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We introduce one-dimensional lattice models with exact matrix-product groundstates describing the fractional quantum Hall (FQH) states in Laughlin series(given by filling factors $\nu=1/q$) on torus geometry. Surprisingly, theexactly solvable Hamiltonian has the same mathematical structure as that of thepseudopotential for the Laughlin wave function, and naturally derives thegeneral properties of the Laughlin wave function such as the $Z_2$ propertiesof the FQH states and the fermion-boson relation. The obtained exact groundstates have high overlaps with the Laughlin states and well describe theirproperties. Using the matrix product method, density functions and correlationfunctions are calculated analytically. Especially, obtained entanglementspectra reflects gapless edge states as was discussed by Li and Haldane.

One-dimensional lattice model with an exact matrix-product ground state describing the Laughlin wave function