Three-component superconductivity: the effect of second-order Josephson couplings

Recently, a three-component Ginzburg-Landau (GL) model compatible with the 3Q pair-density-wave state has been proposed to explain the fractional quantum magnetic resistance oscillations of period $\phi_0/3 = hc/6e$ observed in vanadium-based kagome superconductors. The physics of this model is governed by second-order Josephson-type couplings, which break both time-reversal symmetry and discrete $\pi$-phase flip symmetry. In this work, we theoretically derive the complete set of ground-state solutions and construct a comprehensive phase diagram in the GL parameter space, characterized by analytically determined phase boundaries. We identify five distinct ground states: an 8-fold degenerate frustrated state and four 4-fold degenerate non-frustrated phase-locked states. Four of these states spontaneously break time-reversal symmetry. Numerical analysis of the collective modes reveals the emergence of a Higgs-Leggett mode unique to the frustrated region, accompanied by mode softening near the phase boundaries. Our findings provide a comprehensive theoretical framework for understanding the multifaceted physics of multicomponent superconductivity.