Stability criterion for superfluidity based on the density spectral function

Abstract

We study a stability criterion hypothesis for superfluids expressed in terms of the local density spectral function I n (r,ω) that is applicable to both homogeneous and inhomogeneous systems. We evaluate the local density spectral function in the presence of a one-dimensional repulsive or attractive external potential within Bogoliubov theory, using solutions for the tunneling problem. We also evaluate the local density spectral function using an orthogonal basis, and calculate the autocorrelation function C n (r,t). When superfluids in a d-dimensional system flow below a threshold, I n (r,ω) ∝ ω d holds in the low-energy regime and C n (r,t) ∝ 1/t d+1 holds in the long-time regime. However, when superfluids flow with the critical current, I n (r,ω) ∝ ω β holds in the low-energy regime and C n (r,t) ∝ 1/t β+1 holds in the long-time regime with β < d. These results support the stability criterion hypothesis recently proposed.

Identities and Many-Body Approaches in Bose-Einstein Condensates

“…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]…”

“…where ⊗ is the Kronecker product. Since the normal and anomalous Green's function has the opposite sign in the low-energy regime, i.e., g ij (p) ≃ (−1) i+j mc 2 0 /(ω 2 − c 2 0 p 2 ), where the Bogoliubov phonon sound speed is c 0 = U n 0 /m, the infrared divergence of this correlation function Π has a simple structure, which can be extracted by the following matrix [15][16][17]…”

“…The correlation function Π R ≡ Π − Π IR converges to a finite value in the lowenergy limit. Using this correlation function Π, the four point vertex Γ and the regular part of the density response function χ R are respectively given by [15][16][17]…”

“…7 Finally, we discuss many-body approaches satisfying these identities. [15][16][17] Development of approximation frameworks for BEC is not completed, in the sense that there is not a practical approximation satisfying all the exact relations in BECs. We hope that this paper helps to guide readers towards theories on BECs, and is useful for developing them.…”