Small-World Disordered Lattices: Spectral Gaps and Diffusive Transport

We investigate the dynamic behavior of lattices with disorder introduced through non-local network connections. Inspired by the Watts-Strogatz small-world model, we employ a single parameter to determine the probability of local connections being re-wired, and to induce transitions between regular and disordered lattices. These connections are added as non-local springs to underlying periodic one-dimensional (1D) and two-dimensional (2D) square, triangular and hexagonal lattices. Eigenmode computations illustrate the emergence of spectral gaps in various representative lattices for increasing degrees of disorder. These gaps manifest themselves as frequency ranges where the modal density goes to zero, or that are populated only by localized modes. In both cases, we observe low transmission levels of vibrations across the lattice. Overall, we find that these gaps are more pronounced for lattice topologies with lower connectivity, such as the 1D lattice or the 2D hexagonal lattice. We then illustrate that the disordered lattices undergo transitions from ballistic to super-diffusive or diffusive transport for increasing levels of disorder. These properties, illustrated through numerical simulations, unveil the potential for disorder in the form of non-local connections to enable additional functionalities for metamaterials. These include the occurrence of disorder-induced spectral gaps, which is relevant to frequency filtering devices, as well as the possibility to induce diffusive-type transport which does not occur in regular periodic materials, and that may find applications in dynamic stress mitigation.

The physics of condensed matter systems based on small-world networks has also been explored in the form of Ising models, [14], to explore the onset of localization [15,16] along with transport in quantum lattices [17][18][19].
In this paper, we investigate the dynamics of simple elastic lattices where non-local connections are inspired by the small-world network model. In the context of phononics and elastic metamaterials, [20] the role of disorder has attracted considerable interest. For example, rainbow-based materials have been investigated for band gap widening and energy trapping [21][22][23][24][25][26][27], for wave localization [28], to study the occurrence of topological phase transitions [29] and for signal signal processing applications [30]. Also, the role of disorder has been explored in the context of phonon and thermal transport [31,32], and as part of the exploration of quasi-periodic lattices. [33][34][35][36][37][38][39][40][41] So far, most of prior studies have considered disorder introduced to local parameters or couplings, and the introduction of disorder through non-local connections defined by a network model has not yet been explored. To this end, we consider regular mono-atomic spring-mass lattices in the form of 1D lattices and of 2D hexagonal, square and triangular topologies, where connections are added based on the small-world model characterized by a chosen level of disorder. We first investigate the emergence of spectral gaps as a function of disorder through eigenmode computations performed on large lattices, and for multiple disorder realizations. Robust gaps appear in the form of frequency regions not populated by any modes, or populated only by localized modes, both resulting in low transmission levels across the lattice. These gaps appear to be more pronounced for topologies of lower connectivity, i.e. 1D and 2D hexagonal lattices which exhibit larger gaps than square and triangular lattices. Also, the analysis of transient wave behavior allows the characterization of the transport properties and the identification of transitions from ballistic to super-diffusive and diffusive transport. These are similar to the transitions experienced by quantum and photonic lattices in the presence of on-site disorder [42,43]. The investigations presented herein identify a new route for introducing disorder in metamaterials in the form of non-local connections, which holds potential for the generation of disorder-resilient spectral gaps and for applications related to impact mitigation that leverage diffusive transport, as opposed to ballistic spreading observed in periodic materials.
This paper is organized as follows: following this introduction, section II describes the modeling of the small-world phononic lattices and the associated equations of motion. Next, section III describes the spectral properties of the lattices and the emergence of band gaps through the non-local disordered links. Section IV then describes the transport properties of the network lattices and characterizes the transition from ballistic to diffusive transport.
Finally, section V summarizes the main findings of this work and outlines future research directions. The equation of motion for a mass of index n is expressed as: where r runs over the nearest neighbors, and s runs over the network links that connect to that mass, while u n and f n represent the displacement and the force applied to the n − th mass. For a finite lattice of N masses, the equations of motion can be assembled in matrix form: the corresponding modes, which is defined as: where u n is the n th component of the eigenvector. The IPR varies from 0 to 1 and signals whether a mode is localized or not when its value is high or low, respectively. A few modes are marked in Fig. 3(a) and have their mode shapes displayed in Fig. 3(b). When p = 0, the frequencies of the lattice define a continuous band with only non-localized modes, as expected of a periodic monoatomic lattice [20]. However, the lattice with p = 0.6 supports a series of localized modes, with two examples displayed in Fig. 3(b). While localized modes are expected to appear due to the presence of disorder, we also note that a few frequency Although the spectral properties of these lattices are largely influenced by the topology, in the next section we illustrate that the transport properties are qualitatively similar and that transitions to diffusive transport can be observed in all cases.

IV. TRANSIENT BEHAVIOR: FROM BALLISTIC TO DIFFUSIVE TRANS-PORT
The transient behavior of the small-world lattices is investigated next. We characterized wave motion in disordered lattices by relying on a approach [43,44] that considers the dynamic evolution of the lattice motion resulting from an initial perturbation. Such evolution is quantified by computing the Mean Square Displacement (MSD), which is defined as Here, denotes the averaging operation across multiple lattice realizations, n 0 is the site where the initial perturbation is applied, and d n,n 0 is the distance between the generic site n and n 0 . The MSD is found to scale as t γ , where t denotes time while the exponent γ quantifies the rate of perturbation spreading. Thus, γ is used to classify the type of transport that occurs along the medium. For example, quantum and photonic periodic lattices, which are governed by similar equations of motion, exhibit ballistic transport which is characterized by γ = 2 [43,44]. In the presence of disorder, decreasing values of γ quantify the slower spread that occurs in comparison to regular periodic materials. For instance, depending on the amount of disorder, lattices may exhibit super-diffusive (γ = 1.5) or diffusive (γ = 1) transport, or absence of transport altogether for γ ≈ 0, which corresponds to the onset of Anderson Localization [42,43,45]. This approach has been recently applied to other types of aperiodic systems, for example in fractal lattices where γ is found to be related to the fractal dimension of the lattice, [46] and also for the characterization of wave packets spreading in disordered non-linear architected materials [47].
Here, we adopt the MSD to characterize the transport in the small-world lattices. Starting from the 1D case, we consider a large lattice with N = 1000 masses and apply a perturbation to the n 0 = 500 site. The perturbation is in the form of initial conditions u n 0 (0) = 0,u n 0 (0) = 1, which are equivalent to an impulse excitation f (t) = δ(t) applied to the chosen site. This excitation involves the entire spectrum of the lattice and is similar to the excitation applied to photonic lattices at z = 0, where z is the propagation dimension [43]. The results for the 1D lattices are summarized in Fig. 6. Figures 6(a,b)  corresponds to a transition from ballistic to super-diffusive transport as γ approaches 1.5 for increasing disorder levels. The second example in Fig. 6(b) illustrates the transition from ballistic to diffusive transport as γ becomes closer to 1.
Next, the transport behavior for the 2D hexagonal, square and triangular lattices is illustrated in Fig. 7. The procedure for the 1D case is extended to 2D lattices of sizes 29 × 52, 41 × 41 and 49 × 57 , respectively. The number of cells has been chosen in order to form domains of similar length along the x and y directions, and which are sufficiently large to observe the spreading of the perturbation applied as initial conditions to the center mass.
Similar to the 1D case, the results are obtained upon averaging again over 200 realizations and extracting γ from the M SD curves for each p, α combination. Figures 7(a,b,c) display the resulting variation of γ with respect to p for the hexagonal, square and triangular lattices, with α ranging from 0.5 to 6. As in the 1D case, we again find that periodic lattices for p = 0 are characterized by ballistic transport properties associated with γ = 2. This is verified for any value of α for all the lattice topologies considered, and it is expected for periodic lattices in general. [44] The figures also illustrate how multiple transport regimes are achieved in all lattice topologies by different choices of p, α. Two transitions with α = 2 and α = 6 are exemplified for the hexagonal lattice, where the points marked in Fig. 7(a) have their corresponding averaged displacement fields displayed in panels (d,e). The plots show the absolute value of displacement across the lattice in the x, y plane at 5 subsequent time instants, with time varying along the vertical axis. The displacement for a sectional x, t plane defined for the center y coordinate is also plotted to improve the visualization of the wave spreading as a function of time. Also, due to the amplitude decrease resulting from wave spreading, the color axis in each plot is restricted to a range corresponding to 10% of the maximum displacement value along the entire time history. For α = 2, a transition to super-diffusive behavior (γ ≈ 1.5) is observed, while α = 6 produces a transition to diffusive transport (γ ≈ 1). The associated decrease in the spreading can be clearly observed in the displacement plots by observing how the wave front propagates shorter distances in the p = 1 cases when compared to the ballistic p = 0 cases. Overall, the transport transitions are very similar for all the considered 2D lattice topologies, with small qualitative differences in the γ variations with p and α.
These results illustrate how the disorder introduced through the network connections modify the type of transport for all the considered lattice topologies, causing a transition to super-diffusive or diffusive transport when the strength of the network connections (α) is sufficiently strong. These transitions are reminiscent of those experienced by quantum and photonic lattices with on-site disorder [42,43]. However, we note that for the considered range of α values, γ reaches a plateau close to 1, and Anderson Localization (γ ≈ 0) does not occur. Higher values α > 7.5 are not considered herein since the network connections become much stronger and overcome the couplings of the underlying lattice. Our preliminary investigations showed that the transport in that case was not well captured by the MSD computations, similarly to findings in quantum lattices with distance-independent coupling and absence of an underlying lattice [17]. Such high α regime may be further investigated in the future.

V. CONCLUSIONS
In this paper, we investigate the dynamics of phononic lattices with small-world network connections. Our results illustrate the emergence of spectral gaps due to increasing degrees of disorder, which are persistent across multiple lattice realizations. Lattices of different topologies, such as 1D and 2D hexagonal, square and triangular lattices were shown to feature transitions from ballistic to super-diffusive or diffusive transport. These results motivate a new route for the introduction of disorder in metamaterials through network connections, potentially leading to novel functionalities enabled by disorder such as spectral gaps and diffusive transport, which could be exploited in impact mitigation applications for example. The initial investigations presented here may be expanded in multiple directions in future studies. For example, a variety of other network modeling strategies may be