[Paper Review] Directed polymer in a random medium - an introduction

Somendra M. Bhattacharjee|arXiv (Cornell University)|Feb 4, 2004

Advanced Physical and Chemical Molecular Interactions1 references3 citations

TL;DR

This paper introduces the directed polymer in a random medium as a solvable model for understanding disorder effects in statistical mechanics. It combines intuitive physical pictures with systematic methods like replica theory and renormalization group (RG) techniques to analyze how disorder alters critical behavior, showing that in 1+1 dimensions the system remains in a strong-disorder phase at all temperatures, while in higher dimensions a phase transition to a weak-disorder, pure-like phase occurs at low temperatures.

ABSTRACT

This is a set of introductory lectures on the behaviour of a directed polymer in a random medium. Both the intuitive picture that helps in developing an understanding and systematic approaches for quantitative studies are discussed.

Motivation & Objective

• To provide a pedagogical introduction to the directed polymer model in a random medium for researchers in statistical physics.
• To clarify how disorder modifies universal scaling behavior and critical exponents compared to the pure polymer case.
• To investigate the relevance of disorder using renormalization group (RG) methods and analyze phase transitions in different dimensions.
• To explore the role of rare events and non-self-averaging behavior in disordered systems through free energy fluctuations and overlap statistics.
• To establish connections between the directed polymer problem and the KPZ equation, highlighting exact results and scaling behavior.

Proposed method

• Formulates the directed polymer Hamiltonian with elastic and random potential terms, using a lattice or continuum description.
• Applies the replica trick with $ n \to 0 $ limit to compute disorder-averaged moments of the partition function.
• Employs the Bethe ansatz for exact solutions in 1+1 dimensions, particularly for the free energy distribution.
• Uses the Kardar-Parisi-Zhang (KPZ) equation to describe the evolution of the free energy, linking it to stochastic growth processes.
• Applies renormalization group (RG) techniques to analyze the relevance of disorder, including flow equations for coupling constants.
• Introduces a ghost polymer with weak interaction to define the overlap $ q $, enabling study of ground state degeneracy and replica symmetry breaking.

Experimental results

Research questions

• RQ1 How does quenched disorder affect the universal scaling behavior of a directed polymer, and does it change the critical exponent $ \nu $?
• RQ2 Is disorder relevant in the thermodynamic sense, and does it drive a phase transition in $ d+1 $ dimensions?
• RQ3 What is the nature of the free energy distribution in 1+1 dimensions, and how does it differ from the typical Gaussian behavior?
• RQ4 How does the overlap between two polymers in the same random medium behave, and what does it reveal about ground state degeneracy?
• RQ5 What is the role of rare events in disordered systems, and why do higher moments of the partition function matter?

Key findings

• In 1+1 dimensions, the directed polymer remains in a strong-disorder phase at all temperatures, with no phase transition to a pure-like phase.
• For $ d > 2 $, a phase transition occurs from a low-temperature strong-disorder phase to a high-temperature weak-disorder, pure-like phase, as shown by RG analysis.
• The free energy fluctuation exponent is $ \zeta = 1/3 $ in 1+1 dimensions, consistent with the Tracy-Widom distribution, indicating non-Gaussian fluctuations.
• The size exponent remains $ \nu = 1/2 $, indicating Gaussian scaling, but the system exhibits non-trivial overlap and rare-event effects.
• Overlap $ q $ vanishes at the transition point in $ d > 2 $ as $ q \sim |T - T_c|^{\Sigma \zeta} $ with $ \Sigma < 0 $, signaling the onset of replica symmetry breaking.
• The RG flow for the interaction parameter $ v $ in the overlap problem confirms that $ \Sigma = 0 $ at the stable fixed point in $ d=1 $, but $ \Sigma < 0 $ at the transition in $ d > 2 $, indicating non-trivial critical behavior.