## 1 Introduction
We choose the components of the \(\Psi_{2N\times 1}\) vector in the form of plane waves:  
\[\psi_{I}=a_{B}e^{ikx}e^{i\omega t} \tag{5}\]  
with \(I=1,\ldots,2N;k\) and \(\omega\) being a wave number and an angular frequency, respectively. The amplitude vector \((a_{2N\times 1})\) can be written as:  
\[a_{2N\times 1}=e_{n,N\times 1}\otimes s_{2N\times 1}, \tag{6}\]  
where \(e_{n,N\times 1}\) is the eigen vector of the matrix \(\sqrt{M_{N\times N}}\) (i.e., \(M_{N\times N}\)) with eigen value \(\lambda_{n}\). The operato \(M_{N\times N}\), its eigen values and eigen vectors are representative of the spatial modes across the planar array of the waveguides. Inserting this tensor product solution into Equation (3) yields:  
\[e_{n}\otimes\left\{([\omega I_{2\times 2}-\beta k\sigma_{z}]\pm\alpha\lambda_{ n}\sigma_{x})s_{2\times 1}\right\}=0 \tag{7}\]  
For non-trivial eigen vectors \(e_{n}\), the problem in the space of the directions of propagation reduces to finding solutions of  
\[\left([\omega I_{2\times 2}-\beta k\sigma_{z}]\pm\alpha\lambda_{n}\sigma_{x})s_{2 \times 1}=0 \tag{8}\]  
Figure 1: Schematic of the two- and three-rods experimental systems. The rods are composed of aluminum and are elastically coupled with epoxy. In obtaining Equation (7), we have also used the fact that \(e_{n}\) is an eigen vector of \(I_{N\times N}\) with eigen value 1 and we note that Equation (8) is the 1D Dirac equation for an elastic system which solutions,  
\[s_{2\times 1}=\left( \begin{array}{c}s_{1}\\ s_{2}\end{array} \right) \tag{9}\]  
have the properties of Dirac spinors [18]. This eigen equation gives the dispersion relation  
\[\omega_{n}^{2}=(\beta k)^{2}+(\alpha\lambda_{n})^{2} \tag{10}\]  
and the following spinor eigen vectors projected into the space of directions of propagation:  
\[s_{2\times 1}^{(n)}(\pm)=s_{0}\left( \begin{array}{c}\sqrt{\omega_{n}+\beta k} \\ \pm\sqrt{\omega_{n}-\beta k}\end{array} \right) \tag{11}\]  
The upper script in \(s_{2\times 1}^{(n)}\) refers to the spinor associated with band \"n\". The spinors components represent elastic wave amplitude propagating in the forward and backward directions of propagation [18]; hence,  
\[\Psi_{2N\times 1}(k)=\left\{ \begin{array}{c}s_{2\times 1}^{(n)}(+k)u_{N \times 1}\\ s_{2\times 1}^{(n)}(-k)u_{N\times 1}\end{array} \right\} \tag{12}\]  
However, due to the finite length of the experimental setup (i.e. the acoustic waveguides) that only supports standing wave modes, the components of the spinors has to be converted which includes equi-amplitudes superposition of forward and backward components. Nonetheless, the wave vector, \(k\), is still a good number to characterize the standing wave state in the system. By means of laser vibrometry, one can experimentally map the elastic standing waves states and their corresponding pseudospin states (as is done in [26; 46]). Since Equation (3) is linear, any superposition of modes is also a solution. Thus, it is possible to create a nonseparable superposition of elastic states in an array of coupled waveguides [42; 43; 44]. It is true that at steady states, the displacement field, \(u_{N\times 1}\), takes the form of a linear combination of frequency modes (see Supplementary Materials and [45]). However, it is possible to choose a particular isofrequency driver, which will minimize the amplitude of the \(e_{1,N\times 1}\) spatial mode compared to that of \(e_{2,N\times 1}\) and \(e_{3,N\times 1}\) (\(e_{2,N\times 1}\) and \(e_{3,N\times 1}\) correspond to two pseudospin bands with non-zero spatial eigen values [42; 43; 44]). For instance, exploiting the orthogonality of \(e_{1,N\times 1}\), \(e_{2,N\times 1}\) and \(e_{3,N\times 1}\), one may employ a driving force such that it will be a linear combination of \(e_{2,N\times 1}\) and \(e_{3,N\times 1}\).