Matrix product states as thin torus limits of conformal correlators

We introduce one-parameter families of spin chain ansatz wavefunctions constructed from chiral conformal field theory correlators on a torus, with the modular parameter $τ$ serving as the deformation parameter. In the cylinder limit $τ\to\infty$, these wavefunctions reduce to infinite dimensional matrix product states. In contrast, in the thin torus limit $τ\to0$, they become finite bond dimension matrix product states (MPS). Focusing on families derived from the SU(2)$_1$ and SU(2)$_2$ Wess-Zumino-Witten models, we show that in the thin torus limit they reproduce known MPS ground states, such as those of the Majumdar-Ghosh and AKLT spin chains.