss field
5
Solving the DMFT equations

• Wide variety of computational tools (QMC,
  NRG,ED.)Analytical Methods

6
Comments on the DMFT construction

• Exact in large dimensions Metzner and Vollhardt
  89
• Trick to sum all LOCAL skeleton graphs, Muller
  Hartman 89.
• Can be used for susceptibilities, ordered states
  etc..
• Non perturbative construction, works even when
  skeleton expansion fails.

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Good 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.
• Solution of this problem should lead to advances
  in electronic structure theory (LDA DMFT)

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A time-honored example Mott transition in V2O3
under pressure or chemical substitution on V-site
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Phase Diag Ni Se2-x SxG. Czek et. al. J. Mag.
Mag. Mat. 3, 58 (1976)
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Mott transition in layered organic conductors
Ito et al. (1986) Kanoda (1987) Lefebvre et
al. (2001)
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Schematic DMFT phase diagram one band Hubbard
(half filling, semicircular DOS, role of partial
frustration) Rozenberg et.al PRL (1995)
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Insights into the Mott phenomena

• The Mott transition is driven by transfer of
  spectral weight from low to high energy as we
  approach the localized phase
• Control parameters doping, temperature,pressure

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Evolution of the Spectral Function with
Temperature
Anomalous transfer of spectral weight connected
to the proximity to an Ising Mott endpoint
(Kotliar et.al.PRL 84, 5180 (2000))
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Expt. Ni Se S Matsuura et. Al.
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Ising character of Mott endpoint

• Singular part of the Weiss field is proportional
  to h a Max (p-pc) ,(T- Tc)1/d d3 in mean
  field and 5 in 3d
• h couples to all physical quantities which then
  exhibit a kink at the Mott endpoint. Resistivity,
  double occupancy,photoemission intensity,
  integrated optical spectral weight, etc.
• Divergence of the the compressibility ,in
  particle hole asymmetric situations e.g.
  Furukawa and Imada

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Phase diagram 1 band model
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Divergent compressibility U2.4
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Compressibility

• QMC two band model, U3

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Mott transition endpoint

• Rapid variation has been observed in optical
  measurements in vanadium oxide (Thomas) and Ni
  mixtures(Miyasaka and Takgai)
• Experimental questions width of the critical
  region. Ising exponents or classical exponents,
  validity of mean field theory
• Building of coherence in other strongly
  correlated electron systems.
• condensation of doubly occupied sites and onset
  of coherence .

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Optical Conductivty Miyasaka Takagi (2000)
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Insights from DMFT think in term of spectral
functions , the density is not changing!
Resistivity near the metal insulator endpoint (
Rozenberg et.al 1995) exceeds the Mott limit
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Insights from DMFT

• High temperature behavior around Mott endpoint,
  more universal regime, captured by simple models
  treated within DMFT
• Low temperatures several competing phases .
  Their relative stability depends on chemistry
  and crystal structure, LRO etc..

23
Two Roads for calculations of the electronic
structure of correlated materials
Crystal Structure atomic positions
Model Hamiltonian
Correlation functions Total energies etc.
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LDADMFT

• 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

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Spectral 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)

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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 GDFTr(r) exist, e.g.
  LDA, GGA
• A useful approximation to the exact functional
  can be constructed, the DMFT LDA functional.

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LDADMFT functional
F Sum of local 2PI graphs with local U matrix and
local G
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LDADMFTConnection with atomic limit
Weiss field
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Functional approach
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Comments on LDADMFT

• Static limit of the functional 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.

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LDADMFT 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 GK full self consistent
  implementation cond-mat 0106308. Application to
  Pu, S.Savrasov GK and E. Abrahams
• Nature 410, 793 (2001)

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LDADMFT Self-Consistency loop
E
U
DMFT
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Case study Fe and Ni

• Archetypical itinerant ferromagnets
• LSDA predicts correct low T moment
• Band picture holds at low T

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Iron and Nickel crossover to a real space
picture at high T
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Photoemission Spectra and Spin Autocorrelation
Fe (U2, J.9ev,T/Tc.8) (Lichtenstein,
Katsenelson,GK prl 2001)
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Photoemission and T/Tc.8 Spin Autocorrelation
Ni (U3, J.9 ev)
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Iron and Nickelmagnetic properties
(Lichtenstein, Katsenelson,GK cond-mat 0102297)
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Ni 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)

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Fe 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.

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Case study in f electrons, Mott transition in
the actinide series
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Small amounts of Ga stabilize the d phase
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Delocalization-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.

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Problems 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.

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Problems 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

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Pu DMFT total energy vs Volume
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Lda vs Exp Spectra
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Pu Spectra DMFT(Savrasov) EXP (Arko et. Al)
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Conclusion

• 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

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Outlook

• 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 GK
  Palsson and Biroli )
• 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)

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Outlook

• 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,

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ARPES measurements on NiS2-xSexMatsuura et. Al
Phys. Rev B 58 (1998) 3690
.