Scale-Invariant Elastic Matter Model

• Scale-invariant elastic matter models are frameworks in which constitutive laws remain unchanged under spatial rescaling, ensuring self-similarity in material responses.
• These models employ both manifest and spontaneously broken scale invariance to capture nonlinear hyperelastic responses with universal power-law behaviors.
• They underpin robust computational methods that simulate complex deformation, wave propagation, and astrophysical phenomena in a scale-independent manner.

A scale-invariant elastic matter model is a theoretical and computational framework in which the constitutive response of elastic materials—at both microscopic and macroscopic levels—remains unchanged under scaling transformations of spatial coordinates and relevant physical variables. Such models are characterized by either exact (manifest) or spontaneously broken scale invariance and have applications ranging from quantum field theory and gravitating systems to classical continuum mechanics, network elasticity, and computational methods for simulating deformation and wave propagation.

1. Mathematical Foundations of Scale Invariance in Elasticity

The fundamental property of a scale-invariant elastic matter model is that the underlying equations (constitutive relations, energy functionals, or field equations) and their solutions remain form-invariant under a rescaling of spatial and material variables. Mathematically, for a spatial scale transformation such as $x \to \lambda x$, relevant field quantities (e.g., displacement $u$, energy density $W$, stress tensor $\sigma$) transform according to definite scaling laws.

For classical elastic systems, scale invariance is implemented by constructing the stored-energy density $W(F)$ (with $F$ the deformation gradient) such that, under $F \to \alpha F$, $W(\alpha F)$ transforms homogeneously or even remains invariant, depending on the specific model. For example, in relativistic elastic stars, all matter fields—energy density $\rho$, radial and tangential pressures $p_{\rm rad}, p_{\rm tan}$—scale homogeneously under $r \to A^{-1} r$ as:

$$(\rho(r), p_{\rm rad}(r), p_{\rm tan}(r)) \mapsto A^2 (\rho(\tilde r), p_{\rm rad}(\tilde r), p_{\rm tan}(\tilde r)).$$

This property ensures that structural relations, such as the mass-radius relationship for stars, remain self-similar across scales (Alho et al., 2023).

In field-theoretic or particle physics variants, all mass parameters are eliminated at the level of the classical Lagrangian. Only quartic (scale-invariant) interactions are permitted, with all physical scales generated dynamically via quantum effects (radiative symmetry breaking), as in the Coleman–Weinberg mechanism [(Foot et al., 2010); (Jung et al., 2019); (Kim et al., 2022)].

2. Manifestation and Breaking of Scale Invariance

Two distinct realizations of scale invariance are prominent in elastic matter models:

• Manifest (exact) scale invariance: The action or Lagrangian is precisely invariant under scale transformations. In such models, the absence of intrinsic length or energy scales implies that material response is governed exclusively by dimensionless parameters, with physical scales introduced only through boundary conditions or external constraints.
• Spontaneously broken scale invariance: The symmetry is a property of the bare Lagrangian but is broken by the choice of vacuum or by quantum corrections (as in anomalies). The classical potential is flat along a certain direction ("flat direction"), and quantum corrections dynamically select a nontrivial vacuum expectation value, generating physical scales (dimensional transmutation) [(Foot et al., 2010); (Jung et al., 2019)].

In continuum elasticity, the transition from finite to linearized elasticity under small strains and rescaling highlights the emergence of a quadratic, scale-invariant energy density as the $\Gamma$-limit of more general nonlinear elastic energies (Mainini et al., 2020).

3. Nonlinear and Hyperelastic Scale-Invariant Response

For large deformations, the stress–strain relations of scale-invariant hyperelastic materials cannot be reduced to linear relationships. Instead, the energy density or stress exhibits universal power-law scaling at large strains:

$\sigma \sim \varepsilon^\nu,$

with the exponent $\nu$ determined by the form of the potential or energy function (Baggioli et al., 2020).

In theories where scale invariance is manifest (e.g., holographic models based on AdS/CFT), stress–strain curves exhibit two or more distinct power-law regimes:

• Intermediate strain regime: $\nu_1 = 2\mathfrak{a}$
• Asymptotically large strain regime: $\nu_2 = 3\mathfrak{a}/\mathfrak{b}$, with parameters $\mathfrak{a}, \mathfrak{b}$ specified by the benchmark potential $$W(X,Z) = X^{\mathfrak{a}} Z^{(\mathfrak{b} - \mathfrak{a}/2)}$$ (Baggioli et al., 2020).

In field-theoretic models, scale invariance is imposed at the potential level for scalar fields—such as $$V(H, S, \phi) = \lambda_h (H^\dagger H)^2 + \lambda_{hs} (H^\dagger H) S^2 + \cdots$$—with massless fields only acquiring mass through radiative corrections along flat directions (Jung et al., 2019, Kim et al., 2022).

4. Network Models and Microscopic Scale Invariance

Discrete network formulations for elasticity and elasto-plasticity can also be made scale-invariant by encoding all bond and vertex parameters (such as edge weights) in a manner that is homogeneous under network scaling. The tension tensor $\mathcal{T}(X, \Phi)$, defined as

$$\mathcal{T}(X, \Phi) = \frac{1}{2}\sum_{e=(v,u) \in E} w(e) (\Phi(u) - \Phi(v)) \otimes (\Phi(u) - \Phi(v)),$$

transforms covariantly under affine transformations, encoding the anisotropic redistribution of elastic energy irrespective of absolute scale (Kodama et al., 2021).

Plasticity is modeled by local moves—contractions and splittings—that irreversibly alter connectivity. The energy loss ratio after such moves, when edge weights are chosen via appropriately scaling functions, becomes independent of absolute scale, establishing a type of universality for the elasto-plastic response (Kodama et al., 2021).

5. Computational and Discretization Approaches

A crucial aspect for practical simulation is the formulation of discretization schemes that maintain scale invariance:

• Variational Discrete Element Methods (DEM): For Cosserat elasticity, DEMs are formulated variationally, requiring only macroscopic continuum parameters and allowing both displacement and rotational degrees of freedom at cell centers. Reconstruction techniques ensure that constant strains and stresses per cell are maintained and the role of the characteristic length scale $\ell$ is explicit but does not introduce an "absolute" material scale (Marazzato, 2021).
• Mixed Variational Finite Elements: By introducing local rotations and deformation fields as independent variables and enforcing consistency with the deformation gradient via Lagrange multipliers, mixed FEM frameworks achieve uniform stability and efficiency over a wide range of scales and material stiffnesses, essential for robust computation of scale-invariant elastic matter (Trusty et al., 2022).
• Numerical Robustness with Initial Stress: For incompressible or complex initially stressed media, models using deviatoric-volumetric splitting in the energy density—expressing it as a function of isochoric (volume-preserving) parts of the deformation gradient and deviatoric parts of the initial stress tensor—yield more robust numerical results and facilitate scale-invariant discretizations (Magri et al., 13 Mar 2024).

6. Physical Phenomena and Implications

Scale-invariant elastic matter models have both phenomenological and fundamental implications:

• Stellar Structure: In General Relativity, scale-invariant elastic stars support linear mass–radius relations ($M \propto R$), lack an upper mass limit, and allow for highly compact, radially-stable configurations with radii comparable to the Schwarzschild photon sphere, challenging traditional maximum-mass paradigms for material bodies (Alho et al., 2023). This implies the possible existence of black-hole mimickers without event horizons.
• Particle Physics: All mass scales—including electroweak, Planck, and neutrino mass scales—are dynamically generated from dimensionless couplings and quantum effects, providing a technically natural solution to the hierarchy problem. Hidden and mirror sectors within such models naturally furnish candidates for dark matter while preserving the stability of the electroweak scale (Foot et al., 2010).
• Nonlinear Elastodynamics: In strongly coupled, critical systems or field-theoretic models with holographic duals, scale invariance produces unique nonlinear elastic responses, with stress-strain curves exhibiting universality classes distinguished by the manner in which scale invariance is realized (manifest vs. spontaneously broken). In these frameworks, "black rubber"-type models display complex power-law regimes reminiscent of natural rubber behavior, and elasticity bounds (maximum reversible deformation) are governed by the symmetry structure (Baggioli et al., 2020).

7. Model Construction and Symmetry Constraints

Construction of a scale-invariant elastic matter model requires careful implementation of symmetry and material constraints:

• Constitutive structure: For initially stressed incompressible solids, the strain energy must depend only on reduced invariants—isochoric deformation gradient and deviatoric initial stress—to ensure both physical consistency and robustness under rescaling (Magri et al., 13 Mar 2024).
• Invertibility and reference independence: The stress response function must be invertible (modulo the material symmetry group), and constitutive laws should respect initial stress compatibility and reference independence to guarantee that the physics does not depend on arbitrary choices of reference configuration (Magri et al., 13 Mar 2024).
• Energetic and stability considerations: All couplings must remain perturbative (e.g., below $4\pi$), potentials are required to be bounded below, and physical states must respect energetic, causality, and stability constraints (e.g., subluminality of sound speeds in relativistic elastic media) (Alho et al., 2023, Kim et al., 2022).

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In summary, scale-invariant elastic matter models constitute a broad and mathematically rich class of frameworks that encode elasticity—both linear and nonlinear—with or without quantum and relativistic effects, in such a way that all physical properties scale homogeneously with changes in length or energy units. These models underpin novel mechanism for hierarchical mass generation, describe elasto-plastic networks with universal features, and furnish robust numerical simulation schemes—all while maintaining a direct lineage to the fundamental symmetries of the action or energy functional.