= Solution
For a uniform time-independent saddle, extremizing the <action> gives $(-\mu+g|\psi_0|^2)\psi_0=0$. Its grand-potential density is $f(\rho)=-\mu\rho+g\rho^2/2$, with $\rho=|\psi_0|^2\ge0$. Since $g>0$, the minimum at positive density for $\mu>0$ is
$$
\boxed{\rho_0=\mu/g,\qquad
\psi_0=\sqrt{\rho_0}\,e^{i\phi_0},\qquad
f_{\min}=-\mu^2/(2g).}
$$
For $\mu\le0$ the uniform mean-field minimum is the vacuum, rather than a positive-density condensate. At fixed nonzero density in the weakly interacting condensed regime, the saddle relation sets $\mu\simeq g\rho_0$. The mean-field one-body <density matrix> factorizes as $\langle a^\dagger(\mathbf r)a(\mathbf r')\rangle\simeq\rho_0$, giving macroscopic occupation of the uniform mode. Depletion corrections are omitted at this stage.
The <action> has the global particle-number <symmetry> $\psi\mapsto e^{i\alpha}\psi$, $\bar\psi\mapsto e^{-i\alpha}\bar\psi$. Choosing one saddle phase breaks this $U(1)$ <symmetry>; the degenerate mean-field minima form the circle \b[$U(1)\simeq S^1$]. There is no energetic preference for a uniform value of $\phi_0$, but spatially varying phase costs <gradient energy>. Consequently a neutral condensate has a gapless <Goldstone mode> and a nonzero <phase stiffness>. The mode becomes sound once density dynamics is included. With $\hbar=1$ the <superfluid velocity> is $\nabla\phi/m$, so a single-valued field allows circulation in units $2\pi/m$ around a closed loop that avoids zeros of the amplitude. This phase coherence and rigidity are the physical content of the broken-symmetry description.
The claim is a mean-field saddle statement, not an assertion that every $d$ and <temperature> in the question supports true <long-range order>. In a finite number-conserving system $\langle a\rangle=0$; the symmetry-breaking <order parameter> is defined with the <thermodynamic limit> taken before an infinitesimal source is removed. In a homogeneous infinite short-range gas at nonzero <temperature>, infrared phase fluctuations prohibit true condensation in $d\le2$. In one dimension they also destroy true condensate <long-range order> at zero <temperature>. A two-dimensional gas can nevertheless have <superfluidity> and algebraically decaying correlations below a <Berezinskii–Kosterlitz–Thouless transition>. A finite trap can have substantial coherence without contradicting these thermodynamic statements.<Existence of Long-Range Order in One and Two Dimensions by Pierre Hohenberg>[abstract and pp.383–386; finite-temperature restriction]{ref}
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