Green's-function formalism for a condensed Bose gas consistent with infrared-divergent longitudinal susceptibility and Nepomnyashchii-Nepomnyashchii identity

Abstract

We present a Green's function formalism for an interacting Bose-Einstein condensate (BEC) satisfying the two required conditions: (i) the infrared-divergent longitudinal susceptibility with respect to the BEC order parameter, and (ii) the Nepomnyashchii-Nepomnyashchii identity stating the vanishing off-diagonal self-energy in the low-energy and low-momentum limit. These conditions cannot be described by the ordinary mean-field Bogoliubov theory, the many-body T -matrix theory, as well as the random-phase approximation with the vertex correction. In this paper, we show that these required conditions can be satisfied, when we divide many-body corrections into singular and non-singular parts, and separately treat them as different self-energy corrections. The resulting Green's function may be viewed as an extension of the Popov's hydrodynamic theory to the region at finite temperatures. Our results would be useful in constructing a consistent theory of BECs satisfying various required conditions, beyond the mean-field level.

Identities and Many-Body Approaches in Bose-Einstein Condensates

“…It is clearly seen that the anomalous self-energy is gauge dependent, and we can find Σ 12 (0) = 0. 16 On the other hand, the normal self-energy is gauge independent, and Σ 11 (0) = 0. It is also the case for the chemical potential.…”

“…9 Although almost all infrared divergences are canceled out each other, some contributions remains, which is the same contribution as that in (47). 9,16 On the other hand, as discussed in the topic of the Hugenholtz-Pines identity, the vertex functions in the zero-energy limit is generated from the system energy by eliminating the condensate wave-functions. As a result, three-point vertices in the low-energy limit can be related to the two-point vertices (i.e., self-energies), and one can obtain an equation with respect to the anomalous self-energy.…”

“…By solving this equation, one can find the anomalous self-energy vanishes in the zero-momentum and zero-energy limits, which is caused by the inverse of the weak-infrared divergence shown in (47). 9,16 As related to the fact that the self-energy in the zeroenergy limit can be generated from the energy functional, the self-energy with the small but nonzero momentum and that with the small but nonzero energy is also generated by the similar way. 7 The self-energy contribution in the low-energy regime is thus related to the thermodynamic quantities.…”

“…This section focuses on many-body approaches at nonzero temperatures, such as the random-phase approximation, and the many-body T -matrix theory. [15][16][17] For simplicity, we take the convention V = = k B = 1 in this section. It may be convenient to construct building blocks for many-body contributions from the single-particle Green's function in the Hartree-Fock-Bogoliubov-Popov approximation (Shohno model), 10 given by g(p) = g 11 g 12…”

“…This gives the Bogoliubov excitation -the gapless phonon excitation in the low-energy limit -, which captures properties of the exact single-particle Green's function. One of the building blocks is the correlation function, given by [15][16][17]…”