Infrared Behavior of Dipolar Bose Systems at Low Temperatures

Abstract

We rigorously discuss the infrared behavior of the uniform three-dimensional dipolar Bose systems. In particular, it is shown that low-temperature physics of the system is controlled by two parameters, namely isothermal compressibility and intensity of the dipole–dipole interaction. By using a hydrodynamic approach, we calculate the spectrum and damping of low-lying excitations and analyze the infrared behavior of the one-particle Green’s function. The low-temperature corrections to the anisotropic superfluid density as well as condensate depletion are found. Additionally, we derive equations of the two-fluid hydrodynamics for dipolar Bose systems and calculate velocities of first and second sound.

Appendix

In this section, we present the explicit expressions for the imaginary parts of the exact second-order vertices in the low-length limit. The leading-order contribution is determined by all diagrams with only two vertices, namely $D_{\varphi \varphi \rho }(K,Q|P)$ and $D_{\rho \rho \rho }(K,Q,P)$. Feynman diagrams contributing to $\mathfrak {R}\Pi _{\varphi \rho }(\omega ,\mathbf{k})$ and $\mathfrak {I}\Pi _{\rho \rho }(\omega ,\mathbf{k})$ are given in Figs. 2 and 3, respectively. In the same manner, the low-energy behavior of $\mathfrak {I}\Pi _{\varphi \varphi }(\omega ,\mathbf{k})$ can be easily obtained from Fig. 1. For convenience terms contributing to the Beliaev damping

Fig. 2

Diagrammatic representation of the leading-order contribution to $\mathfrak {R}\Pi _{\varphi \rho }(\omega ,\mathbf{k})$. Crosses denote the spectral weights of the appropriate pair correlation functions (see Ref. [55] for details)

Fig. 3

Exact low-energy asymptotics of $\mathfrak {I}\Pi _{\rho \rho }(\omega ,\mathbf{k})$

$$\begin{aligned} \mathfrak {I}\Pi ^B_{\varphi \varphi }(\omega ,\mathbf{k})= & {} \frac{\pi }{4V}\sum _{\mathbf{q}\ne 0} \frac{\hbar ^4\mathbf{kq}}{m^2}\left[ \mathbf{kq}\frac{c_\mathbf{q}}{q} \frac{|\mathbf{q}+\mathbf{k}|}{c_\mathbf{q+k}}-\mathbf{k}(\mathbf{q}+\mathbf{k})\right] \nonumber \\&\delta (E_{\mathbf{q}+\mathbf{k}}+E_\mathbf{q}-\omega ), \end{aligned}$$

(7.1)

$$\begin{aligned} \mathfrak {I}\Pi ^B_{\rho \rho }(\omega ,\mathbf{k})= & {} \frac{\pi }{8V}\sum _{\mathbf{q}\ne 0} \frac{\hbar ^2q|\mathbf{q}+\mathbf{k}|}{\rho ^2c_\mathbf{q}c_\mathbf{q+k}}\left[ \frac{\mathbf{q}(\mathbf{q}+\mathbf{k})}{q|\mathbf{q}+\mathbf{k}|}c_\mathbf{q}c_{\mathbf{q}+\mathbf{k}}-\rho ^2\frac{\partial }{\partial \rho }\frac{c^2}{\rho }\right] ^2\nonumber \\&\delta (E_{\mathbf{q}+\mathbf{k}}+E_\mathbf{q}-\omega ), \end{aligned}$$

(7.2)

$$\begin{aligned} \mathfrak {R}\Pi ^B_{\varphi \rho }(\omega ,\mathbf{k})= & {} -\frac{\pi }{4V}\sum _{\mathbf{q}\ne 0} \frac{\hbar ^3\mathbf{kq}}{m\rho }\left[ \frac{\mathbf{q}(\mathbf{q}+\mathbf{k})}{q}c_\mathbf{q} -\frac{|\mathbf{q}+\mathbf{k}|}{c_\mathbf{q+k}}\rho ^2\frac{\partial }{\partial \rho }\frac{c^2}{\rho }\right] \nonumber \\&\delta (E_{\mathbf{q}+\mathbf{k}}+E_\mathbf{q}-\omega ), \end{aligned}$$

(7.3)

and the Landau damping

$$\begin{aligned} \mathfrak {I}\Pi ^L_{\varphi \varphi }(\omega ,\mathbf{k})= & {} -\omega \frac{\pi }{V}\sum _{\mathbf{q}\ne 0} \left[ \frac{\hbar ^2\mathbf{kq}}{m}\right] ^2\left\{ \frac{\partial }{\partial E_\mathbf{q}}n(\beta E_\mathbf{q})\right\} \delta (E_{\mathbf{q}+\mathbf{k}}-E_\mathbf{q}-\omega ), \end{aligned}$$

(7.4)

$$\begin{aligned} \mathfrak {I}\Pi ^L_{\rho \rho }(\omega ,\mathbf{k})= & {} -\omega \frac{\pi }{4V}\sum _{\mathbf{q}\ne 0}\left( \frac{E_\mathbf{q}}{\rho }\right) ^2 \left[ 1+\frac{\rho ^2}{c^2_\mathbf{q}}\frac{\partial }{\partial \rho }\frac{c^2}{\rho }\right] ^2\left\{ \frac{\partial }{\partial E_\mathbf{q}}n(\beta E_\mathbf{q})\right\} \nonumber \\&\delta (E_{\mathbf{q}+\mathbf{k}}-E_\mathbf{q}-\omega ), \end{aligned}$$

(7.5)

$$\begin{aligned} \mathfrak {R}\Pi ^L_{\varphi \rho }(\omega ,\mathbf{k})= & {} \omega \frac{\pi }{2V}\sum _{\mathbf{q}\ne 0}\frac{\hbar ^2 \mathbf{kq}}{m}\frac{E_\mathbf{q}}{\rho }\left[ 1+\frac{\rho ^2}{c^2_\mathbf{q}}\frac{\partial }{\partial \rho }\frac{c^2}{\rho }\right] \left\{ \frac{\partial }{\partial E_\mathbf{q}}n(\beta E_\mathbf{q})\right\} \nonumber \\&\delta (E_{\mathbf{q}+\mathbf{k}}-E_\mathbf{q}-\omega ), \end{aligned}$$

(7.6)

are written separately.