Dynamical%20Mean%20Field%20Theory%20,%20Mott%20transition%20and%20Electronic%20Structure%20of%20Actinides

1
Dynamical Mean Field Theory , Mott transition
and Electronic Structure of Actinides

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

SCES 2001 Ann Arbor August 6th-10th 2001
2
Outline

• Introduction to Pu
• Background DMFT study of the Mott transition
  in a toy model
• DMFT as an electronic structure method.
• DMFT results for delta Pu, and some qualitative
  insights into the Mott transition across the
  actinide series

3

Mott transition in the actinide series (Smith
Kmetko phase diagram, Johanssen 1974)
4
Phase transition with Large Volume changes!
Small amounts of Ga stabilize the d phase (A.
Lawson LANL)
5
The bonding problem

• 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.
  Full potential and ASA methods 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.

6
Conventional viewpoint

• Alpha Pu is a simple metal, it can be described
  with LDA pert. corrections. 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, one 5f electrons are band-like while
  four 5f electron is localized.
• LDA U (Savrasov andGK Phys. Rev. Lett. 2000 ,
  Bouchet et.al 2000) predicts correct volume of
  Delta Pu with U4,Alpha Pu has U 0.

7
Problems with the conventional viewpoint of Pu

• U/W is not so different in alpha and delta
• LDAU, LDA, constrained LDA are not good starting
  points to describe the transport and
  thermodynamics, Pu is a light heavy fermion.
• 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.

8
Pu Specific Heat
9
Anomalous Resistivity
10
Outline

• Introduction to Pu
• Background DMFT study of the Mott transition
  in a toy model
• DMFT as an electronic structure method.
• DMFT results for delta Pu, and some qualitative
  insights into the Mott transition across the
  actinide series

11
Theoretical approach to the Mott transition
problem

• Mean field approach to quantum many body systems,
  constructing equivalent impurity models embedded
  in a bath to be determined self consistently.
• Use and compare exact and approximate
  numerical techniques (QMC, RG, ED) as well as
  semianalytical approaches (interpolative schemes)
  to solve the self consistent impurity model.
• Formulation the DMFT equations as saddle points
  of a functional of the spectral function . Deeper
  understanding of the validity of the DMFT
  results.

12
Schematic DMFT phase diagram one band Hubbard
model (half filling, semicircular DOS, partial
frustration) Rozenberg et.al PRL (1995)
13
Phase Diagrams V2O3, Ni Se2-x Sx Mc Whan et. Al
1971,. Czek et. al. J. Mag. Mag. Mat. 3, 58
(1976),
14
Mott transition in layered organic conductors
S Lefebvre et al. Ito et.al, Kanodas talk
Bourbonnais talk
Magnetic Frustration
15
Insights from DMFT

• The Mott transition is driven by transfer of
  spectral weight from low to high energy as we
  approach the localized phase. Fixed density.
• Control parameters doping, temperature,pressure
• The laws that govern the transfer of spectral
  weight can be formulated around special points in
  the phase diagram, where bifurcations take place

16
Mott endpoint Transfer of spectral weight at
fixed density.
Anomalous transfer of spectral weight connected
to the proximity to an Ising Mott endpoint
(Kotliar Lange and Rozenberg PRL 84, 5180 (2000))
17
Anomalous transfer of optical spectral weight in
NiSeS(Miyasaka and Takagi 2000),Photoemission
Matsuura et. Al. 1998
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Outline

• Introduction to Pu
• Background DMFT study of the Mott transition
  in a toy model
• DMFT as an electronic structure method.
• DMFT results for delta Pu, and some qualitative
  insights into the Mott transition across the
  actinide series

19
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

20
Spectral Density Functional effective action
construction (Chitra and GK 2000).

• 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)

21
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.
  Motivated by LDAU

22
LDADMFT functional
F Baym Kadanoff functional of an ATOM . Sum of
local 2PI graphs with local U matrix and local G
23
LDADMFT Introduction of a Weiss field, mapping
onto impurity models
Weiss field
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Comments 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.
• Luttinger theorem is obeyed.

25
Outline

• Introduction to Pu
• Background DMFT study of the Mott transition
  in a toy model
• DMFT as an electronic structure method.
• DMFT results for delta Pu, and some qualitative
  insights into the Mott transition across the
  actinide series

26
Pu DMFT total energy vs Volume(Savrasov
et.al.2001)
27
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 (see also Majumdar and
  Krishnmurthy 1995).
• This result resolves one of the basic paradoxes
  in the physics of Pu.

28
Lda vs Exp Spectra
29
Pu Spectra DMFT(Savrasov) EXP (Arko et. Al)
30
PU (cubic ALPHA AND DELTA
31
Double well structure and d Pu

• Qualitative explanation
  of negative thermal expansion
• Sensitivity to impurities which easily raise the
  energy of the a -like minimum.

32
Minimum in melting curve and divergence of the
compressibility at the Mott endpoint
33
Superconductivity in Am

• Atomic state J0
• How to go from a metal to a closed shell
  insulator by increasing U.
• Entropy has to increase, as U increases, but the
  insulator has zero entropy! Something has to
  happen
• DMFT study of the problem (Capone Fabrizio and
  Tossatti) Superonductivity intervenes!

34
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,
  (Savrasov, Kotliar, Abrahams Nature 2001 Pu),
  Lichtenstein Katsenelson Kotliar (PRL 2001 Fe and
  Ni).
• Outlook compounds, C-DMFT ..

35
References

• Review of DMFT A. Georges, G. Kotliar, W.
  Krauth and M. Rozenberg Rev. Mod. Phys. 68,13
  (1996)
• 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 funcional formulation
  for full self consistent implementation of a
  spectral density functional.
• Application to Pu S. Savrasov G. Kotliar and
  E. Abrahams (Nature 2001).

36
Acknowledgements

• Useful discussions with A. Lichtenstein, J.
  Thompson and R. Schrieffer
• NSF-DMR , DOE (Basic Energy Sciences)

37
DMFT Review A. Georges, G. Kotliar, W. Krauth
and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)
Weiss field
38
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 Krishnamurthy et.al) , CDMFT (
  GK Savrasov 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)

39
Outlook

• 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, phonon
  entropies,

40
Wilson and Kadowaki Woods Ratio
41
Vanadium Oxide
42
ARPES measurements on NiS2-xSexMatsuura et. Al
Phys. Rev B 58 (1998) 3690
.
43
T_MIT.013 Rozenberg et.al 2001
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Realistic DMFT loop
45
V2O3
46
Theoretical Foundations functionalsG. Kotliar
and R. Chitra PRB 1999,2000G. Kotliar and S.
Savrasov 2001

• LDA Fukuda et.al, Aliev and Fernando
• LDAU
• LDADMFT

47
LDA functional
Conjugate field, VKS(r)
48
Minimize LDA functional
49
LDAU functional
50
Double counting term (Lichtenstein et.al)
Problem What is the LDAU functional, a
functional of? What is nab ?
51
Functional Approach
G. Kotliar EPJB (1999)
52
Functional 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 .

53
Solving 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)

54
Schematic DMFT phase diagram one band Hubbard
model (half filling, semicircular DOS, partial
frustration) Rozenberg et.al PRL (1995)
55
Recent QMC phase diagram of the frustrated Half
filled Hubbard model with semicircular DOS ( Joo
and Udovenko 2001).
56
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).

57
NiSeS
58
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.

59
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 .

60
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

61
Cerium
62
Pu Anomalous thermal expansion (J. Smith LANL)
63
MAGNETIC
64
Specific heat and susceptibility.
65
Remarks on the literature

• The qualitative features found by the Rutgers
  ENS groups were challenged in a series of
  publications Logan and Nozieres (1987) S Kehrein
  Phys. Rev Lett. 3192 (1998),R. Noack and F.
  Gebhardt, Phys. Rev. Lett. 82, 1915 (1999), J.
  Schlipf et. al. Phys. Rev. Lett 82, 4890 (1999).
• These works missed subtle non perturbative
  aspects of the Mott metal to insulator transition
  such as the singular behavior of the self energy

66
Two Roles for DMFT in calculations of the
electronic structure of correlated materials
Crystal Structure atomic positions
Model Hamiltonian
Correlation functions Total energies etc.
67
the 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 elemental
  plutonium B. Johanssen Phil Mag. 30,469 (1974)
• Revisit the problem using a new insights and new
  techniques from the solution of the Mott
  transition problem within dynamical mean field
  theory in a simple model Hamiltonian (one band
  Hubbard, semicircular density of states).
• Use the ideas and concepts that resulted from
  this development to give physical insights into
  real materials.
• Turn the technology developed to solve the toy
  model into a practical electronic structure
  method.

68
More on DFT

• LSDA predicts magnetic long range (Solovyev
  et.al.0
• Experimentally d Pu is not magnetic.
• 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
  Eriksson and Wills) is used. This is taken as an
  indication that a Pu is a weakly correlated
  system

69

Mott transition in the actinide series (Smith
Kmetko phase diagram)
70
Minimum of the melting point

• 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

71
Anomalous Resistivity and Mott transition Ni
Se2-x Sx
Miyasaka and Tagaki (2000)
72
LDADMFT Self-Consistency loop
E
U
DMFT