Quench Dynamics in Interacting and Superconducting Nanojunctions (eBook)

Supervisors’ Foreword 7
Abstract 9
Publications Related to this thesis Build-up of vibron-mediated electronic correlations in molecular electronics Rémi Avriller, Rubén Seoane Souto, Álvaro Martín Rodero and Alfredo Levy Yeyati. Physical Review B, 99 (12), 121403 (2019).Transient dynamics in interacting nanojunctions within self-consistent perturbation theory Rubén Seoane Souto, Rémi Avriller, Alfredo Levy Yeyati and Álvaro Martín Rodero. New J. Phys. 20, 083039 (2018).Quench dynamics in superconducting nanojunctions: metastability an dynamical Yang-Lee zeros Rubén Seoane Souto, Álvaro Martín Rodero and Alfredo Levy Yeyati. Physical Review B, 96 (16), 165444 (2017).Analysis of universality in transient dynamics of coherent electronic transport Rubén Seoane Souto, Álvaro Martín Rodero and Alfredo Levy Yeyati. Fortschritte der Physik, 65 1600062 (2017).Andreev bound states formation and quasiparticle trapping in quench dynamics revealed by time-dependent counting statistics Rubén Seoane Souto, Álvaro Martín Rodero and Alfredo Levy Yeyati. Physical Review Letters, 117 (26), 267701 (2016).Transient dynamics and waiting time distribution of molecular junctions in the polaronic regime Rubén Seoane Souto, Rémi Avriller, Rosa Carmina Monreal, Álvaro Martín Rodero and Alfredo Levy Yeyati. Physical Review B, 92 (12), 125435 (2015).Dressed tunneling approximation for electronic transport through molecular transistors Rubén Seoane Souto, Alfredo Levy Yeyati, Álvaro Martín Rodero and Rosa Carmina Monreal. Physical Review B, 88 (8), 085412 (2014). 11
Acknowledgements 12
Contents 14
Acronyms 18
1 General Introduction 19
1.1 Theory of the Quantum Transport 20
1.2 Superconductivity at the Nanoscale 21
1.3 Interactions at the Nanoscale 23
1.4 Time Dependent Transport 24
1.4.1 Dynamics of Interacting Nanojunctions 25
1.5 Current Fluctuations 25
1.5.1 Full Counting Statistics 26
1.5.2 Analogy to Equilibrium Statistical Mechanics: Yang–Lee Zeros 28
1.6 Outline 29
References 31
2 Theoretical Framework in the Stationary Regime 35
2.1 Impurity Level Hamiltonian 35
2.2 Green Function Formalism 37
2.2.1 Equilibrium Green Functions 37
2.2.2 Time-Dependent Green Functions 39
2.2.3 Non-equilibrium Green Functions 41
2.2.4 Transport Properties 43
2.2.5 Interaction Picture 43
2.3 Electron-Electron Interaction: Anderson Model 46
2.3.1 Mean Field Approximation 47
2.3.2 Effects Beyond the Mean Field 48
2.4 Electron-Phonon Interaction: The Spinless Anderson–Holstein Model 49
2.4.1 Second Order Perturbation Expansion 50
2.4.2 Self-consistent Approximations 52
2.4.3 Polaron-Like Approximations 54
2.5 Superconducting Nanojunctions 58
2.5.1 AC-Josephson Effect 62
2.6 Full Counting Statistics 64
2.6.1 Non-interacting System 66
2.6.2 Interaction Effects 68
2.6.3 Factorial Cumulants 69
2.6.4 Dynamical Yang–Lee Zeros 70
References 72
Part I Transient Dynamics in Normal Nanojunctions 76
3 Transient Dynamics in Non-interacting Junctions 77
3.1 Introduction 77
3.2 Mean Transport Properties 78
3.3 Full Counting Statistics 80
3.3.1 Discretized Dyson Equation and the Determinant Formula 81
3.4 Universal Relation Between Cumulants and Zeros 83
3.5 Analysis of the Short Time Universality 86
3.5.1 Single Electrode Junction 86
3.5.2 Two Electrodes Junction: Coherent Effects 88
3.5.3 Bidirectional Transport 90
3.6 Conclusions 92
References 92
4 Polaron Effects in Quench Dynamics 94
4.1 Introduction 94
4.2 Basic Theoretical Formulation 95
4.2.1 Single Pole Approximation 98
4.2.2 Short Time Tunnel Limit 99
4.3 Evolution of System Population and Current 101
4.4 Transient Statistics and Waiting Time Distribution 104
4.5 Conductance and Fano Factor Dynamics at V=n?0 108
4.6 Conclusions 110
References 111
5 Self-consistent Approximations 114
5.1 Introduction 114
5.2 Self-consistent Procedure 115
5.3 Electron–Electron Interaction: The Anderson Model 117
5.3.1 Hartree–Fock Approximation 117
5.3.2 Effects of Correlation Beyond Mean-Field 121
5.4 Electron–Phonon Interaction: Spinless Anderson–Holstein Model 125
5.4.1 Hartree Approximation 125
5.4.2 Effects of Correlation Beyond Hartree Approximation 128
5.5 Electron–Electron and Electron–Phonon Interactions 133
5.6 Calculation of the Steady State Properties 135
5.7 Conclusions 137
References 138
Part II Transient Dynamics in Superconducting Nanojunctions 141
6 Quench Dynamics in Superconducting Nanojunctions 142
6.1 Introduction 142
6.2 Model and Formalism 143
6.2.1 Single Pole Approximation 145
6.3 Quench Dynamics 148
6.4 AC-Josephson Effect 155
6.5 Voltage Pulse Initialization 159
6.6 Conclusions 161
References 161
7 Counting Statistics in Superconducting Nanojunctions 165
7.1 Introduction 165
7.2 Formalism 167
7.2.1 Coarse Grained Statistics 168
7.3 Quench Dynamics 169
7.3.1 Yang–Lee Zeros and Phase Coexistence 173
7.4 Finite Bias Voltage Dynamics 176
7.4.1 Dynamical Yang–Lee Zeros 179
7.5 Voltage Pulse Initialization 180
7.6 Coupling to a Bosonic Mode 183
7.6.1 Model and Formalism 183
7.6.2 Single Particle Properties 185
7.6.3 Counting Statistics 186
7.7 Conclusions 188
References 189
Part III General Conclusions and Outlook 191
8 General Conclusions and Outlook 192
8.1 Normal Nanojunctions 192
8.2 Superconducting Nanojunctions 194
References 197
Appendix A Numerical Renormalization Group 199
A.1 Algorithm 199
A.1.1 Logarithmic Discretization 200
A.1.2 Recursive Diagonalization 202
A.1.3 Spectral Properties 205
A.1.4 Discarded States 206
A.1.5 Averaging Between Different Band Discretizations 207
A.2 Holstein Model 208
Appendix B Toeplitz Matrix Theory 213
B.1 Basic Mathematical Theory 213
B.1.1 Block Toeplitz Matrices 215
B.2 Transport Properties 217
Appendix C Inverse Free Boson Propagator 220
Appendix D Interpretation in Terms of Rate Equations 223
Appendix E Bidirectional Poisson Distribution 225