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, $H^{b'}_{c2} \simeq 6 \ T$, 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 momenta 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$_4$.

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