We study the finite-temperature vestigial superconducting phases of a two-dimensional system of fluctuating spin-triplet pairing and spin magnetism. Denoting the respective primary order parameters by $\mathbf{ d}$ and $\mathbf{ N}$, which are not long-range ordered at finite temperature, the composite fields $\phi_{ dd} = \mathbf{ d}\cdot\mathbf{ d}$ and $\phi_{ dN} = \mathbf{ d}\cdot\mathbf{ N}$ are spin-rotation invariant and can condense at finite temperature. Using a large-$N$ approach that respects the Mermin-Wagner theorem, we here derive the phase diagram which features two vestigial superconductors: $(A)$ a charge-$4e$ superconductor with $\phi_{ dd}\neq 0$ and $\phi_{ dN} =0$ and $(B)$ a charge-$2e$ state with $\phi_{ dN} ,\phi_{ dd}\neq 0$. We analyze the temperature and coupling-constant dependent properties of these two superconductors using a perturbative approach and a variational Hartree-Fock study. This reveals non-trivial spectra in the superconductors, which result from the fundamental building blocks being distinct from the usual Cooper pairs--in phase $(A)$, the elementary bosons are bound states of four electrons and, in phase $(B)$, of three electrons and a hole. This work complements the previous study [Nat. Commun. 15, 1713 (2024), arXiv:2301.01344], which focused on the properties of phase $(B)$.
Categories: cond-mat.supr-con