  We consider ultracold polar molecules trapped in a unit-filled
one-dimensional chain in real space created with an optical lattice or a
tweezer array and illuminated by microwaves that resonantly drive transitions
within a chain of rotational states. We describe the system by a
two-dimensional lattice model, with the first dimension being a lattice in real
space and the second dimension being a lattice in a synthetic direction
composed of rotational states. We calculate this system's ground-state phase
diagram. We show that as the dipole interaction strength is increased, the
molecules undergo a phase transition from a two-dimensional gas to a phase in
which the molecules bind together and form a string that resembles a
one-dimensional object living in the two-dimensional (i.e, one real and one
synthetic dimensional) space. We demonstrate this with two complementary
techniques: numerical calculations using matrix product state techniques and an
analytic solution in the limit of infinitely strong dipole interaction. Our
calculations reveal that the string phase at infinite interaction is
effectively described by emergent particles living on the string and that this
leads to a rich spectrum with excitations missed in earlier mean-field
treatments.
