Time-Reversal Symmetry Breaking Superconducting State and Collective Modes in Kagome Superconductors

TL;DR

This work develops a four-pocket multiband model for the kagome superconductors $AV_3Sb_5$ to investigate pairing symmetry and collective modes. It shows that inter-pocket interactions and DOS variations can drive a transition from conventional pairing to a time-reversal symmetry breaking state $s+is$, mediated by phase frustration among Fermi pockets. The analysis reveals a nearly massless Leggett mode at the TRSB transition and a massless BAG-like mode in the long-wavelength limit that becomes a plasma via the Anderson-Higgs mechanism, offering a clear experimental signature and a framework to distinguish TRSB superconductivity from TRSB charge orders. These results propose Raman and terahertz probes as viable routes to detect TRS-breaking pairing in kagome metals and connect microscopic interactions to observable collective dynamics.

Abstract

We comprehensively study the unconventional pairing and collective modes in the multiband kagome superconductors AV$_3$Sb$_5$ (A=$\mathrm{K},\mathrm{Cs},\mathrm{Rb}$). By solving gap equations at zero temperature, we identify a transition from normal $s++/s\pm$-wave pairing to time-reversal symmetry (TRS) breaking pairing with a variation of inter-pocket interactions or density of states. This TRS breaking pairing originates from the superconducting phase frustration of different Fermi pockets and can account for experimental TRS breaking signal in kagome superconductors. Moreover, we investigate collective modes, including the Higgs, Leggett, and Bogoloubov-Anderson-Goldstone modes, arising from fluctuations of the amplitude, relative phase, and overall phase of the superconducting order parameters, respectively. Remarkably, due to the presence of multibands, one branch of the Leggett modes becomes nearly massless near the TRS breaking transition, providing a compelling smoking-gun signature of TRS-breaking superconductivity, in clear contrast to TRS-breaking charge orders. Our results elucidate the rich superconducting physics and its associated collective modes in kagome metals, and suggest feasible experimental detection of TRS breaking pairing.

Figures (4)

• Figure 1: Schematic superconducting phase diagram with pressure $p$ and temperature $T$ from the multiband scenario in the kagome superconductors AV$_3$Sb$_5$. With increasing pressure (doping), the system undergoes a transition from a sign changing $s\pm$ to a trivial $s++$ state without sign change between two pockets below superconducting transition temperature $T_c(p)$. At the low temperature limit, there exists a time-reversal symmetry breaking superconducting state $s+is$ between the $s++$ and $s\pm$ state. In the vicinity of each superconducting transition, the spectrum of one branch of the Leggett mode exhibits a distinct minimum structure, which provides a characteristic signature for detecting TRS pairing in Kagome metals.
• Figure 2: Scattering processes of the four Fermi pockets as denoted by $\alpha$, $\beta$ and $\gamma$, respectively. Intra-pockets interactions $V_{\alpha}$ (a) and $V_{\beta}$ (d) and inter-pocket interaction $V_{\gamma}$ (b) between electrons at different valleys. (c) The pair-hopping process from $\alpha$ pocket contributed by the $p$ orbital of Sb atoms to $\beta$ pocket contributed by $d$ orbitals of V atoms. We set $V_{\alpha\beta}=-u_{\alpha\beta}/V$. (e) The pair-hopping process from $\alpha$ pocket to $\gamma_{K,K^{\prime}}$ pocket contributed by $d$ orbitals of V atoms. (f) The pair-hopping process from $\beta$ pocket to $\gamma_{K,K^{\prime}}$.
• Figure 3: The evolution of amplitudes and phases of the $\Delta_{\alpha,\beta,\gamma}$ as changing inter-pocket scattering strength or DOS $V_0N_\beta(0)$ with initial intrapocket interaction as $V_{\alpha,\beta,\gamma}=-V_0$. (a) Superconducting state evolves from $s++$ to $s\pm$ as increasing $V_{\alpha\beta}$. In the middle region with repulsive interaction, there exists a TRS breaking $s+is$ state. The initial parameters are chosen as $V_{\alpha\gamma}= V_{\beta\gamma}=-0.1V_0$. (b) As increasing $V_{\beta\gamma}$, there exists a phase transition from $s\pm$ to TRS breaking state. When the repulsive interaction between $\beta$ and $\alpha$ pocket is strong enough, the phase goes into another type of $s\pm$ state, which is denoted by $s^{\prime}\pm$ marked by light green line. The initial parameters are chosen as $V_{\alpha\beta}=V_{\alpha\gamma}=0.1V_0$. In (a) and (b), we set $V_0N_{\alpha,\gamma}(0)=0.4$ and $V_0N_{\beta}(0)=0.6$. (c) By tuning DOS of $\beta$ pocket $V_0N_{\beta}(0)$, a TRS breaking state separate $s++$ and $s\pm$ state; we chose $V_0N_{\alpha,\gamma}(0)=0.3$, $V_{\alpha\beta,\alpha\gamma}=0.1V_0$ and $V_{\beta\gamma}=0.2V_0$. In these figures, blue, red and green line represents the region of $s++$, $s+is$ and $s\pm$ superconducting state, respectively. Here we only illustrate one typical solution of $\phi_{2,3}$, and other solutions $\tilde{\phi}_{2,3}$ can be obtained from $\cos(\tilde{\phi}_{2,3})=\cos(\phi_{2,3})$ and $\cos(\tilde{\phi}_2-\tilde{\phi}_3)=\cos(\phi_2-\phi_3)$.
• Figure 4: Collective modes as functions of interpocket interaction or DOS at the zero momentum ${\bf q}=0$. (a) and (b) illustrate the evolution of masses of BAG and two LMs with increasing interpocket interaction $V_{\alpha\beta}$, and $V_{\beta\gamma}$, respectively. Green arrows represent relative oscillations of the corresponding phase. Here we set $V_0N_{\alpha,\beta,\gamma}=0.5$. (c,d,e) indicates the collective modes resolved from solving $y=-\ln(|\mathrm{Det}{\bf Q}(\omega,0)|)$ as varing $V_{\alpha\beta}$, $V_{\beta\gamma}$ or $V_0N_{\beta}(0)$. Here we plot two-quasiparticel continuum setted by $2\Delta_{1,2,3}$. In (a) and (c), we set parameters $V_{\alpha\gamma}=V_{\beta\gamma}=-0.1V_0$; in (b) and (d) we set $V_{\alpha\beta}=V_{\alpha\gamma}=0.1V_0$. In (c) and (d), we also have $V_0N_{\alpha,\gamma}(0)=0.4$ and $V_0N_{\beta}(0)=0.6$. In (e), we chose $V_0N_{\alpha,\gamma}(0)=0.3$, $V_{\alpha\beta,\alpha\gamma}=0.1V_0$ and $V_{\beta\gamma}=0.2V_0$. The solid black arrows indicate the TRS breaking transition points where mass of LM $\Omega_{\mathrm{LM}-}$ becomes vanishing small.