Hidden Reentrant and Larkin-Ovchinnikov-Fulde-Ferrell Superconducting Phases in a Magnetic Field in (TMTSF)$_{2}$ClO$_{4}$

We solve a long-standing problem about a theoretical description of the upper critical magnetic field, parallel to conducting layers and perpendicular to conducting chains, in (TMTSF)$_{2}$ClO$_{4}$ superconductor. In particular, we explain why the experimental upper critical field, H$^{b}_{c2}$ = 6T, is higher than both the quasi-classical upper critical field and Clogston paramagnetic limit. We show that this property is due to the coexistence of the hidden Reentrant and Larkin-Ovchinnikov-Fulde-Ferrell phases in a magnetic field in a form of three plane waves with non-zero momentums of the Cooper pairs. Our results are in good qualitative and quantitative agreement with the recent experimental measurements of H$^{b}_{c2}$ and support a singlet d-wave-like scenario of superconductivity in (TMTSF)$_{2}$ClO$_{4}$.