# Pseudo-Spin Polarized One-Way Elastic Wave Eigenstates in One-Dimensional Phononic Superlattices
## 1 Introduction  
Breaking reciprocity is one principle for forming acoustic or elastic waves traveling only in one direction (that is, one-way wave propagation). A variety of mechanisms can be used to break reciprocity, which range from symmetry breaking to exploiting non-linearity and spatiotemporal modulations [1; 2]. For example, breaking time-reversal symmetry by using gyroscopic inertial effects in lattice structures can lead to the topologically protected one-way propagation of elastic waves localized at the edges of the lattice [3]. The dependence on the amplitude of the dynamic behavior of the non-linear waves may also lead to non-reciprocity [4]. Moreover, the modulation in space and time of the physical properties of elastic media introduces a bias that breaks time-reversal symmetry and subsequently reciprocity, leading to the one-way propagation of bulk elastic waves [5; 6; 7].  
While breaking reciprocity is one route to one-way propagation [8; 9], other topologically protected systems that preserve reciprocity have been demonstrated. For example, static one-dimensional superlattices that do not break time-reversal symmetry and obey linear elasticity have been shown to support robust, topologically protected elastic eigenvalues with non-zero amplitude in the forward propagation direction but with zero amplitude in the opposite direction [10]. These one-way-propagating, topologically protected eigenwaves occur along elastic bands with a non-zero Berry phase. Due to time-reversal symmetry, these eigenwaves possess a pseudo-spin that breaks mirror symmetry, and aneigenwave with opposite handedness is expected to be supported by the system in the reverse propagation direction. Hence, differing from non-reciprocal one-way systems, these responses do not generally isolate but can be used to efficiently direct signals towards desired directions and are robust against back-reflections, as long as disorder or defects do not couple the two pseudo-spin hands.  
Here, we develop a model for one-dimensional discrete phononic superlattices, which enables us to bridge continuum superlattices that support reciprocal one-way eigenwaves and the elastic analogue of the well-known one-dimensional Su-Schrieffer-Heeger (SSH) model [11; 12], which does not generally support topologically protected, one-way-propagating bulk waves. The discrete superlattice model is investigated using the transfer matrix method to calculate its band structure and corresponding eigenmodes. We show that the transfer matrix exhibits, under specific conditions, eigenvalues that do not span the complete unit circle in the complex plane. Subsequently, such eigenmodes do not satisfy the translational invariance of the elastic modes in the form of Bloch waves over the entire Brillouin zone, unless their amplitude vanishes for a certain wave number. This condition occurs on one side of the Brillouin zone and not the other, leading to one-way wave propagation. In the case of the continuous limit of a finite superlattice, we show that reciprocity is obeyed, as expected, and the one-way nature is associated with an inherent handedness of the eigenmodes (pseudo-spin), which breaks the mirror symmetry. One-way propagation is identified as the combined effect of translational invariance, pseudo-spin, and Fabry-Perot resonances [13] in one of the constitutive media of the superlattice. These results offer an original view of unidirectional propagation, which can be probed via exciting one-way modes by spin-selective sources in 1D periodic systems.