Finite-temperature perturbation theory for the random directed polymer problem
- 57 Downloads
We study the random directed polymer problem—the short-scale behavior of an elastic string (or polymer) in one transverse dimension subject to a disorder potential and finite temperature fluctuations. We are interested in the polymer short-scale wandering expressed through the displacement correlator 〈[δu(X)]2〉, with δu(X) being the difference in the displacements at two points separated by a distance X. While this object can be calculated at short scales using the perturbation theory in higher dimensions d > 2, this approach becomes ill-defined and the problem turns out to be nonperturbative in the lower dimensions and for an infinite-length polymer. In order to make progress, we redefine the task and analyze the wandering of a string of a finite length L. At zero temperature, we find that the displacement fluctuations 〈[δu(X)]2〉 ∝ LX 2 depend on L and scale with the square of the segment length X, which differs from a straightforward Larkin-type scaling. The result is best understood in terms of a typical squared angle 〈α2〉 ∝ L, where α = ∂ x u, from which the displacement scaling for the segment X follows naturally, 〈[δu(X)]2〉 ∝ 〈α2〉 X 2. At high temperatures, thermal fluctuations smear the disorder potential and the lowest-order results for disorder-induced fluctuations in both the displacement field and the angle vanish in the thermodynamic limit L → ∞. The calculation up to the second order allows us to identify the regime of validity of the perturbative approach and provides a finite expression for the displacement correlator, albeit depending on the boundary conditions and the location relative to the boundaries.
KeywordsThermal Fluctuation Perturbative Expansion Random Potential Short Scale Perturbative Approach
Unable to display preview. Download preview PDF.
- 16.M. Schweizer, Weak Pinning of a Directed Polymer in a Random Environment: Semester Thesis (Eidgenössische Technische Hochschule, Zurich, Switzerland, 2010).Google Scholar