  We study the $J_1$-$J_2$ spin-$1/2$ chain using a path integral constructed
over matrix product states (MPS). By virtue of its non-trivial entanglement
structure, the MPS ansatz captures the key phases of the model even at a
semi-classical, saddle-point level, and, as a variational state, is in good
agreement with the field theory obtained by abelian bosonisation. Going beyond
the semi-classical level, we show that the MPS ansatz facilitates a
physically-motivated derivation of the field theory of the critical phase: by
carefully taking the continuum limit -- a generalisation of the Haldane map --
we recover from the MPS path integral a field theory with the correct
topological term and emergent $SO(4)$ symmetry, constructively linking the
microscopic states and topological field-theoretic structures. Moreover, the
dimerisation transition is particularly clear in the MPS formulation -- an
explicit dimerisation potential becomes relevant, gapping out the magnetic
fluctuations.
