  We study a classical model of thermally fluctuating polymers confined to two
dimensions, experiencing a grooved periodic potential, and subject to pulling
forces both along and transverse to the grooves. The equilibrium polymer
conformations are described by a mapping to a quantum system with a
non-Hermitian Hamiltonian and with fermionic statistics generated by
noncrossing interactions among polymers. Using molecular dynamics simulations
and analytical calculations, we identify a localized and a delocalized phase of
the polymer conformations, separated by a delocalization transition which
corresponds (in the quantum description) to the breakdown of a band insulator
when driven by an imaginary vector potential. We calculate the average tilt of
the many-body system, at arbitrary shear values and filling density of polymer
chains, in terms of the complex-valued non-Hermitian band structure. We find
the critical shear value, the localization length, and the critical exponent by
which the shear modulus diverges in terms of the branch points (exceptional
points) in the band structure at which the bandgap closes. We also investigate
the combined effects of non-Hermitian delocalization and localization due to
both periodicity and disorder, uncovering preliminary evidence that while
disorder favours localization at high values, it encourages delocalization at
lower values.
