Holographic superfluid flows with a localized repulsive potential

We investigate a holographic model of superfluid flows with an external repulsive potential. When the strength of the potential is sufficiently weak, we analytically construct two steady superfluid flow solutions. As the strength of the potential is increased, the two solutions merge into a single critical solution at a critical strength, and then disappear above the critical value, as predicted by a saddle-node bifurcation theory. We also analyze the spectral function of fluctuations around the solutions under a certain decoupling approximation.

Figure 1

The plot of $gl$ as a function of $\kappa l$ and $v_{0}l$. When $v_0$ is constant, there are two solutions for $\kappa$ as long as $g$ is less than a critical value $g_c$.

Figure 2

The band structure given by the repulsive potential (5). The shaded region is the allowed region in which Eq. (45) holds, and the blank region is the forbidden one. The dot-dashed (blue) curve represents the function $g$ given by Eq. (20) with $v_{0}l=\pi /2$, while the dashed (black) and solid (green) curves represent the ones with $v_{0}l=0.421\pi$ and $v_{0}l=0.225\pi$, respectively. The horizontal black dotted line expresses the local maximum value ($gl=0.663\pi$) of the blue dot-dashed curve. The six points on the horizontal dotted line correspond to ${\kappa }_{*}l/\pi ≔Tl{\widehat{\kappa }}_{*}(\widehat{\mu })$ for the value of the chemical potential $\mu l/\pi$ respectively, 2.060 (filled circle, red), 2.120 (filled square, green), 2.215 (filled diamond, black), 2.321 (filled triangle, blue), 2.400 (filled nabla, magenta), 2.519 (open circle, orange) when $Tl=1$ and $gl=0.663\pi$.

Figure 3

The log-log plot of the spectral function $\rho (\widehat{\omega })$ is shown for various chemical potentials $\mu l/\pi$ [2.060 (filled circle, red), 2.120 (filled square, green), 2.215 (filled diamond, black), 2.321 (filled triangle, blue), 2.400 (filled nabla, magenta), 2.519 (open circle, orange)] when $Tl=1$ and $gl=0.663\pi$. Note that these points with the colors red, green, black, blue, magenta, and orange correspond respectively to the points on the horizontal black dotted line with the same colors in Fig. 2.

Figure 4

Plots of ${\rho }_{\widehat{\kappa }}(\widehat{\omega })$ for various different values of $\widehat{\kappa }$ with ${\widehat{\kappa }}_{*}=0.35$, corresponding to the red plot (filled circle) in Fig. 3. The location of the peak shifts to the right, as $\widehat{\kappa }$ increases from $\widehat{\kappa }=0.423$ with the difference $\mathrm{\Delta }\widehat{\kappa }=0.08$ between every adjacent two peaks. When $Tl=1$, $\widehat{\kappa }\simeq 0.423$, which gives the peak closest to the origin, corresponds to, ${\kappa }_{1}l/\pi \sim 0.423$, the bottom of the first allowed band for $gl=0.663\pi$. One can find the disappearance of peaks around $\widehat{\omega }=0.32$ due to the band gap for $\widehat{\kappa }=1.063$ and 1.143.

Figure 5

Plots of ${\rho }_{\widehat{\kappa }}(\widehat{\omega })$ for various different values of $\widehat{\kappa }$ with ${\widehat{\kappa }}_{*}=0.50$, corresponding to the green plot (filled square) in Fig. 3. The plots with the same color in this figure and Fig. 4 are for the same values of $\widehat{\kappa }$. Due to the large value of ${\widehat{\kappa }}_{*}$ (or $\widehat{\mu }$) compared to that in Fig. 4, every peak is shifted to the left compared to the location of the corresponding peak in Fig. 4. The disappearance of peaks due to the band gap is also shifted to around $\widehat{\omega }=0.29$. Note also that there is a mode that has a peak arbitrarily close to $\omega =0$, as $0.423<{\widehat{\kappa }}_{*}$.

Figure 6

Plots of ${\rho }_{\widehat{\kappa }}(\widehat{\omega })$ for various different values of $\widehat{\kappa }$ with ${\widehat{\kappa }}_{*}=0.95$, corresponding to the magenta plot (filled nabla) in Fig. 3. The location of every peak is shifted further to the left, and the disappearance of peaks due to the band gap is now around $\widehat{\omega }=0.06$, which forms the slope around $\log (\omega /\pi T)=-1.25$ in Fig. 3.

Figure 7

Plots of ${\rho }_{\widehat{\kappa }}(\widehat{\omega })$ for various different values of $\widehat{\kappa }$ with ${\widehat{\kappa }}_{*}=1.10$, corresponding to the magenta plot (open circle) in Fig. 3. The disappearance of peaks due to the band gap is now around $\widehat{\omega }\sim 0$, and $\rho (\widehat{\omega }\sim 0^{+})$ takes vanishingly small values, corresponding to the behavior of the orange plot in Fig. 3.