  We show that the nonlinear transport of bosonic excitations in a
two-dimensional honeycomb lattice of spin-orbit coupled Rydberg atoms gives
rise to disordered quantum phases which are candidates for quantum spin
liquids. As recently demonstrated in [Lienhard et al. Phys. Rev. X, 10, 021031
(2020)] the spin-orbit coupling breaks time-reversal and chiral symmetries and
leads to a tunable density-dependent complex hopping of the hard-core bosons or
equivalently to complex XY spin interactions. Using exact diagonalization (ED)
we numerically investigate the phase diagram resulting from the competition
between density-dependent and direct transport terms. In mean-field
approximation there is a phase transition from a quasi-condensate to a
120{\deg} phase when the amplitude of the complex hopping exceeds that of the
direct one. In the full model a new phase with a finite spin gap emerges close
to the mean-field critical point as a result of quantum fluctuations induced by
the density-dependence of the complex hopping. We show that this phase is a
genuine disordered one, has a non-vanishing spin chirality and is characterized
by a non-trivial many-body Chern number. ED simulations of small lattices with
up to 28 lattice sites point to a non-degenerate ground state and thus to a
bosonic integer-quantum Hall (BIQH) phase, protected by U(1) symmetry. The
Chern number of C = 1, which is robust to disorder, is however different from
the even Chern numbers found in BIQH phases. For very strong, nonlinear
hoppings of opposite sign we find another disordered regime with vanishing spin
gap. This phase also has a large spin chirality and could be a gapless
spin-liquid but lies outside the parameter regime accessible in the Rydberg
system.
