Eq. (\[e2d\]) with the last term neglected. This agreement provides an important and independent test of our approach.

*LHY energy at the 3D-1D crossover.*—We now focus on the 3D-1D crossover regime where the LHY energy is: $$\label{LHY_1d}
  e_\mathrm{LHY}^\mathrm{1d}(\xi) \!=\!   \lim_{r \rightarrow 0} \frac{\partial }{\partial r} \left(\! r \frac{1}{2}\sum_{q_x,q_y} \int {\rm d} q_z 
  \, e^{i{\bf q} {{\bf r}}} \left( \varepsilon_{\bf q} \!-\! A_{\bf q} \right) \!\right),$$ where $q_{x,y}$ are integers, $q_{x,y}=(L_{\perp}/2\pi)k_{x,y}$, and $q_z$ is a real-valued dimensionless momentum, $q_z=(L/2\pi) k_z$. The Bogoliubov’s energies expressed in the units of $\varepsilon_0$ have the same form as in the 3D-2D system. Similarly as before, we took $L_{\perp} \rightarrow \infty$, and, thus, we substituted: $\frac{1}{V}\sum_{{\bf k}}\rightarrow \frac{1}{2\pi} \int d k \frac{1}{L^2} \sum_{k_x,k_y} $. For small $\xi$, we obtain (see the Supplementary Material) $$\label{e1d}
  e_\mathrm{LHY}^\mathrm{1d} =  - \frac{2\sqrt{2}}{3} \xi^{3/2} + c_2 \xi^2 + c_3 \xi^3,$$ where $c_2 = \frac{1}{4} \left(  \int {\rm d} {\bf n} \,  1/n^2 - \sum_{n_y,n_z \neq 0} \int {\rm d} n_x   1/n^2    \right) 
\simeq 3.06$ and $c_3 = \frac{\pi}{8} \sum_{n_x,n_y \neq 0} (n_x^2+n_y^2)^{-3/2} \simeq 3.55 $.

In Fig. \[fig2\], we plot the ratio $ s^\mathrm{1d}(\xi)= e_\mathrm{LHY}^\mathrm{1d}(\xi)/e_\mathrm{LHY}^\mathrm{3d}(\xi)$. As before, the analytic approximate expression for the 3D-1D LHY term almost perfectly matches the full 