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The coupling constant $g_{1d}$ is related to $g$ by $g_{1d} \simeq g(1+C a/L)$, where $C$ is a constant of the order of unity. It turns out that the correction to the energy density, given by the $g C a/L$ term in the coupling constant, is negligible in the limit $\xi \ll 1$.

SUPPLEMENTAL MATERIAL
=====================

In this Supplemental Material we discuss the details of the derivation of Eqs. (7) and (15) from the main paper. We show how to properly handle the sums and integrals in the Lee-Huang-Yang energy in order to arrive at a correct finite result.

Crossover from 3D to quasi-2D {#crossover-from-3d-to-quasi-2d .unnumbered}
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We start from the main equation for the Lee-Huang-Yang energy, i.e., Eq. (6) from the main. We thus have: 