Impact of elastic inhomogeneity on collective dynamical properties investigated by field theoretical description in real space

1. Problem Statement

The collective dynamical properties of crystalline solids are well-described by Debye's theory, which relies on translational symmetry. However, this paradigm fails for amorphous solids, which exhibit mesoscopic-scale inhomogeneity. This inhomogeneity leads to anomalous vibrational properties, most notably:

• An excess of low-frequency vibrational density of states (DOS) known as the Boson Peak (BP).
• Deviations in specific heat at low energies.
• The emergence of localized, non-phononic vibrational modes.

Previous theoretical attempts to model these phenomena using field theory in wave-vector space (k-space) relied on perturbative approaches. These approaches treat elastic inhomogeneity as a small perturbation to plane waves. However, they suffer from two major limitations:

1. Underestimation of interactions: Perturbation theory may fail to capture strong microscopic interactions.
2. Loss of spatial information: K-space descriptions cannot easily visualize real-space processes (e.g., localized "hot spots" or string-like excitations) that are crucial for comparing with recent particle-level simulations and experiments.

The core problem addressed is how to rigorously model elastic inhomogeneity to explain the origin of the Boson Peak and vibrational inhomogeneity without relying on perturbative approximations or losing real-space resolution.

2. Methodology

The authors propose a non-perturbative, exact real-space field theoretical approach to solve the scalar wave equation for an inhomogeneous elastic medium.

• Governing Equation: The study utilizes the scalar wave equation in an inhomogeneous medium:
  $$\partial_t^2 u(\mathbf{r}, t) - \nabla \cdot [\kappa(\mathbf{r}) \nabla u(\mathbf{r}, t)] = 0$$
  where $\kappa(\mathbf{r})$ is the spatially fluctuating elastic modulus and $u$ is the vibrational intensity.
• Green's Function Approach: Instead of transforming to k-space (which introduces complex convolutions and integral equations), the authors solve for the Green's function $G_\omega(\mathbf{r}, \mathbf{r}')$ directly in real space:
  $$[\omega^2 + \kappa(\mathbf{r})\nabla^2 + \nabla\kappa(\mathbf{r}) \cdot \nabla] G_\omega(\mathbf{r}, \mathbf{r}') = -\delta(\mathbf{r} - \mathbf{r}')$$
• Numerical Implementation:
  • The differential operator is discretized using the finite difference method, converting the problem into a system of linear equations ($H G = \delta$).
  • The spectral function $A_\omega(\mathbf{r}, \mathbf{r}')$ is derived from the imaginary part of the Green's function.
  • The Local Density of States (LDOS) is obtained as $A_\omega(\mathbf{r}, \mathbf{r})$, and the Total DOS is the spatial integral of the LDOS.
• Modeling Inhomogeneity: The elastic modulus $\kappa(\mathbf{r})$ is modeled as a random field with a specific correlation length ($L$) and inhomogeneity intensity. This is generated by applying a filter in k-space to uncorrelated random numbers and transforming back to real space.

3. Key Contributions

1. Shift from Perturbative K-Space to Exact Real-Space: The paper demonstrates that solving the field theory directly in real space avoids the mathematical complications of convolutions in k-space and allows for numerically exact solutions without perturbative assumptions.
2. Mechanism of the Boson Peak: The study identifies the origin of the Boson Peak not as a flat dispersion relation (as suggested by some string-like excitation theories), but as a selective scattering effect induced by elastic inhomogeneity.
3. Visualization of Vibrational Inhomogeneity: The method provides a direct visualization of the Local DOS in real space, revealing "highly excited spots" (soft regions) that correspond to the anomalous modes observed in particle simulations.
4. Correlation Length Scaling: The work establishes a quantitative link between the correlation length of the elastic inhomogeneity and the frequency of the Boson Peak.

4. Key Results

• Selective Scattering Mechanism:
  • Elastic inhomogeneity acts as a filter: it suppresses the spectral function for short wavelengths (high wave vectors $q > 1/2L$) due to strong scattering.
  • Conversely, it enhances or leaves unaffected the spectral function for long wavelengths ($q < 1/2L$).
  • This redistribution of spectral weight leads to a reduction in DOS at high frequencies and a corresponding excess (peak) at low frequencies, violating Debye's law ($g(\omega) \propto \omega^2$).
• Inverse Relationship with Correlation Length:
  • The frequency of the Boson Peak ($\omega_{BP}$) is inversely proportional to the correlation length ($L$) of the elastic fluctuations:
    $$\omega_{BP} \sim \frac{\bar{\kappa}^{1/2}}{2L}$$
  • This finding aligns with recent experimental and simulation results where the BP frequency scales with the size of localized excitations (e.g., string-like modes).
• Emergence of Highly Excited Spots:
  • At low frequencies (within the BP regime), the LDOS is highly non-uniform. It concentrates in specific "highly excited spots" located in soft regions (low $\kappa$) surrounded by stiff cages (high $\kappa$).
  • As frequency increases, these spots "explode" into ring-like structures, and the LDOS becomes more uniform.
  • This confirms that vibrational inhomogeneity (spatially varying LDOS) is a direct consequence of elastic inhomogeneity.
• Absence of Flat Dispersion: Unlike theories relying on "flat modes," this model reproduces the Boson Peak without a flat dispersion relation, suggesting that the BP can arise purely from scattering mechanisms in disordered elastic media.

5. Significance

• Bridging Theory and Simulation: The results provide a theoretical framework that bridges continuum elastic theory with particle-level simulations. The visualization of "highly excited spots" offers a concrete real-space interpretation of the "string-like excitations" and localized modes observed in granular and colloidal systems.
• Refining the Origin of the Boson Peak: The paper challenges the notion that the Boson Peak requires a flat dispersion relation. It proposes that selective scattering due to elastic inhomogeneity is a sufficient mechanism to generate the BP, offering a unified explanation for systems where flat modes are not observed (e.g., strain glasses).
• Methodological Advancement: By demonstrating the efficacy of real-space field theory for disordered systems, the paper offers a robust alternative to perturbative k-space methods, enabling the study of strong disorder and non-linear effects that were previously inaccessible.

In summary, this work establishes that elastic inhomogeneity alone is sufficient to explain the low-frequency vibrational anomalies in amorphous solids, characterized by a correlation-length-dependent Boson Peak and the emergence of spatially localized, highly excited vibrational spots.