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We introduce an exactly solvable fermion chain that describes a $\nu=1/3$fractional quantum Hall (FQH) state beyond the thin-torus limit. The groundstate of our model is shown to be unique for each center of mass sector, and ithas a matrix product representation that enables us to exactly calculate orderparameters, correlation functions, and entanglement spectra. The ground stateof our model shows striking similarities with the BCS wave functions andquantum spin-1 chains. Using the variational method with matrix product ansatz,we analytically calculate excitation gaps and vanishing of the compressibilityexpected in the FQH state. We also show that the above results can be relatedto a $\nu=1/2$ bosonic FQH state.

Exactly Solvable Fermion Chain Describing a $\nu=1/3$ Fractional Quantum Hall State