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Table of contents
General introduction
Contents
1 Bosons in the path-integral formalism
1.1 Ideal Bose gas
1.1.1 Free density matrix and permutations
1.1.2 Path integrals
1.2 Interacting Bose gas
1.2.1 Trotter approximation
1.2.2 Pair-product approximation
1.2.3 Interaction boxes
Appendix 1.A Metropolis algorithm
2 The repulsive weakly-interacting Bose gas
2.1 The mean-field weakly-interacting Bose gas
2.1.1 Scattering length
2.1.2 Trapped bosons
2.2 Beyond-mean-field corrections
2.2.1 The Lee–Huang–Yang equation of state
2.2.2 Path-integral simulation and comparison to experiments .
2.2.3 Numerical and experimental grand-canonical equation of state .
2.3 Conclusion
3 Efimov trimers in the path-integral formalism
3.1 Unitary limit
3.1.1 Square-well interaction potential
3.1.2 Zero-range interaction potential
3.2 Efimov effect
3.2.1 Regularization of the zero-range interaction potential
3.2.2 Path-integral argument
3.2.3 Numerical simulation of a single Efimov trimer
3.2.4 Universality of Efimov trimers
3.3 Conclusion
Appendix 3.A Path-integral argument for the dimer state
Appendix 3.B Calculation of the pair-product zero-range correction factor
Appendix 3.C Hyperspherical coordinates
4 The unitary Bose gas
4.1 High-temperature equation of state
4.1.1 Many-body simulation
4.1.2 Equation of state
4.2 Phase diagram of the unitary Bose gas
4.2.1 Transition to the Efimov liquid phase
4.2.2 Simple model for the transition to the Efimov liquid phase
4.2.3 Homogeneous phase diagram
4.3 Unitary liquid in experimental systems
4.3.1 Experimentally accessible regions of the phase diagram
4.3.2 Experimental observables
4.4 Conclusion
Appendix 4.A Path integrals and momentum distribution
4.A.1 Off-diagonal density matrix and momentum distribution
4.A.2 Momentum distribution of two interacting bosons
4.A.3 Simple algorithm to obtain the momentum distribution
General conclusion
Bibliography
