= Infrared phase fluctuations of a homogeneous Bose gas
{title2=$\langle|\phi_{\mathbf q}|^2\rangle\propto T/q^2\ (T>0),\quad\propto1/|q|\ (T=0)$}
The <Bose-gas phase-only action> gives a thermal infrared integral $\int d^dq/q^2$ and a zero-temperature equal-time integral $\int d^dq/|q|$. Thus an infinite uniform short-range interacting gas has no true finite-temperature condensate in $d\le2$, and the one-dimensional <ground state> also lacks true condensate <long-range order>. This does not exclude local density coherence, a finite-size condensate description or two-dimensional algebraic order and <superfluidity>.<Existence of Long-Range Order in One and Two Dimensions by Pierre Hohenberg>[abstract; nonzero-temperature theorem]{ref}
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