[Paper Review] Chaos in the thermal regime for pinned manifolds via functional RG

Olaf Duemmer, Pierre Le Doussal|ArXiv.org|Sep 10, 2007

Theoretical and Computational Physics3 citations

TL;DR

This paper investigates chaos in pinned elastic manifolds at finite temperature using one-loop functional renormalization group (FRG) methods. It analytically and numerically determines the chaos exponent $ a $ for the random periodic class in $ d=2 $, finding $ a = \epsilon(1/3 - 1/(2\ln(1/T))) $ with non-trivial logarithmic corrections at low temperature, and estimates $ a \approx 0.083346(6)\epsilon $ for short-range (random bond) disorder with significant finite-size effects.

ABSTRACT

The statistical correlations of two copies of a d-dimensional elastic manifold embedded in slightly different frozen disorder are studied using the Functional Renormalization Group to one-loop accuracy, order O(eps = 4-d). Determining the initial (short scale) growth of mutual correlations, i.e. chaos exponents, requires control of a system of coupled differential (FRG) equations (for the renormalized mutual and self disorder correlators) in a very delicate boundary layer regime. Some progress is achieved at non-zero temperature, where linear analysis can be used. A growth exponent a is defined from center of mass fluctuations in a quadratic potential. In the case where temperature is marginal, e.g. a periodic manifold in d=2, we demonstrate analytically and numerically that a = eps (1/3 - 1/(2 log(1/T)) with interesting and unexpected logarithmic corrections at low T. For short range (random bond) disorder our analysis indicates that a = 0.083346(6) eps, with large finite size corrections.

Motivation & Objective

• To understand the initial growth of mutual correlations in disordered elastic manifolds under small perturbations, particularly the chaos exponent $ \alpha $, in the thermal regime $ T > 0 $.
• To address the challenge of solving coupled FRG equations for mutual and self-disorder correlators in the delicate boundary layer where $ D(u) $ and $ \Delta(u) $ differ significantly at short scales.
• To apply linear analysis to the FRG flow equations in the thermal regime, where $ \Delta(u) $ is smoothed within the thermal boundary layer (TBL), enabling tractable computation of the chaos exponent.
• To determine the chaos exponent $ a $ for the random periodic class in $ d=2 $, where temperature is a marginal parameter, and for the random bond class with short-range disorder.

Proposed method

• Uses one-loop functional renormalization group (FRG) to study the flow of two correlation functions: $ \Delta(u) $ (self-disorder correlator) and $ D(u) $ (mutual disorder correlator) between two replicas.
• Applies linearized FRG equations around a fixed-point solution $ \Delta_T(u) $ for the random periodic class at $ d=2 $, where temperature is marginal and the system has a line of fixed points.
• Performs a perturbative expansion in $ T $ to analyze the thermal boundary layer (TBL), where $ \Delta(u) $ is rounded off, enabling linear analysis of $ D(u) $.
• Solves the linearized FRG equations for the growth exponent $ a $ by matching solutions inside and outside the TBL, using polynomial and logarithmic eigenfunctions in the variable $ u $.
• Uses matched asymptotic expansions and boundary layer analysis to handle the non-analytic cusp in $ \Delta(u) $ and the analyticity of $ D(u) $, ensuring consistency across scales.
• Performs numerical checks and compares with known results, including corrections to earlier FRG work and high-precision numerical data at $ T=0 $.

Experimental results

Research questions

• RQ1 What is the chaos exponent $ a $ for the random periodic class in $ d=2 $ at finite temperature, and how do logarithmic corrections emerge at low $ T $?
• RQ2 How does the finite-temperature FRG framework allow for the analytical determination of the initial growth of mutual correlations in the chaotic regime?
• RQ3 What is the value of the chaos exponent $ a $ for the random bond class with short-range disorder, and how are finite-size effects quantified?
• RQ4 How does the structure of the thermal boundary layer (TBL) influence the solution of the coupled FRG equations for $ D(u) $ and $ \Delta(u) $?

Key findings

• For the random periodic class in $ d=2 $, the chaos exponent is analytically derived as $ a = \epsilon(1/3 - 1/(2\ln(1/T))) $, revealing unexpected logarithmic corrections at low temperature.
• The result is confirmed numerically and shows that chaos persists at finite $ T $, with the exponent diverging logarithmically as $ T \to 0 $, consistent with the marginal nature of temperature in $ d=2 $.
• For the random bond class, the chaos exponent is estimated as $ a = 0.083346(6)\epsilon $, with large finite-size corrections, indicating strong sensitivity to system size in numerical realizations.
• The analysis confirms that $ D(u) $ remains analytic while $ \Delta(u) $ develops a cusp at $ u=0 $, justifying the use of linearized FRG in the TBL regime.
• The eigenvalue problem for the linearized FRG flow yields a discrete set of solutions, and only specific eigenvalues satisfy the zero-mean condition required for physical consistency.
• The derived solution matches known results from earlier FRG work when corrected for a misprint, validating the method and the perturbative expansion in $ T $.