Beliaev technique for a weakly interacting Bose gas

Aiming for simplicity of explicit equations and at the same time controllable accuracy of the theory we present results for all thermodynamic quantities and correlation functions for the weakly interacting Bose gas at short-to-intermediate distances obtained within an improved version of Beliaev's diagrammatic technique. With a small symmetry breaking term Beliaev's diagrammatic technique becomes regular in the infrared limit. Up to higher-order terms (for which we present order-of-magnitude estimates), the partition function and entropy of the system formally correspond to those of a non-interacting bosonic (pseudo-)Hamiltonian with a temperature dependent Bogoliubov-type dispersion relation. Away from the fluctuation region, this approach provides the most accurate--in fact, the best possible within the Bogoliubov-type pseudo-Hamiltonian framework--description of the system with controlled accuracy. It produces accurate answers for the off-diagonal correlation functions up to distances where the behaviour of correlators is controlled by generic hydrodynamic relations, and thus can be accurately extrapolated to arbitrarily large scales. In the fluctuation region, the non-perturbative contributions are given by universal (for all weakly interacting U(1) systems) constants and scaling functions, which can be obtained separately--by simulating classical U(1) models--and then used to extend the description of the weakly interacting Bose gas to the fluctuation region. The theory works in all spatial dimensions and we explicitly check its validity against first-principle Monte Carlo simulations for various thermodynamic properties and the single-particle density matrix.