Attempted Bethe ansatz solution for one-dimensional directed polymers in random media

TL;DR

This paper maps the problem of one-dimensional directed polymers in random media to a quantum boson system, finds its eigenstates, and derives a universal distribution for free energy fluctuations.

Contribution

It provides an exact solution for the eigenvalues and eigenfunctions of the associated quantum system and derives a universal free energy distribution for the polymers.

Findings

Derived the full spectrum of the quantum boson system

Obtained an explicit universal distribution for free energy fluctuations

Addressed the analytic continuation ambiguity in the replica method

Abstract

We study the statistical properties of one-dimensional directed polymers in a short-range random potential by mapping the replicated problem to a many body quantum boson system with attractive interactions. We find the full set of eigenvalues and eigenfunctions of the many-body system and perform the summation over the entire spectrum of excited states. The analytic continuation of the obtained exact expression for the replica partition function from integer to non-integer replica parameter N turns out to be ambiguous. Performing the analytic continuation simply by assuming that the parameter N can take arbitrary complex values, and going to the thermodynamic limit of the original directed polymer problem, we obtain the explicit universal expression for the probability distribution function of free energy fluctuations.