In previous studies, we proposed a scaling ansatz for electron-electron interactions under renormalization group transformation. With the inclusion of phonon-mediated interactions, we show that the scaling ansatz, characterized by the divergent logarithmic length l d and a set of renormalizationgroup exponents, also works rather well. The superconducting phases in a doped two-leg ladder are studied and classified by these renormalization-group exponents as demonstration. Finally, nontrivial constraints among the exponents are derived and explained.PACS numbers: 71.10. Fd, 71.10.Hf, 71.27.+a, 71.10.Pm Even though superconductivity has been observed [1] for one hundred years, the mechanism turning electrons into pairs remains an open question. In conventional superconductors, phonons mediate effective attractions [2] between electron in the low-energy limit and lead to Cooper pair formation near the Fermi surface. On the other hand, for unconventional superconductors[3] like cuprates, phonon-mediated interactions seem to play a secondary role while the electronic correlations are believed to reigns. It is speculated that the spin fluctuations is the key to the pairing mechanism (and perhaps the pairing symmetry as well). However, it hasn't been fully understood how Coulomb repulsion eventually glues electrons into pairs.The discovery of iron-based superconductors keeps the puzzling charm going. Collecting from experimental observations, it is believed that electronic correlations in these materials are important but much weaker than those in cuprates. Though the renormalization-group (RG) analysis[4-6] for electron-electron interactions delivers the correct pairing symmetry, the isotope effects measured in laboratories show conflicting results. In addition, the iron-based superconductors are distinct from the cuprates due to the presence of multiple active bands.Inspired by the anomalous isotope effects in iron-based superconductors, it is helpful to investigate the competitions between electron-electron [7,8] and electron-phonon interactions [9][10][11] by RG analysis. The major difficulty lies in the enormous couplings for all allowed interactions. Besides, as the number of couplings grow, reading out the desired messages from RG flows can be challenging as well. Recently, we found a scaling ansatz [12], characterized by a set of RG exponents, for electron-electron interactions in many correlated systems. These RG exponents build a clear hierarchy of relevant couplings and serve as an unambiguous indicator for the ground-state instabilities. We are curious whether similar scaling ansatz exists even when the retarded interactions mediated by phonons are included.In this Rapid Communications, we show that the scaling ansatz works in the presence of both electron-electron and electron-phonon interactions and the patterns of the extracted RG exponents describe the ground states and also the quantum phase transitions between them. To make the discussions concrete, we study the exemplar two-leg ladder shown in Fig....
