Helical Rashba-exchange gauge field drives a uniaxial pair density wave in EuRbFe$_4$As$_4$

The recent discovery of an intrinsic, zero-field pair density wave (PDW) in the iron-pnictide superconductor EuRbFe$_4$As$_4$ poses a fundamental puzzle: how does a unidirectional, nanometer-scale superconducting modulation arise spontaneously below the magnetic ordering temperature? Here we show that the interplay of Rashba spin-orbit coupling -- induced by the locally non-centrosymmetric FeAs layers -- and the period-four helical Eu$^{2+}$ exchange field generates a layer-rotating effective $U(1)$ gauge field for the Cooper pairs. Because this gauge field shares the symmetry of the Fe $3d_{xz}/3d_{yz}$ orbital doublet, it drives an orbital-selective, finite-momentum pairing instability. Using a Ginzburg-Landau theory on the magnetic unit cell, we demonstrate that this mechanism naturally stabilizes a strictly uniaxial Bloch superconducting state at the experimentally observed wavelength, accompanied by spontaneous interlayer loop currents accessible to muon-spin relaxation or scanning SQUID microscopy.