Ultrasound evidence for multicomponent superconducting order parameter in Ba$_{1-x}$K$_x$Fe$_2$As$_2$ with electron quadrupling phase

TL;DR

The paper addresses the existence and symmetry of a multicomponent electron quadrupling order parameter in Ba$_{1-x}$K$_x$Fe$_2$As$_2$ near the magic doping. It combines ultrasound experiments with a minimal Ginzburg-Landau theory that couples a two-component superconducting order parameter to a quadrupole order parameter $\Psi$, and analyzes how strain couples to different order-parameter symmetries. The results show strong signatures in the transverse $c_{66}$ mode and jumps at the superconducting transition, consistent with a time-reversal-symmetry-breaking superconducting state in the quadrupling phase; however, reconciling all observations with a simple $s+is$ scenario requires incorporating strain, defects, or nematic effects. The study provides a framework and experimental strategy to diagnose order-parameter symmetry in emergent multicomponent condensates, highlighting the role of lattice coupling and stress in revealing hidden symmetries. The findings advance understanding of complex superconducting states and suggest that high-quality, strain-controlled crystals are essential to definitively determine the pairing symmetry in this system.

Abstract

Experiments have pointed to the formation of the electron quadrupling condensate in Ba$_{1-x}$K$_x$Fe$_2$As$_2$ at $x \sim 0.8$. The state spontaneously breaks time-reversal symmetry and is sandwiched between two critical points, separating it from the broken time-reversal symmetry (BTRS) superconducting state at $T_{\rm c}^{U(1)}$ and normal-metal state at $T_{\rm c}^{\rm Z2}$. We report a theory of the acoustic effects spectroscopy of systems with an electron quadrupling phase based on ultrasound-velocity measurements. We show that the experimental results are consistent with BTRS superconductivity at $x \sim 0.8$, fulfilling the necessary condition for the formation of electron quadrupling in Ba$_{1-x}$K$_x$Fe$_2$As$_2$. We provide the theoretical basis and the experimental strategy to study the order parameter symmetry of emerging quadrupling condensates in superconductors.

Figures (8)

• Figure 1: A schematic plot of our phase diagram with BTRS dome and quartic phase constructed according to experimental data Grinenko2020superconductivityGrinenko2021stateshipulin2023calorimetric
• Figure 2: Experimental results demonstrating the change of relative sound velocity as a function of temperature for Ba$_{1-x}$K$_x$Fe$_2$As$_2$ with doping $x\approx0.78$, for a longitudinal $c_{11}$ and (c) transverse $c_{66}$ acoustic modes. The measurements were done using a transit acoustic signal (zero echo) at zero field (ZF) and filed applied along the $c$-axis. Temperature dependence of the relative change of the sound velocity (b) for the longitudinal $c_{11}$ and (d) transverse $c_{66}$ acoustic modes (left) with subtracted in-field data and AC magnetic susceptibility (right) measured at $B$ = 1 Oe with $f$ = 417 Hz applied along the $c$-axis.
• Figure 3: Temperature dependence of the relative change of the sound velocity of KFe$_2$As$_2$ for (a) the longitudinal $c_{11}$, (b) transverse $c_{66}$, (c) longitudinal $(c_{11}+c_{12}+2c_{66})/2$, and (d) transverse $(c_{11}-c_{12})/2$ acoustic modes. The measurements were done using a transit acoustic signal (zero echo) at zero field (ZF) and filed applied along the $c$-axis for the sound propagation along the [100] direction. Only zero-field measurements were performed for the sound propagation along the [110] direction.
• Figure 4: Temperature dependence of the relative change of the sound velocity of KFe$_2$As$_2$ with subtracted normal state contribution for the data shown in Fig. \ref{['fig:data2']} for (a) the longitudinal $c_{11}$, and transverse $c_{66}$ acoustic modes (left) and DC magnetic susceptibility (right) measured at $B$ = 5 Oe applied along the $c$-axis and (b) for the longitudinal $(c_{11}+c_{12}+2c_{66})/2$ and transverse $(c_{11}-c_{12})/2$ acoustic modes.
• Figure 5: The ultrasound response in the longitudinal, transverse $B_{1g}$ [ $(c_{11}-c_{12})/2$], and transverse $B_{2g}$$c_{66}$ mode for the three OP cases considered, with the free energy defined in \ref{['eq:free_energy']}. The $s, d_{xy}$ model reproduces: a linear change with temperature in the transverse mode below the quadrupling transition $T=T_c^{Z_2}\approx 2$, a minor response in the longitudinal mode at the quadrupling transition, and jumps in both modes at the superconducting transition $T=T_c^{U(1)}\approx 1$.