  We propose a scheme to simulate one-dimensional XXZ-type Heisenberg spin
models with competing interactions between nearest-neighbors (NNs) and next-NNs
in photon-coupled micro-cavities. Our scheme, for the first time, exploits the
rich resources and flexible controls available in such a system to realize
arbitrarily adjustable ratios between the effective NN and next-NN coupling
strengths. Such powerful capability allows us to simulate frustration phenomena
and disorder behaviors in 1-d systems arising from next-NN interactions, a
large class of problems of great importance in condensed matter physics. Our
scheme is robust due to the lack of atomic excitations which suppresses
spontaneous emission and cavity decay strongly.
