Strong electronic correlations in low-dimensional systems [Elektronische Ressource] / vorgelegt von Andreas Dolfen

Andreas Dolfen

Sujets

• Physik

Informations

• Publié par: rheinisch-westfalischen_technischen_hochschule_-rwth-_aachen
• Publié le: 01 janvier 2010
• Nombre de lectures: 2
• Langue: English
• Poids de l'ouvrage: 9 Mo

Extrait

Strong Electronic Correlations
in Low-Dimensional Systems
Von der Fakult at fur Mathematik, Informatik und
Naturwissenschaften der RWTH Aachen University zur
Erlangung des akademischen Grades eines Doktors der
Naturwissenschaften genehmigte Dissertation
vorgelegt von
Dipl.-Phys. Andreas Dolfen
aus K oln
Berichter: Univ.-Prof. Dr.rer.nat. Erik Koch Eva Pavarini, Ph.D.
Tag der mundlic hen Prufung: 16.09.2010
Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfugbar.Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 Electrostatic Screening in Dielectrics . . . . . . . . . . . . . 5
1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.1 Microscopic Point-Dipole Model . . . . . . . . . . . . . . . . . . 8
1.1.2 Vector-Space Notation . . . . . . . . . . . . . . . . . . . . . . 9
1.1.3 Obtaining the Screening Energy for Coulomb Parameters . . . . . . . 10
1.1.4 Ferroelectric Catastrophe and Reciprocal Space . . . . . . . . . . . 11
1.2 Screening in a One-Dimensional Chain . . . . . . . . . . . . . . . 15
1.2.1 Con nement to One Dimension . . . . . . . . . . . . . . . . . . 15
1.2.2 Analytical Solution . . . . . . . . . . . . . . . . . . . . . . . 16
1.2.3 Screening Peculiarities . . . . . . . . . . . . . . . . . . . . . . 18
1.2.4 Comparison to Cluster Calculations using the Real-Space MPDM Code . 29
1.3 Screening in a Two-Dimensional Square Lattice . . . . . . . . . . 32
1.3.1 Con nement to Two Dimensions . . . . . . . . . . . . . . . . . 32
1.3.2 Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . 33
1.3.3 Microscopic Field and Averaging . . . . . . . . . . . . . . . . . 41
1.4 Screening in a Three-Dimensional Simple Cubic Lattice . . . . . . 42
1.4.1 Clausius-Mossotti Relation from the Microscopic Perspective . . . . . 42
1.4.2 Numerical Solution for Moderate Polarizability in Real Space . . . . . 45
1.4.3 in Reciprocal Space. . . . . . . . . . . . . . . 51
1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53iv Contents
2 Organics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.1 Electronic Structure Theory . . . . . . . . . . . . . . . . . . . . 59
2.1.1 Density-Functional . . . . . . . . . . . . . . . . . . . . 59
2.1.2 Density-Functional Practice with FHI-aims . . . . . . . . . . . . . 62
2.1.3 Hubbard model . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.2 Organics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.2.1 Building Blocks { Organic Molecules . . . . . . . . . . . . . . . . 70
2.2.2 Molecular Crystals . . . . . . . . . . . . . . . . . . . . . . . 71
2.3 Evaluation of Model Parameters . . . . . . . . . . . . . . . . . . 76
2.3.1 Hopping Parameters . . . . . . . . . . . . . . . . . . . . . . . 76
2.3.2 Coulomb Parameters. . . . . . . . . . . . . . . . . . . . . . . 82
2.4 Application to TTF-TCNQ . . . . . . . . . . . . . . . . . . . . 89
2.4.1 The Crystal . . . . . . . . . . . . . . . . . . . . . . . . . . 90
2.4.2 Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . 92
2.4.3 Realistic Description . . . . . . . . . . . . . . . . . . . . . . .103
2.5 Application to (BEDT-TTF) I . . . . . . . . . . . . . . . . . .1082 3
2.5.1 The Crystal . . . . . . . . . . . . . . . . . . . . . . . . . .108
2.5.2 Model Parameters . . . . . . . . . . . . . . . . . . . . . . . .115
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .123
3 Dynamical Lattice Susceptibilities from DMFT . . . . . . .125
3.1 Correlation and Green’s functions . . . . . . . . . . .126
3.1.1 Concepts For A Microscopic Description . . . . . . . . . . . . . .126
3.1.2 Microscopic Response Theory . . . . . . . . . . . . . . . . . . .128
3.1.3 n-body Green’s Function . . . . . . . . . . . . . . . . . . . . .130
3.1.4 Single-Particle Green’s Function . . . . . . . . . . . . . . . . . .131
3.1.5 Two-Particle Green’s Function . . . . . . . . . . . . . . . . . .131
3.1.6 Analytical Structure and Properties of Correlation Functions . . . . .133
3.1.7 (Non-)Interacting Green’s Functions and Self-Energy . . . . . . . . .139
3.1.8 From Green’s to Dynamical Correlations Functions. . . . . . . . . .140
3.1.9 Analytic Continuation and Pade Approximants . . . . . . . . . . .146
3.2 DMFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .155
3.2.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . .155
3.2.2 Self-Consistency Loop in the Hamiltonian Formalism . . . . . . . . .157
3.2.3 Exact Diagonalization as Impurity Solver . . . . . . . . . . . . . .160
3.2.4 Exact DMFT Limits . . . . . . . . . . . . . . . . . . . . . . .166v
3.2.5 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . .167
3.3 DMFT Dynamical Susceptibilities and Vertex Functions . . . . . .178
3.3.1 General Formalism and DMFT Approximation . . . . . . . . . . .178
3.3.2 Explicit Derivation in the ED Framework . . . . . . . . . . . . . .182
3.3.3 Properties of the Two-Particle Green’s Function and the Vertex . . . .189
3.3.4 Overview of the Actual Calculation & Tests . . . . . . . . . . . . .192
3.4 Application to the 3-dimensional Periodic Anderson Model. . . . .199
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .206
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .207
A Ewald Method . . . . . . . . . . . . . . . . . . . . . . . . . .211
A.1 Generalized Derivation of the Ewald Summation . . . . . . . . . .211
A.2 Ewald Summation for the Dipole Matrix . . . . . . . . . . . . . .214
A.3 Results for Body- and Face-Centered Cubic Lattices . . . . . . . .215
B Euler Angles . . . . . . . . . . . . . . . . . . . . . . . . . . .217
B.1 General Introduction . . . . . . . . . . . . . . . . . . . . . . .217
B.2 Extracting Euler angles . . . . . . . . . . . . . . . . . . . . . .219
C FHI-aims: Basis Sets . . . . . . . . . . . . . . . . . . . . . .221
D Atomic units . . . . . . . . . . . . . . . . . . . . . . . . . . .223
E Massively Parallel Lanczos Solver . . . . . . . . . . . . . . .225
F Fourier transforms & Friends . . . . . . . . . . . . . . . . .229
F.1 In nite Continuous Fourier Transform . . . . . . . . . . . . . . .229
F.2 Discrete Fourier T . . . . . . . . . . . . . . . .230
F.3 Relationship between Continuous and Discrete Fourier Transforms .230
F.4 Convolution Theorem and Autocorrelation function . . . . . . . .231
F.4.1 Similar relations used in Condensed Matter Physics . . . . . . . . .231
G Inversion by Partitioning . . . . . . . . . . . . . . . . . . . .233
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .235
Acknowledgements. . . . . . . . . . . . . . . . . . . . . . . . . . .245vi ContentsIntroduction
It has always been a dream of physicists to understand and predict properties of materials
without resorting to direct experimental measurements. In principle, the fundamental
laws describing our world between the nuclear and astronomical scales are well-known.
All we need to do is solve the time-dependent many-body Schr odinger equation providing
direct access to all properties. Neglecting relativistic e ects it takes the form
@
i ji =Hji;
@t
whereji is the many-body wavefunction and
N N N N N Nn e e n e n2 2 2 2 2X X XX X XpP Z e e Z Z ej H = + + + ;
2M 2m jr R j jr rj jR Rj j j k
=1 j=1 j=1 =1 j<k <
denotes the many-body Hamiltonian of the system. Z is the atomic number, M the
mass, R the position, and P the momentum of nucleus . p and r denote the jj
thj electron’s momentum and position. N , N are the number of electrons and nuclei,e n
respectively. This equation does not only describe essentially all everyday phenomena but
also unusual quantum e ects. Striking examples are macroscopic quantum states such as
superconductivity, super uidity, or the entanglement of states that lies at the heart of
quantum computing.
Soon after Schr odinger formulated his equation [ 1], Dirac realized that this "Theory of
Almost Everything" comes with a catch [2]:
The underlying laws necessary for the mathematical theory of a large part
of physics and the whole of chemistry are thus completely known, and the
di culty is only that exact applications of these laws lead to equations
which are too complicated to be soluble. It therefore becomes desirable that
approximate practical methods of applying quantum mechanics should be
developed, which can lead to an explanation of the main features of complex
atomic systems without too much computation.2 Introduction
The complexity, Dirac refers to, arises from the quantum many-body nature of the
problem. As illustration let us consider a simpli ed iron atom. With its N = 26e
electrons the total electronic wavefunction depends on 26 times 3 coordinates. Choosing
a very crude approximation by specifying the wavefunction on a grid with 10 points per
78coordinate yields 10 numbers to store { let alone process. This huge amount of data
cannot even be stored on a hard drive as large as our home galaxy, the milky way.
The most successful, \approximate practical method" of applied quantum mechanics
for condensed matter systems is density-functional theory. It works well for many
classes of materials, where it is a good approximation to think of each electron as an
individual entity, moving in the static mea