  As shown in recent experiments [V. Lienhard et al., Phys. Rev. X 10, 021031
(2020)], spin-orbit coupling in systems of Rydberg atoms can give rise to
density-dependent Peierls Phases in second-order hoppings of Rydberg spin
excitations and nearest-neighbor (NN) repulsion. We here study theoretically a
one-dimensional zig-zag ladder system of such spin-orbit coupled Rydberg atoms
at half filling. The second-order hopping is shown to be associated with an
effective gauge field, which in mean-field approximation is static and
homogeneous. Beyond the mean-field level the gauge potential attains a
transverse quantum component whose amplitude is dynamical and linked to density
modulations. We here study the effects of this to the possible ground-state
phases of the system. In a phase where strong repulsion leads to a density
wave, we find that as a consequence of the induced quantum gauge field a
regular pattern of current vortices is formed. However also in the absence of
density-density interactions the quantum gauge field attains a non-vanishing
amplitude. Above a certain critical strength of the second-order hopping the
energy gain due to gauge-field induced transport overcomes the energy cost from
the associated build-up of density modulations leading to a spontaneous
generation of the quantum gauge field.
