Space-Time Phononic Crystals with Anomalous Topological Edge States

It is well known that an interface created by two topologically distinct structures could host nontrivial edge states that are immune to defects. In this letter, we introduce a one-dimensional space-time phononic crystal and study the associated anomalous topological edge states. A space-decoupled time modulation is assumed. While preserving the key topological feature of the system, such a modulation also duplicates the edge state mode across the spectrum, both inside and outside the band gap. It is shown that, in contrast to conventional topological edge states which are excited by frequencies in the Bragg regime, the time-modulation-induced frequency conversion can be leveraged to access topological edge states at a deep subwavelength scale where the entire phononic crystal size is merely 1/5.1 of the wavelength. This remarkable feature could open a new route for designing miniature devices that are based on topological physics.

There are two major findings in this study: (1) the creation of anomalous topological edge states in the bulk band; (2) the possibility to excite the edge state in the Bragg regime using ultralow frequencies whose wavelengths could be orders of magnitude greater than the periodicity of the PC. These features could bring about new designs of miniature devices based on topological phases for unconventional wave manipulation, excitation, and detection. Although the current study focuses on 1D acoustic systems, the theory developed here can readily be extended to higher dimensions and other wavebased systems.
STM of the medium properties has been proposed as a robust way to break reciprocity which is an intrinsic property of wave dynamics in conventional systems. Consequently, non-identical wave transmission in opposite directions has been achieved. Space-time (ST) photonic crystals were in fact introduced decades ago [ [36][37][38] and the subject on non-reciprocity has recently been reinvigorated in the field of photonics for the control of electromagnetic waves using time-variation of permittivity and/or permeability [39][40][41][42][43]. There has also been numerous works regarding STM in acoustics [44,46] and elastic waves where the mechanical properties of the medium are ST modulated to achieve nonreciprocal wave propagation [47][48][49][50]. Meanwhile, it has been shown that external time-dependent perturbations can be used to achieve topological spectra in initially trivial systems at the equilibrium [6,16,51]. This class of systems is known to be the basis of Floquet topological insulators where timeperiodic potential is introduced to a Hermitian system which leads to nontrivial duplicated edge states in the band structure [52][53][54].
In this work, in contrast to the approach for building Floquet topological insulators, we start from a stationary 1D system possessing nontrivial topological phases as well as topologically protected edge states inside a BG. This initial system is constructed from two space-modulated PCs exhibiting distinct Zak phases, i.e., 0 and π. We then introduce the time modulation into both PCs in a way that the Zak phase difference between the two PCs remains unchanged throughout the entire duration of time modulation [53]. The resulting PC is referred to as the space-time PC: a "living" PC that evolves in time. As will be shown below, frequency conversion occurs in the space-time PC, which gives rise to the duplication of edge states across the entire temporal spectrum. This, as shown later, is the underlying physics for the anomalous edge state discovered in this study.
We first consider two PCs having a sinusoidal space modulation of the medium density ( ). The governing equations in acoustics in 1D read [55]: (1) where , , and are the pressure, the particle velocity, the density, and the compressibility of the medium. This study assumes to be a constant, i.e., space-and-time independent, though we can conduct a similar study where the density is constant and is space-and-time dependent.
The top left panels of Figs phase can be also confirmed by using the following formula [24]: where is the wave number and , the pressure field distribution (real part of the complex pressure) is plotted in Fig.1 (d) during one time period ⁄ . The acoustic energy is clearly confined in the vicinity of the interface between the two PCs (x=0). It is noted that although sinusoidal space modulation is chosen in this study, the theory developed here can be generalized to other types of modulation (e.g., square modulation) and the main physics is expected to be the same.
The time modulation is subsequently introduced to both PCs in the following manner: where and are assumed. A modulation frequency of is chosen which is close to the edge state's frequency of the stationary system (corresponding to β = 0).
This allows us to excite an ultralow frequency edge state mode as will be shown later. Please note that the density here is not assumed to be weakly modulated as was done in [45]. This time modulation is space-decoupled in the sense that the functions (sine and cosine) governing the space and time modulations are independent from each other, which is largely unexplored in the past. One can visualize this as the density of the system only having its amplitude varying with time. Fig. 2 (a) shows the time variation of the reduced density ( ) of the super-cell having 14 unit cells.
This constitutes the domain in which Eq. 1 is to be solved. Another type of ST modulation where the density has a wave-like variation (so-called space-coupled time modulation), which has been explored for non-reciprocal propagation, is discussed in the supplementary material where the fundamental differences between the two modulation are delineated [56]. With the space-decoupled time modulation of the super cell described by Eq. (3), each PC has its Zak phase fixed with time. This is because only the amplitude of the density is changing with time, so that the symmetry of the second band is conserved at each instant. Consequently, the Zak phase difference between the two PCs remains independent of time.
Knowing that the density is periodic in both space and time, one can apply the Floquet theorem: where ̃ and ̃ are periodic both in time and space. Inserting these expressions into Eq. (1), the following eigenvalue problem can be constructed where is the eigenvalue to be solved for each wavenumber k : The time variable t can be considered as a synthetic dimension in addition to the real physical dimension x. Subsequently, Eq. (5) can be solved in the space-time domain displayed in Fig. 2(a) along with the Floquet boundary conditions. Such a method can be generalized to solving wave equation problems involving media with arbitrary space-time periodic properties.
In this study, Eq. within the vicinity of x=0: the first one is located at frequency (Fig 2.(c)) and it corresponds to the edge state inside the BG in the stationary case; the second and third modes are located exactly at (Fig 2.(b)) and (Fig 2. (d)). These modes are anomalous edge state, and their emergence in the bulk band can be attributed to the time-modulationinduced frequency conversion, which changes the topological feature of the structure leading to duplicated edge states at frequencies where n is an integer [52][53][54]. The positions of these modes in the band can be tuned by changing the modulation frequency as shown in Fig 2. (e), where the theory (lines) agrees very well with the simulation (circles). To gain a better insight on these modes, Fig. 3(a) illustrates the calculated transmission spectrum through the STM system. The calculation is performed in the time domain where we analyze the Fourier spectrum of the transmitted wave through the structure using a wideband incident pulse (a Gaussian-modulated cosine function). A transmission peak corresponding to the edge state is observed inside this BG at the normalized frequency which corresponds exactly to the stationary case. Interestingly, the system remains reciprocal and the BG does not experience frequency shift from the stationary case, which is a major departure from the commonly studied space-coupled time modulation where reciprocity is broken [45][46][47]. In these systems, the space-time perturbation or modulation of the intrinsic property of the medium (density, compressibility or stiffness) has a driven or wavelike directional motion which breaks the reciprocity. This is not the case for our spacedecoupled time modulation which preserves reciprocity. We also observe two less visible transmission peaks at 0.014 and 0.9821 (insets of Fig. 3(a)) corresponding to the excitation of the first-order edge states outside the BG at , which confirms the prediction of the earlier eigenvalue analysis.
These peaks appear less pronounced in the spectrum compared to the one at because they are located outside the BG. The bottom panel of Fig. 3(b) shows the wave pressure field for a monochromatic excitation at the frequency where one can observe the localized edge state mode at the interface x=0. This mode, however, is not pure in its Fourier components, although the excitation is strictly monochromatic. To this end, we apply temporal Fourier transform to the wave field shown in Fig. 3(b) and the results (referred to as the spatial-frequency spectrum of the wave field) are presented in Fig. 3(c) for the case of monochromatic excitations at and 0.014, respectively plotted in the left and right panels. In other words, these are the pressure fields along the PC across the frequency spectrum. The spatial domain has a range of ⁄ . In the case of an incident wave at (left panel of Fig. 3(c)), edge state modes confined at the center of the system (in the vicinity of x=0) are excited both at this same frequency and at , albeit with the latter having a much smaller amplitude.
Meanwhile, if we choose a monochromatic excitation exactly at 0.014, the wave field plotted in the right panel of Fig. 3(c) suggests that it is possible to excite the edge state at inside the BG. We also would like to point out that it is likely that the edge state at is also excited but its characteristics (e.g., energy confinement) is not clearly visible as it is "drown" by the wave field excitation at this frequency. This has happened partially due to the fact that this frequency, again, is located outside the BG. Remarkably, for the case of monochromatic excitation at 0.014, the incident wavelength is at and the frequency conversion allows the excitation of the edge state inside the BG at in the Bragg regime. Giving that the entire phononic crystal has a length of , the functionality of the structure is effective though its size is only 1/5.1 of the incident wave wavelength (vs. 7×wavelength in the stationary case). Though the peak pressure of the first-order edge state is about 20 dB lower than that of the zeroth order edge state, it is still well within the detectable range under a reasonable signal-to-noise ratio.
Finally, we show that these edge states are remarkably robust to defects both in time and space. For instance, we consider a defect of size ⁄ in the vicinity of the interface located at x=0 between the two PCs, where the density is static. Thus, the whole system has its density following the form: where and .
We conduct a similar study to show that this defect does not affect the characteristic of the PC.  Figure 4(a) presents the transmission spectrum for wave propagation through the structure where we can clearly see that the curve is very similar to the one of the system without the defect in Fig. 3(a). In fact, the two curves would virtually overlap if we were to plot them together. We can also observe the edge state inside this BG (the peak) at the normalized frequency =0.5 along with two transmission peaks located at 0.016 and 0.9841 which correspond to the excitation of the first order edge states outside the BG at . Because of the defect, these frequencies are slightly offset from the original ones (0.014, 0.04981 and 0.9821) by a small amount that is roughly 0.002. We plot in Fig. 4(c) the spatial-frequency spectrum of the wave field for monochromatic excitations at and , respectively in the left and right panels. We can clearly observe the same results as in Fig. 3(c) for the system without the defect.
The edge states, including the anomalous ones, are therefore topologically protected against the defect and can be excited at a deep subwavelength scale where the incident wavelength is in this case. In conclusion, we propose to enrich the functionalities of topological devices via time modulation, which gives rise to the duplication of topological edge states across the entire spectrum.
This feature can be used to achieve deep subwavelength manipulation of the edge state in the Bragg regime, which could be proven useful for designing miniature topological devices for wave guiding, sensing, and excitation, among other intriguing functionalities. Furthermore, the system presented here remains reciprocal while we invite the reader to the supplementary material [56] where we study the space-coupled time modulated system which breaks reciprocity that, in conjunction with the anomalous edge states, could engender exotic unidirectional wave behaviors. Future work will expand the model to higher dimensions where even richer physics can be foreseen. It would be also interesting to construct a real physical model to experimentally validate the theory. Active acoustic metamaterials [45] and piezoelectric materials [49,50] could be viable options in this regard.

Acknowledgments
The authors would like to thank Dr. Meng Xiao for his critical comments on the paper.