Non-Hermitian gap closure and delocalization in interacting directed polymers

Abstract

We examine the interplay of band theory and non-Hermitian mechanics in a classical system of thermally fluctuating directed polymers subject to shear forces and experiencing a continuum periodic potential in 1+1D. 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 find the critical shear value and the critical exponent by which the shear modulus diverges in terms of the branch points in the complex-valued 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.