Benemérita universidad autónoma de Puebla
Instituto de Física Luis rivera Terrazas.
Scattering Properties of Open Systems of Interacting Quantum Particles.
Para obtener el grado de:
Doctor en ciencias (Física).
Presenta:
Suren Sorathia.
Asesor:
Izrailev Felix.
Puebla Pué. Febrero 2010
Contents
Title Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
1 Introduction 1
I Fundamental Concepts 7
2 Internal Chaos 9
2.1 Random Matrix Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Description of the random matrix models . . . . . . . . . . . . . . . . . . . 11
3 Open quantum systems: Coupling to continuum 15
3.1 The scattering matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 The effective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3 Coupling of random matrix ensembles to continuum . . . . . . . . . . . . . 19
3.4 Poles of the S-matrix: Transition to superradiance . . . . . . . . . . . . . . 20
II Scattering properties of chaotic systems: Random matrix approach 25
4 Statistical properties of cross sections 29
4.1 The average scattering matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 The average fluctuating cross sections . . . . . . . . . . . . . . . . . . . . . 32
4.3 The elastic enhancement factor . . . . . . . . . . . . . . . . . . . . . . . . . 37
5 Cross section correlations 41
5.1 Interplay between the internal degree of chaos and coupling to the continuum 41
6 Conductance and fluctuations 47
6.1 The average conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
iv
Contents v
6.2 Variance of the conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.3 The role of correlations: Universal conductance fluctuations . . . . . . . . . 54
7 Summary of Part II 57
III Transport and localisation in disordered tight-binding models 61
8 Open tight-binding models 63
8.1 The tight-binding Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 64
8.2 Effective Hamiltonian for tight-binding models . . . . . . . . . . . . . . . . 67
8.3 The K-matrix method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
8.4 No disorder: Perfect crystal connected to leads . . . . . . . . . . . . . . . . 74
9 The 1D Anderson model: Statistical properties at the band centre 82
9.1 The Thouless localisation length and single parameter scaling . . . . . . . . 82
9.2 The entropic localisation length and the disorder parameter . . . . . . . . . 88
9.3 The phenomenological level spacing distribution . . . . . . . . . . . . . . . . 93
9.4 Statistical properties of scattering for the open Anderson model . . . . . . . 101
9.5 Transmission and fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . 108
9.6 Cross sections and correlations . . . . . . . . . . . . . . . . . . . . . . . . . 116
10 The full energy dependent formalism of the open Anderson model 127
10.1 Energy dependent poles of the S-matrix . . . . . . . . . . . . . . . . . . . . 128
10.2 Ensemble average transmission and variance . . . . . . . . . . . . . . . . . . 133
10.3 The effective localisation length . . . . . . . . . . . . . . . . . . . . . . . . . 136
11 The quasi-1D and 2D Anderson model 141
11.1 The exact energy dependent effective Hamiltonian . . . . . . . . . . . . . . 141
11.2 Definition of Conductance in Quasi-1D and 2D . . . . . . . . . . . . . . . . 146
11.3 Transmission and Variance in Quasi 1D systems . . . . . . . . . . . . . . . 148
12 Summary of Part III 152
13 Conclusions 156
A Historical development of the different correlation terms 160
A.1 With regard to conductance: Segregation of left and right channels . . . . . 161
A.2 All five correlation groups: EE, EI1, EI0, II1 and II0 . . . . . . . . . . . 163
B Normalisation of semi-infinite leads 166
C Calculation of the contour integral 168
Bibliography 172
Introduction:
The problem of quantum transport is generic for all realistic quantum systems interacting
with the environment. A transmission of a signal through a many-body quantum
aggregate of interacting particles is essentially the main instrument in studying such systems
and using them for practical communication purposes. Currently, this is one of the
crucial lines of development of mesoscopic physics with broad applications to quantum
information, electronics and materials science.
To date, much is understood about the conditions for the onset of chaos in closed system
of interacting particles. On the one hand, there are many physical problems related
to the fact that these systems are physically non-isolated, due to the coupling to the
continuum.
As a result, such systems have a non-zero probability for irreversible decay into
the continuum that can be treated in terms of chaotic scattering. Although the theory of
chaotic scattering is well developed, it refers to the situation of the so-called one-body chaos
that neglects the interaction between particles.