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We discuss standard and tighter upper bounds on the critical temperature$T_c$ of two-dimensional (2D) superconductors and superfluids versus particledensity $n$ or filling factor $\nu$ for continuum and lattice systems from theBardeen-Cooper-Schrieffer (BCS) to Bose regime. We discuss only one-bandHamiltonians, where the transition from the normal to superconducting(superfluid) phase is governed by Berezinskii-Kosterlitz-Thouless (BKT)mechanism of vortex-antivortex binding, such that a direct relation between thesuperfluid density tensor and $T_c$ exists. We demonstrate that it isimperative to consider at least the full effect of phase fluctuations of orderparameter for superconductivity (superfluidity) to establish tighter bounds.Using the renormalization group, we obtain phase-fluctuation criticaltemperature $T_c^{\theta}$, a much tighter upper bound to critical temperaturesupremum $T_c^{\rm sup}$ than standard critical temperature upper bound$T_c^{\rm up1}$ from Ferrell-Glover-Tinkham sum rule. We go beyond textbookphase-fluctuation theories and show that chemical potential renormalization,order parameter equation, and Nelson-Kosterlitz relation need to be solvedself-consistently and simultaneously with renormalization group flow equationto produce $T_c^{\theta}$ over the entire BCS-Bose crossover. We note that ananalytic theory including modulus fluctuations of the order parameter validthroughout the BCS-Bose evolution is still lacking, but the inclusion ofmodulus fluctuations can only produce a critical temperature lower than$T_c^{\theta}$ and thus produce an even tighter bound to $T_c^{\rm sup}$. Weconclude by indicating that if the measured critical temperature exceeds$T_c^{\theta}$ in experiments involving 2D single-band systems, then a non-BKTmechanism must be invoked to describe the superconducting (superfluid)transition.

Tighter upper bounds on the critical temperature of two-dimensional superconductors and superfluids from the BCS to the Bose regime