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In the vicinity of continuous quantum phase transitions (QPTs), quantumsystems become scale-invariant and can be grouped into universality classescharacterized by sets of critical exponents. We have found that despitescale-invariance and universality, the experimental data still containinformation related to the microscopic processes and scales governing QPTs. Weconjecture that near QPTs, various physical quantities follow the genericexponential dependence predicted by the scaling theory of localization; thisdependence includes as a parameter a microscopic seeding scale of therenormalization group, $L_0$. We also conjecture that for interacting systems,the temperature cuts the renormalization group flow at the length travelled bya system-specific elementary excitation over the life-time set by the Planckiantime, $\tau_P$=$\hbar/k_BT$. We have adapted this approach for QPTs in severalsystems and showed that $L_0$ extracted from experiment is comparable tophysically-expected minimal length scales, namely (i) the mean free path formetal-insulator transition in doped semiconductors, (ii) the distance betweenspins in Heisenberg and Ising chains, (iii) the period of an optical latticefor cold atom boson gases, and (iv) the period of a moir\'e superlattice forthe Mott QPT in dichalcogenide bilayers. In the first companion paper, we showthat in superconducting films and nanowires, as well as in the high temperaturesuperconductor La$_{1.92}$Sr$_{0.08}$CuO$_4$, $L_0$ is comparable tosuperconducting coherence length. In the second companion paper, we show thatin quantum Hall systems, $L_0$ is comparable to the magnetic length. Thedeveloped theoretical approach quantitatively explains and unifies a large bodyof experimental data and can be expanded to other complex systems

Microscopic scale of quantum phase transitions: from doped semiconductors to spin chains, cold gases and moir\'e superlattices