  The pair density wave (PDW) is a novel superconducting state in which Cooper
pairs carry center-of-mass momentum in equilibrium, leading to the breaking of
translational symmetry. Experimental evidence for such a state exists in high
magnetic field and in some materials that feature density wave orders that
explicitly break translational symmetry. However, evidence for a zero-field PDW
state that exists independent of other spatially ordered states has so far been
elusive. Here, we show that such a state exists in the iron pnictide
superconductor EuRbFe4As4 (Eu-1144), a material that features coexisting
superconductivity (Tc ~ 37K) and magnetism (Tm ~ 15 K). We show from the
Spectroscopic Imaging Scanning Tunneling Microscopy (SI-STM) measurements that
the superconducting gap at low temperature has long-range, unidirectional
spatial modulations with an incommensurate period of ~8 unit cells. Upon
raising the temperature above Tm, the modulated superconductor disappears, but
a uniform superconducting gap survives to Tc. When an external magnetic field
is applied, gap modulations disappear inside the vortex halo. The SI-STM and
bulk measurements show the absence of other density wave orders, showing that
the PDW state is a primary, zero-field superconducting state in this compound.
Both four-fold rotational symmetry and translation symmetry are recovered above
Tm, indicating that the PDW is a smectic order.
