Pair correlations from symmetry-broken states in strongly correlated electronic systems [Elektronische Ressource] / vorgelegt von Falk Günther
185 Pages
English
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Pair correlations from symmetry-broken states in strongly correlated electronic systems [Elektronische Ressource] / vorgelegt von Falk Günther

Downloading requires you to have access to the YouScribe library
Learn all about the services we offer
185 Pages
English

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Pair Correlations from Symmetry-Broken Statesin Strongly Correlated Electronic SystemsVon der Fakult¨at fu¨r Mathematik, Naturwissenschaften und Informatikder Brandenburgischen Technischen Universit¨at Cottbuszur Erlangung des akademischen GradesDoktor der Naturwissenschaften(Dr.rer.nat.)genehmigte Dissertationvorgelegt vonDiplom-Physiker¨Falk Gunthergeboren am 13. Februar 1980 in GroßenhainGutachter: Prof.Dr.G¨otz Seibold, BTUGutachter: Prof.Dr.Marco Grilli, Univ.La Sapienza, RomGutachter: Prof.Dr.Vladimir Hizhnyakov, Univ.TartuTag der mu¨ndlichen Pru¨fung: 15. Oktober 2010Pair Correlations from Symmetry-Broken States inStrongly Correlated Electronic SystemsFalk Gu¨ntherOctober 2010iiiiiNon quia difficilia sunt, non audemus,sed quia non audemus, difficilia sunt.Lucius Annaeus SenecaivContentsKurzfassung xAbstract xii1 Introduction 11.1 Crystal Structure and Electronic Configuration . . . . . . . . . . . . . 92 Hubbard Model -Limits and Approximations 152.1 The Hubbard Model - Exact and Approximative Solutions . . . . . . . 152.2 Hartree Fock Approximation of theHubbard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3 Bogoliubov Transformation. . . . . . . . . . . . . . . . . . . . . . . . . 182.4 Attraction Repulsion Transformation . . . . . . . . . . . . . . . . . . . 213 Gutzwiller Approachand Model Specifications 233.1 Gutzwiller’s Wave-function and Approximation . . . . . . . . . . . . . 243.

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Published 01 January 2010
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Pair Correlations from Symmetry-Broken States
in Strongly Correlated Electronic Systems
Von der Fakult¨at fu¨r Mathematik, Naturwissenschaften und Informatik
der Brandenburgischen Technischen Universit¨at Cottbus
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
(Dr.rer.nat.)
genehmigte Dissertation
vorgelegt von
Diplom-Physiker
¨Falk Gunther
geboren am 13. Februar 1980 in Großenhain
Gutachter: Prof.Dr.G¨otz Seibold, BTU
Gutachter: Prof.Dr.Marco Grilli, Univ.La Sapienza, Rom
Gutachter: Prof.Dr.Vladimir Hizhnyakov, Univ.Tartu
Tag der mu¨ndlichen Pru¨fung: 15. Oktober 2010Pair Correlations from Symmetry-Broken States in
Strongly Correlated Electronic Systems
Falk Gu¨nther
October 2010iiiii
Non quia difficilia sunt, non audemus,
sed quia non audemus, difficilia sunt.
Lucius Annaeus SenecaivContents
Kurzfassung x
Abstract xii
1 Introduction 1
1.1 Crystal Structure and Electronic Configuration . . . . . . . . . . . . . 9
2 Hubbard Model -
Limits and Approximations 15
2.1 The Hubbard Model - Exact and Approximative Solutions . . . . . . . 15
2.2 Hartree Fock Approximation of the
Hubbard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Bogoliubov Transformation. . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Attraction Repulsion Transformation . . . . . . . . . . . . . . . . . . . 21
3 Gutzwiller Approach
and Model Specifications 23
3.1 Gutzwiller’s Wave-function and Approximation . . . . . . . . . . . . . 24
3.2 Charge-Rotationally-Invariant Gutzwiller Approach . . . . . . . . . . . 26
3.3 Extended Hubbard Model with Local On-Site Attraction . . . . . . . . 32
3.4 Unified Slave Boson Representation . . . . . . . . . . . . . . . . . . . . 35
vvi CONTENTS
3.5 Gutzwiller Energy Functional and
Lagrangian Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.6 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.7 Characterization of the Solution . . . . . . . . . . . . . . . . . . . . . . 48
4 Homogeneous SC and CDW Solutions 53
4.1 Stability Analysis in Infinite Dimensions . . . . . . . . . . . . . . . . . 54
4.2 Solutions in the HFA and GA . . . . . . . . . . . . . . . . . . . . . . . 58
4.3 Transformation to an Effective GA-BCS Hamiltonian . . . . . . . . . . 62
4.4 d-Wave Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . 66
5 Inhomogeneous Solutions 73
5.1 Homogeneous Charged Stripes . . . . . . . . . . . . . . . . . . . . . . . 75
5.2 Stripe-less Solution for V >0 . . . . . . . . . . . . . . . . . . . . . . . 81
5.3 Stripes with V >0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.4 Vortices and Point-Like Inhomogeneities . . . . . . . . . . . . . . . . . 91
6 Gutzwiller Analysis of the Superfluid Stiffness 105
6.1 Definition and Interpretation of the
Superfluid Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.2 Superfluid Stiffness of the Gutzwiller Solution . . . . . . . . . . . . . . 109
6.3 Superfluid Stiffness and Sum Rules . . . . . . . . . . . . . . . . . . . . 114
Summary and Conclusion 119
A 123
A.1 Notation and Conventions . . . . . . . . . . . . . . . . . . . . . . . . . 123
B 127CONTENTS vii
B.1 Effective Mean Field Hubbard Hamiltonian . . . . . . . . . . . . . . . . 127
B.2 Fourier Transformation of the d-wave Interaction Term . . . . . . . . . 128
B.3 Interaction Kernel of the ph-Channel in the
Free Energy Expansion of the GA Functional. . . . . . . . . . . . . . . 129
B.4 The Constrained GA Energy Functional . . . . . . . . . . . . . . . . . 130
B.5 Derivatives of the GA-functional . . . . . . . . . . . . . . . . . . . . . . 131
B.6 Derivatives of the z-Factors . . . . . . . . . . . . . . . . . . . . . . . . 140
iB.7 Derivatives of the MFA Matrix A . . . . . . . . . . . . . . . . . . . . . 144
B.8 Derivatives of the Inter Site Repulsion Term . . . . . . . . . . . . . . . 150
C 153
C.1 The Bardeen-Cooper-Schrieffer Theory . . . . . . . . . . . . . . . . . . 153
C.2 Attraction-Repulsion Transformation . . . . . . . . . . . . . . . . . . . 155
C.3 Second Order Perturbation Theory . . . . . . . . . . . . . . . . . . . . 156
C.4 Green’s Function and Sum Rule . . . . . . . . . . . . . . . . . . . . . . 157viii LIST OF ABBREVIATIONS
List of Abbreviations
BCS Bardeen-Cooper-Schrieffer
BZ Brillouin zone
CDW Charge density wave
DOS, N(ω) Density of states
GA Gutzwiller Approximation
HFA Hartree-Fock approximation
HTSC High temperature superconductor
Im Imaginary part
KR Kotliar-Ruckenstein
MFA Mean field approximation
PDW Pair density wave
QMC Quantum-Monte-Carlo
Re Real part
RPA Random phase approximation
SC Superconductor
SDW Spin density wave
σ Spin index
TDGA Time dependent Gutzwiller approximation
VAV Vortex-anti-vortex