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Contents
Introduction
Dilute Bose gas
Introduction
Mean-field description: Gross-Pitaevskii equation
Dilute Bose gas
Gross-Pitaevskii equation
Ground state energy
Elementary excitations
Beyond mean-field: Bogoliubov theory
Bogoliubov Hamiltonian and elementary excitations
Ground state energy
Quantum depletion of the condensate
One-body density matrix
Dilute Bose gas with disorder: perturbation expansion
Introduction
Bogoliubov theory in the presence of disorder
Random external potential
Diagonalization of the Hamiltonian
Ground state energy
Quantum depletion of the condensate
One body density matrix
Superfluid density
Connection between and the transverse current-current response function
Superfluid fraction in the presence of disorder
calculation of the superfluid fraction from GPE
Quantum Monte Carlo Method
Diffusion Monte Carlo
Introduction
Schrödinger equation
Green's function
DMC algorithm
Parallel DMC algorithm
Homogeneous Bose Gas
Trial Wavefunction
Comparison between VMC and DMC methods
Outputs of the calculation
Energy
Superfluid density
One body density matrix and condensate fraction
Extrapolation technique from mixed and variational estimators
Systematic errors
Population of walkers
Time step
Finite size errors
Other sources of errors
Dilute Bose gas with disorder: a Diffusion Monte Carlo study
Introduction
Trial wavefunction
Average over disorder
Distribution of impurities
Dependence on the number of disorder realizations
Results
Ground-state energy
Superfluid density and condensate fraction
Scaling behaviour
Shape of the one-body density matrix
Quantum phase transition
Bibliography
Appendix
Notation for Fourier transforamtion
useful formulae
Series expansion of the OBDM
Acknowledgements
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