Theoretical Treatments of Correlation Effects

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Theoretical Treatments of Correlation Effects

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

Workshop on Chemical Physics of Emerging
Materials Schloss Rinberg May 29th 2001
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What can theory contribute to materials research
?
Summary

• Some universal aspects can be gleaned from
  simple models. Example, recent DMFT study of
  the Mott transition endpoint.
• Non universal physics requires detailed
  modeling. Case study Recent LDADMFT study of d
  Pu.

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Why 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.

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Mott transition in layered organic conductors
S Lefebvre et al. cond-mat/0004455
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A time-honored example Mott transition in V2O3
under pressure or chemical substitution on V-site
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Kuwamoto Honig and AppellPRB (1980)
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Phase Diag Ni Se2-x SxG. Czek et. al. J. Mag.
Mag. Mat. 3, 58 (1976)
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Theoretical Approach

• Mean field approach to quantum many body systems,
  constructing equivalent impurity models embedded
  in a bath to be determined self consistently.
  Use exact numerical techniques as well as
  semianalytical approaches to study this problem.
  (DMFT).
• Exact in infinite dimensions (Metzner and
  Vollhardt ) , can be improved systematically
  using cluster methods (DCA, CDMFT).
• Study simple model Hamiltonians (such as the one
  band model on simple lattices)
• Understand the results physically in terms of a
  Landau theory certain high temperature aspects
  are independent of the details of the model and
  the approximations used. Other results are
  approximate, and very sensitive on solid state
  aspects.

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Reviews of DMFT

• Prushke T. Jarrell M. and Freericks J. Adv.
  Phys. 44,187 (1995)
• A. Georges, G. Kotliar, W. Krauth and M.
  Rozenberg Rev. Mod. Phys. 68,13 (1996)

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

• 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 Lange and Rozenberg PRL 84, 5180 (2000))
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Insights from DMFT think in term of spectral
functions (branch cuts) instead of well defined
QP (poles )
Resistivity near the metal insulator endpoint (
Rozenberg et. Al 1995) exceeds the Mott limit
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Anomalous Resistivity and Mott transition Ni
Se2-x Sx
Miyasaka and Tagaki (2000)
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ARPES measurements on NiS2-xSexMatsuura et. Al
Phys. Rev B 58 (1998) 3690
.
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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 specific heat.

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Mott transition endpoint

• Rapid variation has been observed in optical
  measurements in vanadium oxide and nises mixtures
• 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.
• Unify concepts from different theoretical
  approaches, condensation of d and onset of
  coherence .

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Insights from DMFT

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

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Delocalization Localization across the actinide
series
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Small amounts of Ga stabilize the d phase
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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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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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effective action construction (Fukuda, Valiev
and Fernando , Chitra and GK).

• Select a set of local orbitals.
• Define a frequency dependent, local Greens
  function by projecting onto the local orbitals.
• The exact free energy can be expressed as a
  functional of the local Greens function and of
  the density
• A useful approximation to the exact functional
  road to total energy calculations.

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LDADMFT

• 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 and E. Abrahams full
  self consistent implementation ( Nature, 2001)

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LDADMFT Self-Consistency loop
DMFT
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Pu DMFT total energy vs Volume (S. Savrasov )
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Lda vs Exp Spectra
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Pu Spectra DMFT(Savrasov et. al ) EXP (Arko et.
al)
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Outlook

• Some universal aspects can be gleaned from
  simple models. Recent DMFT study of the Mott
  transition endpoint.
• Many more simple qualitative pictures of little
  corners in the space of all materials, are still
  to be found.
• Non universal physics requires detailed
  modeling. Recent LDADMFT study of d Pu.
• New developments in many body and electronic
  structure methods, predictions of new compounds?
  More interactions with chemical physics and
  material science.

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Mean-Field Classical vs Quantum
Quantum case
Classical case
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Landau Functional
G. Kotliar EPJB (1999)
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Double counting correction
Simplest case F0 only. Generalization
Lichtenstein et.al in The context of LDAU