Delocalization of interacting directed polymers on a periodic substrate: Localization length and critical exponents from non-Hermitian spectra

TL;DR

This paper investigates the delocalization transition of interacting directed polymers on a periodic substrate using non-Hermitian quantum models, revealing critical behavior and effects of disorder through simulations and analytical methods.

Contribution

It introduces a novel mapping of polymer conformations to non-Hermitian quantum systems and characterizes the localization transition with critical exponents and band structure analysis.

Findings

Identified localized and delocalized phases of polymers.

Calculated the critical shear value and localization length.

Discovered disorder's contrasting effects on localization at different regimes.

Abstract

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.