Vestigial pairing from fluctuating magnetism and triplet superconductivity

TL;DR

The paper investigates vestigial order arising from fluctuating spin-triplet superconductivity and spin magnetism in a two-dimensional system. It develops a bosonic effective theory with composite order parameters $φ_{dd}$ and $φ_{dN}$, then applies a large-$N$ expansion to derive a phase diagram with two vestigial superconductors: (A) a charge-4e state with $φ_{dd}\\neq 0$, $φ_{dN}\\=0$, and (B) a charge-2e state with $φ_{dN}\\neq 0$ and $φ_{dd}\\neq 0$, ensuring the primary orders do not condense at finite temperature. The electronic sector is treated perturbatively and via Hartree-Fock mean-field theory to reveal distinctive spectral features: in phase (A) the four-electron bound-state pairing leads to unique self-energies and possible subgap structures, while phase (B) exhibits a conventional-like but still vestigial two-electron pairing with a hard gap in certain regimes. Finite-momentum transfer is explored to show how spectral features are washed out, with implications for moiré-material platforms and guidance for future momentum-dependent extensions and numerical studies.

Abstract

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 $φ_{dd} = \mathbf{d}\cdot\mathbf{d}$ and $φ_{dN} = \mathbf{d}\cdot\mathbf{N}$ 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 $φ_{dd}\neq 0$ and $φ_{dN} =0$ and $(B)$ a charge-$2e$ state with $φ_{dN} ,φ_{dd}\neq 0$. 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 $(B)$.

Figures (8)

• Figure 1: (a) Visual representation of Eq. (\ref{['SEToSolve']}) with red (blue) referring to left-hand (right-hand) side. (b) Solution of Eq. (\ref{['SEToSolve']}) with lower free energy as a function of $b_2/b_1$ for $r_d'=1$ and $b_1=1$. For $b_2>b_1$, the theory is unstable. The inset is a zoom-in on the transition region. (c) Phase diagram for $\phi^0_{dN} = 0$, where solid (dashed) red lines refer to first (second) order phase transitions. Here, we have rescaled all parameters in Eq. (\ref{['SEToSolve']}) so that the phase diagram depends only on the single parameter $b_1/b_2$. As explained in the text, $r_d'$ can be thought of as (being in a monotonic relation with) temperature within our theory.
• Figure 2: Comparison between (a) the large-$N$ phase diagram in the classical limit and (b) the phase diagram derived by minimizing $V$ in Eq. (\ref{['Potential']}). We set $b_1=1$, ${b_3=4.3}$, ${r_d'=0.8}$ and ${r_{N}'=0.7}$. Solid (dashed) red lines refer to first (second) order phase transitions. Lowering temperature in (a) leads to an expansion of the superconducting phases $(A)$ and $(B)$ and the transition between $(A)$ and $(C)$ can become second order as well.
• Figure 3: Diagrams for the perturbative analysis of the electronic spectral function, see text.
• Figure 4: Numerical results for the spectral functions, where the self energy was evaluated including all Matsubara frequencies in the limit of high temperatures and long correlation lengths. (a) Shows the spectral function for the action $S_2$ which describes superconducting triplet fluctuations. One can see an increase in the Fermi velocity as well as life-time broadening near the Fermi surface. (b) Shows the spectral function for the action $S_1$ which describes spin fluctuations. There is no renormalization of the Fermi-velocity, but a life-time broadening for all momenta. (c) Shows the combined effect of $S_1+S_2$ on the spectral function. As expected, there is life-time broadening for all momenta, along with a renormalization of the Fermi-velocity near the Fermi-surface. We chose $\tilde{r}_d/\tilde{r}_N=1$, $v_{d}/v_F=v_{N}/v_F=0.4$, and $\phi^0_{dd}=0$.
• Figure 5: Results for phase $(A)$ with $|g_2/g_{\phi}|=2$, i.e., $T^* > T_c$. In (a) we show the behavior of the maximum value of the superconducting order parameter $\Delta_{\boldsymbol{k}}$ with respect to the temperature. The solutions for the self-consistent Hartree-Fock calculations are shown in (b)-(d). The superconducting order parameter (b) is non zero for larger regions around the Fermi surface for lower temperatures. The renormalized dispersion (c) has a discontinuity for temperatures below $T^{*}$, which can also be seen in the dispersion of the excitations (d) where a gap is found for $T<T^{*}$. For temperatures below $T_c$ one can see that the dispersion becomes completely flat in the region where $\Delta_{\boldsymbol{k}}\neq 0$. (e) The DOS for temperatures in the three regimes $T>T^*$ (black), $T_c<T<T^*$ (green) and $T<T_c$ (red). For $T<T_c$ one obtains $\delta$-like coherence peaks (regularized when including finite $\boldsymbol{q}$, see Fig. \ref{['fig:selfconsistencyq']}) inside the larger region of completely suppressed DOS. Due to the gap in the dispersion, the DOS still vanishes in an entire energy range around the Fermi level for $T_c<T<T^*$; when $T>T^*$, one only finds a partial suppression of the DOS at low energies, leading to a $V$-shaped behavior.