Flow equations for cold Bose gases

Abstract

We derive flow equations for cold atomic gases with one macroscopically populated energy level. The generator is chosen such that the ground state decouples from all other states in the system as the renormalization group flow progresses. We propose a self-consistent truncation scheme for the flow equations at the level of three-body operators and show how they can be used to calculate the ground state energy of a general N-body system. Moreover, we provide a general method to estimate the truncation error in the calculated energies. Finally, we test our scheme by benchmarking to the exactly solvable Lieb-Liniger model and find good agreement for weak and moderate interaction strengths.

Impurities in a one-dimensional Bose gas: the flow equation approach

“…For convenience of the reader, we shall present in this chapter the main ingredients of the IM-SRG method for bosons, see also Ref. [1], Appendix A, and Fig. 1.…”

“…To normal order a Hamiltonian with one-and two-body operators, we define the contractions 2 , following Ref. [1] :…”

“…The function f 1b is the ground-state wave function of the one-boson Hamiltonian as in Refs. [1,2]. The second function is obtained within a mean-field approximation, i.e., f GP is the solution of the Gross-Pitaevskii equation.…”

“…A related method, the in-medium similarity renormalization group (IM-SRG), was recently developed and successfully used in nuclear physics (see, e.g., Refs. [4,5]) 1 . The main concept behind both methods is the same.…”

“…IM-SRG was recently extended to cold Bose gases [1,2]. It was tested by calculating the ground-state energies of the Lieb-Liniger model and a one-dimensional (1D) Bose gas with an impurity atom ('Bose polaron') [1,2]. Those works demonstrate that flow equations allow one to go beyond mean-field approximation without relying on many-body perturbation theory.…”