# Pseudo-Spin Polarized One-Way Elastic Wave Eigenstates in One-Dimensional Phononic Superlattices
## 2 Model and Method
### Conditions for One-Way Propagation in Infinite Superlattices  
For the continuous system, we have shown [17] that if a band contains the frequency point \(\omega_{0}\), such that \(\text{sin}\,k_{2}d_{2}=0\), then at \(\omega_{0}(q_{0})\) the dispersion relation becomes \(\cos q_{0}L=\cos(k_{1}d_{1}+k_{2}d_{2})\), \(A_{+}=0\), and \(A_{-}=i[\text{sin}\,(k_{1}d_{1}+k_{2}d_{2})-\text{sin}\,q_{0}L]\). This condition corresponds to a Fabry-Perot resonance of the second medium in the superlattice unit cell [13]. Since \(k_{1}d_{1}+k_{2}d_{2}>0\), the sign of \(q_{0}\) determines whether \(A_{-}\) vanishes or not. From Equation (16b), \(A_{+}=0\) and \(A_{-}=0\), meaning that \(B_{+}=0\) and \(B_{-}=0\). At \(\omega_{0}(+q_{0})\), the amplitudes of the Bloch modes may be finite, while at \(\omega_{0}(-q_{0})\) the Bloch modes equal zero. This behavior was shown to be associated with the non-conventional topology of the Bloch modes of the continuous superlattice [18]. The Berry connection undergoes a \(\pi\) jump at \(q_{0}\), where the amplitudes vanish. The dispersion branches supporting zero-amplitude modes in the band structure of the superlattice exhibit a \(\pi\) Berry phase [18]. This argument also extends to a discrete superlattice with \(p>1\) in the following manner. Let us first find expressions for the eigenvalues of the transfer matrix. These eigenvalues are solutions of the equation \(\lambda^{2}-\lambda\big{(}t_{11}+t_{11}^{*}\big{)}+t_{11}t_{11}^{*}-t_{12}t_{12 }^{*}=0\). The Eigen values are given by:  
\[\lambda=\frac{1}{2}(t_{11}+t_{11}^{*})\pm\frac{1}{2}\sqrt{\big{(}t_{11}-t_{11}^ {*}\big{)}^{2}+4t_{12}t_{12}^{*}} \tag{25}\]  
For vibrational modes to be Bloch waves, we require \(\lambda\propto e^{\pm iqL}\). According to Equation (23), \(A_{+}=-t_{12}=0\) when \(\sin k_{2}d_{2}=0\). In that case, the eigenvalues reduce to:  
\[\lambda=\frac{1}{2}(t_{11}+t_{11}^{*})\pm\frac{1}{2}|t_{11}-t_{11}^{*}| \tag{26}\]or:  
\[\lambda=Real(t_{11})\ \pm i|Imag(t_{11})| \tag{27}\]  
where _Real_ and _Imag_ stand for the real and imaginary parts. The condition for the discrete superlattice to support Bloch waves takes the form of \(Real(t_{11})\pm|Imag(t_{11})|=\cos qL\pm i\sin qL\). This condition is met only for a wave number \(q\) located on one side of the Brillouin zone, where \(|Imag(t_{11})|\) can be identified as a sine function. For \(q\) on the other side of the Brillouin zone, \(|Imag(t_{11})|\) is identifiable to minus one times a sine function, which has the opposite sign of \(\sin qL\). In other words, \(\lambda\) does not span the same region of the unit circle in the complex plane as \(\varepsilon^{\pm iql}\). Under this condition, Equation (20b) leads to \(A_{-}=0\) on one side of the Brillouin zone but \(A_{-}\
eq 0\) on the other side. \(B_{+}=0\) and \(B_{-}=0\) on the side of the Brillouin zone, where \(A_{+}=0\) and \(A_{-}=0\). This result indicates that the two counterpropagating eigenmodes have different modal structures with defined handedness levels, which can be interpreted as a pseudo-spin associated with the propagation direction. In the present scenario, only one pseudo-spin state is allowed to propagate in the lattice in a specific direction, associated with spin momentum locking. It is interesting to note that for the SSH model (\(n=p=1\)), \(\sin k_{2}d_{2}=\sin k_{2}a\), and since in the denominator of Equation (23) the quantity \(\left(\delta_{+}^{(2)}-\delta_{-}^{(2)}\right)\propto\sin k_{2}a\), both terms cancel each other; \(A_{+}\) cannot vanish anymore at \(\sin k_{2}(d_{2}=a)=0\). The component of the transfer matrix \(t_{12}\
eq 0\) for the SSH system. Therefore, the SSH model does not lead to zero-amplitude Bloch modes, as was noted at the end of Section 2.2. This is also true for a superlattice with a segment of harmonic lattice 1 extending beyond one lattice spacing but with a segment of harmonic lattice 2 limited to one lattice spacing (\(n>1\) and \(p=1\)).