Information-Entropic Measure of Energy-Degenerate Kinks in Two-Field Models

Abstract: We investigate the existence and properties of kink-like solitons in a class of models with two interacting scalar fields. In particular, we focus on models that display both double and single-kink solutions, treatable analytically using the Bogomol'nyi--Prasad--Sommerfield bound (BPS). Such models are of interest in applications that include Skyrmions and various superstring-motivated theories. Exploring a region of parameter space where the energy for very different spatially-bound configurations is degenerate, we show that a newly-proposed momentum-space entropic measure called Configurational Entropy (CE) can distinguish between such energy-degenerate spatial profiles. This information-theoretic measure of spatial complexity provides a complementary perspective to situations where strictly energy-based arguments are inconclusive.

• The paper demonstrates that configurational entropy effectively discriminates among energy-degenerate kink solutions.
• It employs a two-field BPS model to illustrate transitions between single- and double-kink profiles using Fourier analysis of energy density.
• The findings show that spatial complexity captured by CE provides a complementary observable to energy minimization for characterizing solitonic configurations.

Information-Entropic Discrimination in Energy-Degenerate Kinks for Two-Field Models

Introduction

The paper "Information-Entropic Measure of Energy-Degenerate Kinks in Two-Field Models" (1409.0029) conducts an information-theoretic analysis of solitonic solutions in (1+1)-dimensional field theories with two coupled real scalar fields. While the BPS formalism enables analytic construction of static, stable, minimum-energy configurations (kinks, lumps, Bloch walls), the work reveals that energy-minimization alone is insufficient to fully characterize the physical content of such models. Specifically, it is demonstrated that these systems admit families of static solutions which are profoundly distinct in spatial structure yet share identical BPS energies across a continuous parameter range. To resolve this ambiguity, the paper rigorously applies the configurational entropy (CE) framework—recently advanced by Gleiser and Stamatopoulos—as a measure sensitive to spatial complexity and modal structure. The analysis substantiates that CE provides a discriminative, quantitative tool that effectively distinguishes among energy-degenerate field configurations, thereby offering a complementary criterion to energetic arguments.

Analytical Structure of Energy-Degenerate Kink and Wall Solutions

The model is defined by a Lagrangian with two scalar fields, $$\mathcal{L} = \frac{1}{2}(\partial_\nu \phi)^2 + \frac{1}{2}(\partial_\nu \chi)^2 - V(\phi, \chi)$$, where the potential is derived from a superpotential $W(\phi, \chi)$ facilitating first-order BPS equations. For the superpotential $$W(\phi,\chi) = -\lambda\phi + \frac{\lambda}{3}\phi^3 + \mu\phi\chi^2$$, the system supports multiple isolated vacua and analytic kink-like solutions connecting these vacua, classified as Degenerate Bloch Walls (DBW) and Critical Bloch Walls (CBW).

It is established that, for specific parameter regimes, the DBW solutions depend continuously on an integration constant $c_0$. Intriguingly, for any choice of $c_0$ within allowed bounds, the solutions interpolate between fixed vacua and share exactly the same energy. As $c_0$ approaches critical values, the morphology of the solutions transitions between single- and double-kink profiles, with the corresponding $\chi$ field solutions exhibiting either lump-like or flat-top characteristics.

The CBW emerge precisely at the critical values of $c_0$, representing limit configurations with unique analytic profiles. Both species arise as exact BPS solutions to the uncoupled first-order system and are characterized analytically in terms of hyperbolic/trigonometric functions.

Configurational Entropy Formalism and Application

Configurational entropy, as constructed by Gleiser and Stamatopoulos, incorporates the Fourier mode decomposition of the field energy density $\rho(x)$ to construct a modal fraction $f(k)$ and entropic functional $S_c = -\int dk\, \tilde{f}(k)\log\tilde{f}(k)$, where $\tilde{f}(k)$ is normalized with respect to the maximal modal fraction. The quantity $S_c$ is sensitive to the distributional complexity of the spatial profile: sharply localized (orderly) solutions yield lower entropy, while spatially delocalized or multimode configurations produce higher entropy.

In the context of the DBW/CBW two-field model, the authors compute the energy density for each analytic solution, perform Fourier analysis, and numerically evaluate the CE as a function of $c_0$. They uncover a rich, nontrivial structure for $S_c$ within the DBW family: as $c_0$ is varied, $S_c$ exhibits sharp minima/maxima aligned with qualitative morphological transitions in the kink profiles (e.g., emergence of double kinks and flattening of lump-like configurations). This information-theoretic observable is unequivocally non-degenerate, in direct contrast with the system's energetic degeneracy with respect to $c_0$. The minimal CE occurs in the vicinity of $c_0$ values where spatial complexity is lowest, i.e., where the profile is most localized or structurally simple.

Key Results

• Infinite Degeneracy Distinguished: Among the DBW solutions, infinite degeneracy with respect to $c_0$ (BPS energy conservation) is sharply resolved via CE, rendering it effective for distinguishing physically distinct configurations.
• Transition Detection: The minima in CE accurately demarcate the transition from single-kink to double-kink profiles and alterations in lump spatial character.
• Numerical Analysis: The modal fraction remains sharply peaked at $k=0$, but its spatial width and decay properties reflect the underlying structural transitions, as captured by $S_c$.

Implications and Future Developments

This analysis strengthens CE as a physical observable that captures spatial organization and complexity beyond what is accessible from the energy functional. In application to multi-field models, supersymmetric wall networks, and higher-dimensional (p-brane) defect structures, CE introduces an additional layer of quantitative classification useful in model selection, effective theory construction, and perhaps in the identification of dynamical stability properties of solitonic configurations. CE could also serve as a diagnostic for solutions arising in cosmological field theories (defect networks in the early universe), and in condensed-matter analogues where spatial localization is paramount.

From a more abstract standpoint, the utility of CE in unambiguously breaking energetic degeneracy motivates further exploration in higher-dimensional or more intricate field theories, and in generalizing the information-theoretic landscape for spatially complex solutions—potentially aligning with recent developments in quantum information and the entropic characterization of field states. Additionally, the results reinforce the perspective that variational principles (minimization of action or energy) do not exhaust the full set of physically meaningful observables in field theoretic models, especially regarding spatial complexity and information content.

Conclusion

The work systematically applies the configurational entropy framework to the problem of energy-degenerate, analytic BPS kink solutions in a prototypical two-field model. The results rigorously demonstrate that configurational entropy is an effective discriminant among solutions with identical energy but different spatial (information-theoretic) content. This approach augments standard energetic analysis and opens pathways for characterization and selection of physically significant solutions in nonlinear field theories with complex vacuum structure (1409.0029).