Unified Picture of Superfluidity: From Bogoliubov’s Approximation to Popov’s Hydrodynamic Theory

Using a nonperturbative renormalization-group technique, we compute the momentum and frequency dependence of the anomalous self-energy and the one-particle spectral function of two-dimensional interacting bosons at zero temperature. Below a characteristic momentum scale $k_G$, where the Bogoliubov approximation breaks down, the anomalous self-energy develops a square-root singularity and the Goldstone mode of the superfluid phase (Bogoliubov sound mode) coexists with a continuum of excitations, in agreement with the predictions of Popov’s hydrodynamic theory. Thus our results provide a unified picture of superfluidity in interacting boson systems and connect Bogoliubov’s theory (valid for momenta larger than $k_G$) to Popov’s hydrodynamic approach.

Figure 1

$\lambda /g$, $Z_C$, $V_A/V_A^{*}$, and $c/c{|}_{k=0}$ vs $\ln (k_G/k)$, where $k_G=\sqrt{(gm{)}^3\overline{n}}/4\pi$, for $\overline{n}=0.01$ and $2mg=0.1$ [$\ln (k_G/k_h)\simeq -5.87$]. The inset shows $k_G$ vs $2mg$ obtained from the criterion $V_A{|}_{k_G}=V_A^{*}/2$ [the green solid line is a fit to $k_G\propto (2mg{)}^{3/2}$]. All figures are obtained with $\Lambda =1$, $2m=1$, and the regulator $R_k(\mathbf{p})=Z_{A,k}{ϵ}_{\mathbf{p}}/(e^{{\mathbf{p}}^2/k^2}-1)$.

Figure 2

Real and imaginary parts of the retarded vertex ${\Gamma }_B^R(\mathbf{p},\omega )={\Sigma }_{\mathrm{an}}^R(\mathbf{p},\omega )/n_0$ for various values of $|\mathbf{p}|$ ranging from $0.3k_G$ up to $170k_G\sim k_h/2$ ($\overline{n}=0.01$, $2mg=0.1$, and $k=0$). The Bogoliubov approximation corresponds to ${\Gamma }_B^R(\mathbf{p},\omega )=g=0.1$. ($\Re [{\Gamma }_B^R]\ge 0$ and $\Im [{\Gamma }_B^R]\le 0$.)

Figure 3

Same as Fig. 2 but for $|\mathbf{p}|/k_G\simeq 0.036$. The dotted lines show the analytical result (9) obtained from the approximation (8). The latter is shown by the blue dash-dotted line, while the blue crosses give the numerical results for ${\Gamma }_B(\mathbf{p},i\omega )$ vs the rescaled Matsubara frequency $\omega /(c^{*}|\mathbf{p}|)$. The inset shows ${\Gamma }_B^R(\mathbf{p},\omega )$ on a much larger energy scale ($\omega \le 10c^{*}{k}_G\sim 280c^{*}|\mathbf{p}|$; the square-root singularity at $\omega \ll c^{*}{k}_G$ is not visible on this plot).

Figure 4

Longitudinal spectral function $A_{\mathrm{ll}}(\mathbf{p},\omega )$ for $|\mathbf{p}|/k_G\simeq 0.036$ and the same parameters as in Fig. 3. The red solid line is the result obtained from the Padé approximant, while the green dashed line is obtained from the analytic expression (12). The inset shows the ratio between $A_{\mathrm{ll}}(\mathbf{p},\omega )$ and the approximate result (12) on a larger energy scale.