Application of many-body perturbation theory to the description of correlated metals [Elektronische Ressource] / Stanislav Chadov

Stanislav Chadov

Informations

• Publié par: ludwig-maximilians-universitat_munchen
• Publié le: 01 janvier 2007
• Nombre de lectures: 5
• Langue: English
• Poids de l'ouvrage: 6 Mo

Extrait

Dissertation zur Erlangung des Doktorgrades
der Fakult¨at fu¨r Chemie und Pharmazie
der Ludwig–Maximilians–Universit¨atMu¨nchen
Application of Many-Body Perturbation
Theory to the Description of Correlated Metals
Stanislav Chadov
aus
Kiew, Ukraine
2007Erkl¨arung
Diese Dissertation wurde im Sinne von§13 Abs. 3 der
Promotionsordnung vom 29. Jan 1998 von Prof. Dr. H. Ebert
betreut.
Ehrenw¨ortliche Versicherung
Diese Dissertation wurde selbstst¨andig, ohne unerlaubte Hilfe
erarbeitet.
Mu¨nchen, am 20.12.2007 Stanislav Chadov
Dissertation eingereicht am 20.12.2007
1. Gutachter: Prof. Dr. H. Ebert
2. Gutachter: Prof. Dr. A. Lichtenstein
/Institut fu¨r Theor. Physik, Uni Hamburg/
Mu¨ndlichen Pru¨fung am 22.02.2008Contents
1 Introduction 7
2 The Hubbard model 13
2.1 The Hubbard Hamiltonian . . . . . . . . . . . . . . . . . 13
2.2 The strong interaction limit . . . . . . . . . . . . . . . . 17
2.3 The weak interaction limit . . . . . . . . . . . . . . . . . 18
3 Perturbational description 23
3.1 The Baym-Kadanoff theory . . . . . . . . . . . . . . . . 23
3.2 The T-matrix approximation. . . . . . . . . . . . . . . . 27
3.3 T-matrix formulation for real energies . . . . . . . . . . . 32
4 The mean-field approximations 35
5 The energy functional description 39
5.1 Density functional theory. . . . . . . . . . . . . . . . . . 41
5.2 The local density approximation . . . . . . . . . . . . . . 44
5.3 The LDA+U method . . . . . . . . . . . . . . . . . . . . 47
5.3.1 The Coulomb interaction matrix. . . . . . . . . . 48
5.3.2 The double-counting term . . . . . . . . . . . . . 51
5.4 Spectral density functional theory . . . . . . . . . . . . . 53
5.5 The dynamical-mean field approximation . . . . . . . . . 58
5.6 LDA+DMFT . . . . . . . . . . . . . . . . . . . . . . . . 61
6 Numerical implementation 65
6.1 DMFT within the KKR method . . . . . . . . . . . . . . 65
6.1.1 Solution of the Kohn-Sham equations . . . . . . . 65
56 Contents
6.1.2 Common features and comparison of the DMFT
solvers . . . . . . . . . . . . . . . . . . . . . . . . 72
6.2 Application to ground-state properties . . . . . . . . . . 77
6.2.1 Orbital magnetic moments . . . . . . . . . . . . . 77
6.2.2 Calculation of the total energy . . . . . . . . . . . 83
6.2.3 Summary . . . . . . . . . . . . . . . . . . . . . . 85
6.3 Applications to photoemission . . . . . . . . . . . . . . . 86
6.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . 86
6.3.2 Theoretical framework . . . . . . . . . . . . . . . 88
6.3.3 Fano-effect in the VB-XPS of Fe and Co . . . . . 93
6.3.4 High-energy angle-resolved VB-XPS of Ni . . . . 95
6.3.5 High-energy angle-integrated VB-XPS of NiMnSb 96
7 Applicationtolinearresponsetheory 99
7.1 Time-dependent perturbation theory . . . . . . . . . . . 99
7.2 Optical conductivity . . . . . . . . . . . . . . . . . . . . 103
7.3 Green’s function within the variational basis formalism:
implementation in the LMTO method . . . . . . . . . . 105
7.4 Accounting for localized correlations . . . . . . . . . . . 108
7.5 Applications to 3d-transition metal systems . . . . . . . 111
7.6 Application to heavy-fermion systems . . . . . . . . . . . 114
8 Summary 119
Bibliography 123
Acknowledgements 137
Curriculum vitae – Lebenslauf 139
List of Publications (2002-2007) 141Chapter 1
Introduction
Electronsystemswithstronginteractionreceivedstrongtheoreticaland
experimental interest for several decades. This interest has been inten-
sified withthe discoveryof heavy-fermionand relatednon-Fermi-liquid
systems and high-T superconductors. An extensive and continuousC
effort in theoretical field is devoted to the investigation of the most
extensively used nowadays prototype for an exactly solvable model of
many-electron systems - the so-called Hubbard model [1]. Being rel-
atively simple, the Hubbard model contains a great manifold of phe-
nomena which up to now is far from being fully investigated, as well
as the rich possibilities for testing various physical ideas and meth-
ods. It was successfully applied to describe the set of new electronic
phenomena where electronic correlations are significant: such as metal
insulator transition [2], itinerant magnetism [3], spin-density waves [4]
and local pair formation [5,6] which plays a key role in the explana-
tion of the high-T superconductivity and the superconductivityin theC
heavy-fermion systems.
However, it is very difficult to solve the Hubbard model in gene-
ral, except of the few tractable limits. One is the so-called weak-
coupling limit which leads to the non-interacting electron gas which
is well-understood. However, even for a weak coupling there is an ex-
ceptional situation occurring at the half-filling. Namely, even the in-
finitesimallysmallCoulombrepulsiondrivesthesystemthroughmetal-
insulatortransition,whichcannotbedescribedbythefree-electronpic-
ture [7]. Much less clear is the limit of strong interaction. At a half
filling the model turns to the Heisenberg antiferromagnetic insulator
78 Chapter 1. Introduction
[7]. However, if the occupation is away from half-filling the behavior of
the system becomes a complete mystery.
Duringthelast40yearsofinvestigationsnumerousapproacheshave
been suggested: decoupling of the Green’s functions [1], the varia-
tional approach [8], linearization of the equations of motion [9], the
self-consistentmomentsmethod[10],thecompositeoperatorsapproach
(COM) [11] and many others. However none of them can be consid-
ered as universal. One of the approaches dealing with the simplest
extension of the Hubbard model, which adequately accounts for a sit-
uation in solid, namely, the strongly interacting dilute electron gas in
theone-particlescalarpotential,istheso-calledBaym-Kadanofftheory
[12]. However,the complexityof the crystalscontainingmanydifferent
atoms per unit cell, interactions between electronic and lattice degrees
of freedom demands a very detailed investigationof the band structure
insuchsystemsanddonotallowtoapplytheBaym-Kadanoffapproach
in a straightforwardway for the real solid.
The only general ab-initio approach which accounts for the specific
features in real compounds is based on the so-called Density Func-
tional Theory (DFT) developed by Hohenberg, Kohn and Sham [13–
15]. Themajorityof practicalnowadaysDFT applicationstomagnetic
solids are based on the local mean-field description provided by the
so-called Local Spin Density Approximation (LSDA) which treats the
exchange-correlation part of an effective single-particle DFT potential
as a density-dependent exchange-correlation potential treated on the
basis of the results for the homogeneous electron gas. There are a lot
of successes of LSDA, however, there are also some failures related to
thefactthatincasewhensomeportionoftheelectronicstructurecould
be better described by the atomic-likeorbitals,the homogeneousgas is
not a suitable starting point.
Thus, a reasonable strategy in this situation would be to have a
simple and accurate approach that could describe the most important
features of the realistic electron structure and at the same time takes
the most important correlation effects into account.
One of the first successful steps in this direction was the so-called
GW approximation (GWA) for quasiparticle spectra in solids [16,17]
whichdeliverstheone-particleGreen’sfunctionaccountingforthenon-
localself-energycalculatedtothelowestorderinthescreenednon-local9
Coulombinteraction(W).Beingappliedforstronglycorrelatedsystems
like NiO, the GWA gives a rather good description of the size of the
band gap and also improves the description of the Op-band compared
to the LSDA. However, the application to more complex systems has
not been feasible up to now due to large computational efforts.
Another approach accounting for correlation effects in localized d-
and f-shells is the so-called LSDA+U method [18,19]. The method
separates the localized d- (or f)-electron subsystem from the rest and
introduces the additional Coulomb repulsion U in the form of a Hub-
bard model term. The rest consisting of the delocalized conduction
electronsisdescribedbytheorbital-independentone-particleLSDApo-
tential. Thisapproachgivesamorereliabledescriptionoftheelectronic
structure within the same charge, spin and orbital ordering than does
theplainLSDA. However,due tothesingleenergyscaleitdoes notde-
scribe themost interestingcorrelationeffects whichareconnectedwith
theenergy(ortime)dependenceoftheself-energy: therenormalization
of the quasiparticle spectra and mass enhancement.
The real breakthrough in this field was made few years later with
the development of the Dynamical Mean-Field Theory (DMFT) [20–
23]. This approach distinguishes the localized interacting electronic
subsystem from the rest and provides the conditions required by the
Baym-Kadanoff description by treating the coupling of the interacting
subsystem with its environment as a local mean-field. Thus the many-
body problem becomes equivalent to the well-known Anderson model
[24] and can be solved by the corresponding impurity solvers based
on various approximating techniques or applying the so-called Quan-
tum Monte Carlo method (QMC) [25]. The combination