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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 todensity-dependent Peierls Phases in second-order hoppings of Rydberg spinexcitations and nearest-neighbor (NN) repulsion. We here study theoretically aone-dimensional zig-zag ladder system of such spin-orbit coupled Rydberg atomsat half filling. The second-order hopping is shown to be associated with aneffective gauge field, which in mean-field approximation is static andhomogeneous. Beyond the mean-field level the gauge potential attains atransverse quantum component whose amplitude is dynamical and linked to densitymodulations. We here study the effects of this to the possible ground-statephases of the system. In a phase where strong repulsion leads to a densitywave, we find that as a consequence of the induced quantum gauge field aregular pattern of current vortices is formed. However also in the absence ofdensity-density interactions the quantum gauge field attains a non-vanishingamplitude. Above a certain critical strength of the second-order hopping theenergy gain due to gauge-field induced transport overcomes the energy cost fromthe associated build-up of density modulations leading to a spontaneousgeneration of the quantum gauge field.

Self-generated quantum gauge fields in arrays of Rydberg atoms