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Density fluctuations in the 1D Bose gas
Panfil, M.K.
Publication date
2013
Link to publication
Citation for published version (APA):
Panfil, M. K. (2013). Density fluctuations in the 1D Bose gas.
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Contents
1 Introduction 1
1.1 The 3D Bose gas and the Bose-Einstein condensation . . . 3
1.2 Long-range Order and the MWH Theorem . . . 7
1.3 A Glimpse of the 1D World . . . 10
1.4 Structure of the Thesis . . . 10
2 Many-body Physics in 1D 13 2.1 1D Bose Gas . . . 13
2.1.1 Coordinate Bethe Ansatz . . . 14
2.1.2 Physical Picture . . . 19
2.2 The Density-density Correlation Function . . . 21
2.2.1 Real Space Correlation Function . . . 21
2.2.2 Momentum Space. . . 25
2.3 How to Compute Correlation Functions . . . 29
2.3.1 The ABACUS. . . 31
2.4 How to Measure Correlation Functions . . . 31
3 The Algebraic and the Thermodynamic Bethe Ansatz 35 3.1 The Algebraic Bethe Ansatz . . . 36
3.1.1 Construction of Eigenstates . . . 40
3.1.2 Repulsive Interactions . . . 42
3.1.3 Attractive Interactions . . . 45
3.1.4 Scalar Products and Form Factors . . . 47
3.1.5 The Density Operator . . . 49
3.2 Thermodynamic Bethe Ansatz . . . 50
3.2.1 Density of Quantum Numbers. . . 51
3.2.2 Density of Rapidities . . . 52
3.2.3 Excitations and the Back-flow Function . . . 56
3.2.4 Thermal Equilibrium . . . 59
4 Exact prefactors of the density-density correlation function in the (non-) linear Luttinger liquid model 65 4.1 Luttinger Liquid Theory . . . 66
4.2 Nonlinear Luttinger Liquid . . . 71
4.3 Thermodynamic Limit of the Density Operator . . . 77
4.3.1 Evaluation ofM1 . . . 86 ii
Contents iii
4.3.2 Obtaining term M2 . . . 87
4.3.3 Fredholm Determinants . . . 101
4.4 Results. . . 104
4.4.1 Thermodynamic Limit of Form Factors. . . 104
5 Finite temperature density-density correlations 113 5.1 ABACUS and the Saddle-point Method . . . 114
5.2 Momentum Space . . . 116
5.3 Real Space . . . 121
5.4 Conclusions . . . 122
6 Correlations in the super Tonks-Girardeau Gas 123 6.1 Interaction Quench and the sTG gas . . . 124
6.2 Excitations in the sTG gas. . . 126
6.3 Momentum Space . . . 127
6.4 Real Space . . . 130
6.5 Conclusions . . . 131
7 Conclusions and Future Ideas 133 A Notation and Convention 135 A.1 Notation . . . 135
A.2 Fourier Transform . . . 135
A.3 Principal Value Integrals . . . 137
B Special Functions and their Properties 139 C Correlations in the weakly and strongly interacting regimes 143 C.1 Bogolyubov approximation . . . 143
C.2 Free Fermionic Gas . . . 147
Bibliography 150
Acknowledgments 159
Summary 161