Dynamical Mean Field Theory or Metallic Plutonium

1
Dynamical Mean Field Theory or Metallic
Plutonium

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

Collaborators S. Savrasov (NJIT) and Xi Dai
(Rutgers)
IWOSMA Berkeley October 2002
2
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)
• Revisit the problem using a new insights and new
  techniques from the solution of the Mott
  transition problem within dynamical mean field
  theory in the model Hamiltonian context.
• Use the ideas and concepts that resulted from
  this development to give physical qualitative
  insights into real materials.
• Turn the technology developed to solve simple
  models into a practical quantitative electronic
  structure method .

3

• 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.
• Identificaton of Kondo resonance n Ce , PRB 28,
  5347 (1983).

4
Outline

• 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

5

Mott transition in the actinide series (Smith
Kmetko phase diagram)
6
Small amounts of Ga stabilize the d phase (A.
Lawson LANL)
7
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 phase Is 35
  lower than experiment
• This is the largest discrepancy ever known in DFT
  based calculations.

8
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

9
Pu Specific Heat
10
Anomalous Resistivity
11
Pu is NOT MAGNETIC
12
Specific heat and susceptibility.
13
Problems with the conventional viewpoint of a
Pu

• U/W is not so different in alpha and delta
• 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.

14
Outline

• 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

15
Dynamical Mean Field Theory(DMFT)Review A.
Georges G. Kotliar W. Krauth M. Rozenberg. Rev
Mod Phys 68,1 (1996)

• 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).

16
DMFT Effective Action point of view.R. Chitra
and G. Kotliar Phys Rev. B.(2000), (2001).

• Identify observable, A. Construct an exact
  functional of ltAgta, 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.

17
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.

18
Outline

• 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

19
Schematic DMFT phase diagram one band Hubbard
model (half filling, semicircular DOS, partial
frustration) Rozenberg et.al PRL (1995)
20
Spectral Evolution at T0 half filling full
frustration
X.Zhang M. Rozenberg G. Kotliar (PRL 1993)
21
Phase Diagrams V2O3, Ni Se2-x Sx Mc Whan et. Al
1971,. Czek et. al. J. Mag. Mag. Mat. 3, 58
(1976),
22
Mott transition in layered organic conductors
S Lefebvre et al. Ito et.al, Kanodas talk
Bourbonnais talk
Magnetic Frustration
23
Cerium
24
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

25
Ultrasound study of
Fournier et. al. (2002)
26
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
• Elastic anomalies, more pronounced with orbital
  degeneracy.

27
Minimum in melting curve and divergence of the
compressibility at the Mott endpoint
28
Cerium
29
Outline

• 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

30
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)

31
LDADMFT approximate functional

• 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

32
LDADMFT-outer loop relax
Edc
U
DMFT
33
Outer loop relax
Edc
G0
Impurity Solver
G,S
Imp. Solver Hartree-Fock
U
SCC
DMFT
LDAU
34
LDADMFT and LDAU

• Static limit of the LDADMFT functional ,
• with Fatom FHF reduces to the LDAU
  functional
• of Anisimov Andersen and Zaanen.
• Crude approximation. Reasonable in ordered Mott
  insulators.
• Total energy in DMFT can be approximated by
  LDAU with an effective U . Extra screening
  processes in DMFT produce smaller Ueff.
• ULDAU lt UDMFT

35
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)
• ..

36
LDADMFT 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)

37
Spectral Density Functional 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,
  Gr(r),G(R,R)(iw)

38
LDADMFT Self-Consistency loop
E
U
DMFT
39
Comments on LDADMFT

• Static limit of the LDADMFT functional , with F
  FHF reduces to LDAU
• 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.

40
References

• 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).

41
References

• 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.

42
Outline

• 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

43
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
• S5/2 Ms5 mB
• Mtot0

44
LDAU bands. (Savrasov GK ,PRL 2000).
45
Magnetic moment

• L5, S5/2, J5/2, MtotMsmB gJ .7 mB
• Crystal fields G7 G8
• GGAU estimate (Savrasov and Kotliar 2000)
  ML-3.9 Mtot1.1
• This bit is quenched by Kondo effect of spd
  electrons DMFT treatment
• Experimental consequence neutrons large
  magnetic field induced form factor (G. Lander).

46
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)/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)

47
Technical details

• Atomic sphere approximation.
• Ignore crystal field splittings in the self
  energies.
• Fully relativistic non perturbative treatment of
  the spin orbit interactions.

48
Pu DMFT total energy vs Volume (Savrasov 00)
49
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.

50
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.

51
Comparaison with the Hartree Fock static limit
LDAU.
52
Dependence on structure
53
Comments on the HF static limit

• Describes only the Hubbard bands.
• No QP states.
• Single well structure in the E vs V curve.
• (Savrasov and Kotliar PRL)

54
Lda vs Exp Spectra
55
Pu Spectra DMFT(Savrasov) EXP (Arko Joyce Morales
Wills Jashley PRB 62, 1773 (2000)
56
Comparaison with LDAU
57
Summary
Spectra
Method
E vs V
LDA
LDAU
DMFT
58
Outline

• 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

59
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.
• Calculations can and should be refined and
  extended.

60
Conclusions

• Outsanding question electronic entropy, lattice
  dynamics.
• In the making, new generation of DMFT programs,
  QMC with multiplets, full potential DMFT,
  frequency dependent Us, multiplet effects ,
  combination of DMFT with GW

61
DMFT EXPERIMENTS
62
Pu Anomalous thermal expansion ( Smith and
Boring )
63
DMFT MODELS.
64
Mean-Field Classical vs Quantum
Classical case
Quantum case
A. Georges, G. Kotliar (1992)
Phys. Rev. B 45, 6497
65
Example Single site DMFT, functional formulation

• Express in terms of Weiss field (G. Kotliar EPJB
  99)

Local self energy (Muller Hartman 89)
66
DMFT Impurity cavity construction
67
DMFT Review A. Georges, G. Kotliar, W. Krauth
and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)
Weiss field
68
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).

69
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.

70
Interfacing DMFT in calculations of the
electronic structure of correlated materials
Crystal Structure atomic positions
Model Hamiltonian
Correlation functions Total energies etc.
71
Combining LDA and DMFT

• The light, SP electrons well described by LDA.
  The heavier D electrons treat by DMFT.
• LDA already contains an average interaction of
  the heavy electrons, subtract this out by
  shifting the heavy level (double counting term,
  Edc , review Anismov Aersetiwan and Lichtenstein
  )
• Atomic physics parameters . UF0 cost of double
  occupancy irrespectively of spin, JF2F4, Hunds
  energy favoring spin polarization , F2/F4.6,..
• Calculations of U, Edc, (Gunnarson and
  Anisimov, McMahan et.al. Hybertsen et.al) or
  viewed as parameters

72
DMFT MODELS RESULTS
73
QMC calculationof n vs m (Kotliar Murthy
Rozenberg PRL 2002, 2 band, U3.0)
k diverges at generic Mott endpoints
74
Compressibilty divergence
75
Cerium
76
E-DMFTGW effective action
G D
77
LDADMFT functional
FAtom Sum of all local 2PI graphs build with
local Coulomb interaction matrix, parametrized
by Slater integrals F0, F2 and F4 and local
G.Express F in terms of AIM model.
78
LDADMFT functional
F Sum of local 2PI graphs with local U matrix and
local G
79
E-DMFT GW P. Sun and G. Kotliar Phys. Rev. B 2002
80
Solving the DMFT equations

• Wide variety of computational tools
  (QMC,ED.)Analytical Methods
• Reviews A. Georges, G. Kotliar, W. Krauth and
  M. Rozenberg Rev. Mod. Phys. 68,13 (1996).
  Prushke T. Jarrell M. and Freericks J. Adv.
  Phys. 44,187 (1995)

81
Density functional theory and Dynamical Mean
Field Theory

• DFT Static mean field, electrons in an effective
  potential.
• Functional of the density.
• DMFT Promote the local (or a few cluster Greens
  functions ) as the basic quantities of the
  theory.
• Express the free energy as a functional of these
  local quantities and the density.
• Provide useful approximations to the functional.

82
Realistic DMFT loop
83
LDADMFTConnection with atomic limit
Weiss field
84
Double counting term (Lichtenstein et.al)
Problem What is the LDAU functional, a
functional of? What is nab ?
85
Plutonium
86
PU (cubic ALPHA AND DELTA