Using Popov’s hydrodynamic approach we derive an effective Euclidean action for the longwavelength phase fluctuations of superfluid Bose gases in D dimensions.We then use this action to calculate the damping of phase fluctuations at zero temperature as a function of D. For D > 1 and wavevectors |k| 2mc (where m is the mass of the bosons and c is the sound velocity) we find that the damping in units of the phonon energy E_k = c|k| is to leading order γ_k/E_k = A_D(k_o~D /2πρ)(|k|/k_0)~(2D?2), where ρ is the boson density and k0 = 2mc is the inverse healing length. For D → 1 the numerical coefficient AD vanishes and the damping is proportional to an additional power of |k|/k0; a self-consistent calculation yields in this case γ_k/E_k = 1.32 (k0/2πρ)~(1/2)|k|/k0. In one dimension, we also calculate the entire spectral function of phase fluctuations.
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机译：使用Popov的流体力学方法，我们得出了D维度上超流体Bose气体的长波相位波动的有效欧几里得作用，然后使用该作用来计算零温度下随D波动的相位波动的阻尼。对于D> 1和波矢| k | 2mc（其中m是玻色子的质量，c是声速），我们发现以声子能量E_k = c | k |为单位的阻尼。是前导γ_k/ E_k = A_D（k_o〜D /2πρ）（| k | / k_0）〜（2D？2），其中ρ是玻色子密度，k0 = 2mc是逆愈合长度。当D→1时，数值系数AD消失，阻尼与| k | / k0的乘方成正比;在这种情况下，自洽计算得出γ_k/ E_k = 1.32（k0 /2πρ）〜（1/2）| k | / k0。在一个维度上，我们还计算了相位波动的整个频谱函数。
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