Spectral Density Functional: a first principles approach to the electronic structure of correlated solids

1
Spectral Density Functional a first principles
approach to the electronic structure of
correlated solids

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

2001 JRCAT-CERC Workshop on Phase Control on
Correlated Electron Systems
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Outline

• Motivation. Some universal aspects of simple
  DMFT the Mott transition endpoint in frustrated
  systems.
• Non universal physics requires detailed
  material modeling. Combining DMFT and band
  structure a new functional for electronic
  structure calculations (S. Savrasov and GK)
• Results d electrons Fe and Ni.
• (Lichtenstein, Katsenelson and GK, PRL in press)

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Outline

• Results f electrons delta Pu ( S. Savrasov G.
  K and E. Abrahams,Nature (2001))
• Conclusions further extensions the approach.

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Importance of 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 and 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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Theoretical Approach to the Mott endpoint.

• DMFT.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
  (QMC, ED ) as well as semianalytical (IPT)
  approaches to solve this simplified problem.
• 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.

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DMFT Review A. Georges, G. Kotliar, W. Krauth
and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)
Weiss field
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DMFT Review A. Georges, G. Kotliar, W. Krauth
and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)
Weiss field
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Schematic DMFT phase diagram one band Hubbard
model (half filling, semicircular DOS, role of
partial frustration) Rozenberg et.al PRL (1995)
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Landau Functional
G. Kotliar EPJB (1999)
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Functional Approach

• The Landau functional offers a direct connection
  to the atomic energies
• 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 .

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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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A time-honored example Mott transition in V2O3
under pressure or chemical substitution on V-site
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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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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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Compressibility
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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.
• condensation of doubly occupied sites and onset
  of coherence .

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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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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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Two Roads for first principles calculations of
correlated materials using DMFT.
Correlation functions etc..
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Insights from DMFT

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

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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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DMFT LDA 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, andf local
  Greens function by projecting onto the local
  orbitals.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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LDADMFT

• The functional can be built in perturbation
  theory in the interaction (well defined
  diagrammatic rules )The functional can also be
  constructed from the atomic limit.
• DFT is useful because e good approximations to
  the exact density functional GDFTr(r) exist,
  e.g. LDA.
• A useful approximation to the exact functional
  can be constructed, the DMFT LDA functional.

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LDADMFT functional
Sum of local 2PI graphs with local U matrix and
local G
Double counting correction
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Spectral density functionalConnection with
atomic limit
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LDADMFT Self-Consistency loop
DMFT
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Realistic DMFT loop
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LDA functional
Double counting correction
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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 G.Kotliar, funcional
  formulation for full self consistent
  implementation (2001)

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Iron and Nickel band picture at low T, crossover
to real space picture at high T
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Photoemission Spectra and Spin Autocorrelation
Fe(U2, J.9ev) (Lichtenstein, Katsenelson,GK
prl in press)
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Photoemission and Spin Autocorrelation Ni (U3,
J.9 ev)
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Iron and Nickelmgnetic 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 3.09 (theory) 3.12 (expt)
• Ni 1.50 (theory) 1.62 (expt)
• Curie Temperature Tc
• Fe 1900 ( theory) 1043(expt)
• Ni 700 (theory) 631 (expt)

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Fe and Ni

• Spin wave stiffness controls the effects of
  spatial flucuations, it is about twice as large
  in Ni and in Fe
• Classical calculations using measured exchange
  constants
• (Kudrnovski Drachl PRB 2001) Weiss mean field
  theory gives right Tc for Ni but overestimates
• Fe , RPA corrections reduce Tc of Ni by 10 only
  but reduce Tc of Fe by nearly factor of 2.

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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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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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Pu DMFT total energy vs Volume (S. Savrasov )
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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. (Rutgers ENS), nice
  qualitative insights.
• This has lead to extensions to more realistic
  models, and a beginning of a first principles
  approach interpolating between atoms and bands.

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Conclusions

• 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)

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Conclusions

• Extensions of DMFT implemented on model systems,
  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.