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Topological state transfer in Kresling origami

Topological mechanical metamaterials have been widely explored for their boundary states, which can be robustly isolated or transported in a controlled manner. However, such systems often require pre-configured design or complex active actuation for wave manipulation. Here, we present the possibility of in-situ transfer of topological boundary modes by leveraging the reconfigurability intrinsic in twisted origami lattices. In particular, we employ a dimer Kresling origami system consisting of unit cells with opposite chirality, which couples longitudinal and rotational degrees of freedom in elastic waves. The quasi-static twist imposed on the lattice alters the strain landscape of the lattice, thus significantly affecting the wave dispersion relations and the topology of the underling bands. This in turn facilitates an efficient topological state transfer from one edge to the other. This simple and practical approach of energy transfer in origami-inspired lattices can thus inspire a new class of efficient energy manipulation devices.

In parallel, recent studies have explored protocols of exploiting the topological properties of several systems, quantum [32][33][34][35][36] and classical [37], for the state transfer of localized states, a process of great importance for quantum technologies. The key advantage topology offers in such process is the inherent protection of the topologically protected boundary against disorder. This is a significant improvement over the conventional systems, which usually involves energy leakage due to the noises and fabrication errors in the system. However, even in topological systems, most systems suffer from the inefficiency in energy management.
They either use pre-configured passive lattices that are not tunable after assembly [22][23][24][25][26], or use a complex setting of active elements spanning the entire structure [29,31], which is cumbersome for practical purposes.
To address such limitations, we employ a mechanical lattice consisting of origami-based mechanical units, which offers a simple strategy to tune wave dispersion relationships insitu and facilitate a robust state transfer. Origami has served as an efficient design principle to tailor kinematic and static responses in mechanical metamaterials. Lately, the origamiinspired metamaterials have also been found to be effective in realizing the wave manipulation in them, leveraging their high reconfigurability and controllability [38,39].
In this study, we specifically consider the 1D dimer lattice composed of the origami unit cells to explore the tunable wave dispersion relationship and the emerging topological edge states. The origami unit is based on the Kresling pattern with opposite chirality, which shows the axial-rotation coupling and nonlinear static responses [40,41]. Interestingly, due to this axial-rotation coupling in our system, alternating chirality along the length of the lattice itself leads to the opening of a lower band gap. We find that this lower band gap is topologically nontrivial, and we demonstrate experimentally that this lower band gap hosts topologically protected edge states.
More interesting mechanism is obtained when the lattice is twisted, which incurs the change of the axial strain and effective stiffness landscape along the lattice. As a result, the linear wave dispersion relationships of the system are altered, and in this process, we witness the emergence of another band gap in higher frequency regime. We report that this upper band gap transitions from topologically trivial to nontrivial states by changing the twist angle in time, and therefore, facilitates a robust state transfer in the lattice from one side of the lattice to the other. We numerically show such efficient boundary state transfer in the higher frequency regime and evaluate the transfer fidelity.
Notably, we find that such state transfer observed in the upper band gap cannot be achieved in the lower band gap, since it preserves its topologically nontrivial characteristics at any twist angle. This implies that by leveraging the coupled dynamics in our origami system, we can realize very distinctive energy management capabilities in different frequency regimes. Thus, our origami system hints an efficient and controllable way to manipulate multiple wave phenomena hosted within the single topological mechanical metamaterial by combining the concepts of topology and origami.

A. Physical set-up and mathematical model
We employ the geometrical and kinematic parameters shown in Fig. 1a to describe the Kresling-patterned unit cell [40]. The unit cell has two degrees of freedom: translation along z-axis (u) and rotation about z-axis (ϕ). As the unit cell gets compressed (i.e., u changes), the top surface rotates (ϕ varies), therefore exhibiting coupled behavior. The resultant forcedisplacement relationships under pure axial compression (i.e., no external torque applied) in the experiment is shown in Fig. 1b, denoted as a red dashed line.
To model this coupled folding behavior, we employ a truss model [39] by replacing the creases along segments AB and AC (denoted by red and blue solid lines in Fig. 1a) with linear spring elements, and segment AD (green solid line in Fig. 1a) with a linear torsion spring. The model gives the total potential energy expressed as, where N p is the number of vertices of the polygonal cross-section, k a and k b are the linear spring coefficients of the truss elements along AB and AC respectively, and k ψ is the torsion spring coefficient of the element along AD. The a and b are the lengths of the element AB and AC, and ψ is the angle between the horizontal surface and the triangular facet (e.g., ACD), which are function of u and ϕ (see Supplementary Note 1 for the explicit expressions). The subscripted values a 0 , b 0 , and ψ 0 correspond to their initial lengths and angle. F is the axial force along z-axis, and T is the torque about z-axis.
By applying the principle of minimum potential energy, we obtain F and T as a function of u and ϕ (see Supplementary Note 1). This analytical force-displacement relationship is shown in Fig. 1b, denoted as blue solid line. Here, the spring coefficients are empirically determined using the least-square method to fit the model curve with the experimental curve. As a result, we can see the model agrees well with the experimental force-displacement one. Note that the slopes of the experimental and analytical force-displacement curvesrepresenting the stiffness of the system-are also in agreement as shown in the inset figure of Fig. 1b.
Having the mathematical model for the unit cell, we now consider the 1D dimer lattice consisting of two different types of unit cells: one with positive chirality θ (1) 0 > 0 and the other with negative chirality θ (2) 0 < 0. Figure 1c shows the dimer lattice, where positive chirality unit cells (red-colored) and negative chirality unit cells (blue-colored) are connected through the polygonal separator, which has mass m and rotational inertia j about z-axis.
If Kresling unit cells serve as inter-polygonal springs while having negligible mass and inertia compared to the separators (see Supplementary Table 1), we obtain the equations of motion of the dimer Kresling lattice, where u n , v n and ϕ n , ϑ n are axial displacement and rotational angle of the odd-and evennumbered polygonal separators respectively. The subscripts 1 and 2 of the force and torque functions correspond to the positive and negative chirality unit cells. Figure 1d shows a schematic of the mathematical model of this coupled system, where the mass and rotational inertia are considered separately, such that the two lattices are connected to each other with nonlinear springs to represent the coupled nature.

B. Tunable wave dispersion relationship
From now on, we consider the dimer Kresling lattice with opposite and equal magnitude chirality, namely odd-and even-numbered Kresling unit cells exhibit θ = h 0 and R (1) = R (2) = R). Our interest is the dynamics of small amplitude elastic waves, and therefore, we start by linearizing Eq. (2): where α and β are the linear coefficients of the positive and negative chirality unit cells, respectively (see Supplementary Note 2 for the detail). Furthermore, we substitute the Bloch wave solution in Eq. (3) and perform the Fourier transformation to get the eigenvalue problem, where k and ω are the wave number and angular frequency, respectively;D k is the dynamical In this coupled 1D system, we can observe four branches: two lower and two upper branches (see the enlarged view in Fig. 2d; two lower branches are almost collapsing onto each other) [41]. Note that at k = π/h 0 there is degeneracy between the two lower and the two upper branches. Between the upper and the lower branches (i.e., second and third branches), we see a wide band gap denoted as BG1 in Fig. 2c. Notably, this band gap emerges due to only the opposite chirality in our system (i.e., α 12 = −β 12 in Eq. (4)), without the necessity of dimerizing axial or rotational stiffness (i.e., without changing any geometric parameters along the lattice to alter α 11 and β 11 ). Alternating chirality itself introduces a band gap in the dispersion relation, and this is a key feature of our system distinctive from previous studies [42,43]. Now, we explore the in-situ tunability of the wave dispersion relationship, without replacing or changing the unit cell design, but by just twisting the lattice. Here, we assume that the lattice has an even number of unit cells in total (see Supplementary Note 3 and 5 for more information). Interestingly, if we quasi-statically twist the lattice along the z-axis without changing the total length of the lattice (i.e., the fixed boundary at both ends), we can change the strain landscape of the lattice. For instance, starting from the natural configuration ( Fig. 2a), as we twist the lattice in the positive ϕ direction, we can see that  Figure 2e shows the linear wave dispersion relationships for three different twist angles per supercell: In the lower frequency regime, either positively or negatively twisting the lattice induces the first branch to shift slightly downward and the second branch to shift significantly upward. As a result, the degenerate point at k = π/h 0 disappears, and the BG1 becomes narrower. In the higher frequency regime, we see that an additional band gap labeled as band gap 2 (BG2) opens between the third and fourth branches (see also Supplementary Video 2). Strikingly, the topological nature of these band gaps are highly distinctive, as will be discussed next.
We note in passing that the analysis above is specific to the lattice with an even number of unit cells, which shows symmetric behavior when twisted positively and negatively. See Supplementary Note 3 and 5, Supplementary Figure 1-2 for the comparison between the lattices with even and odd numbers of unit cells.

C. Topological characterization
For the topological characterization of the system, we first show that the dynamical matrixD k can be written asPD whereP is the symmetry operator (please see Supplementary Note 5 for the detail). In the absence of degeneracies, the Zak phase of a band can be obtained by accumulating the phase resulting from the corresponding eigenvectors all over the first Brillouin zone (BZ; Fig. 2e). If we discretize the BZ with K points, the Zak phase for mth isolated band is defined as [44,45] Here, the band index runs from m = 1 to 4. Except for the case of natural condition (i.e., zero twist angle; see below) the four bands in the dispersion relation do not have a degenerate point aside from the origin (k = 0). Therefore, we directly apply Eq. (6) to obtain the Zak phases of the bands, which are labeled in the left and right panel of Fig. 2e, respectively.
Indeed the Zak phases of the four bands take the values 0 or π, while interestingly enough, these values are switched for the negatively and positively twisted lattices. To topologically characterize the band gaps, we sum the Zak phases below the corresponding band gaps [46].
We find in Fig. 2e that for both negatively and positively twisted lattices, the BG1 holds a sum of Zak phases π, which indicates that it is a topological nontrivial band gap, a property that does not change with the value and sign of the twisting angle. A special treatment is needed for the case of zero twist angle, because the bands are degenerated at the end of BZ (k = π/h 0 ) and therefore Eq. (6) can not be used. For such a case, we obtain the topological index for the bands mth and (m + 1)th (that are degenerated at some point) together via many band Berry phases (Wilson-loop eigenvalues) [45,47], such that In our system, we calculate that φ 1,2 = φ 3,4 = π under the natural condition. We may thus conclude that BG1 is nontrivial even for the case of zero applied twisting.
Now, for BG2, the sum of the Zak phase below BG2 is π for the negatively twisted lattices and 0 for the positively twisted lattices. This implies that the BG2 exhibit a topological phase transition (from nontrivial to trivial) as the twist angle varies from negative to positive.
At the special case of zero twist, the gap closes. This distinctive topological nature of the BG2 as a function of the applied twisting angle enables the topological state transfer across the lattice, as will be discussed and verified in the following sections.
where u is the eigenvector, K is the commutation matrix, and w is the weighting vector (see Within the band gap, linear analysis predicts the localization at the left end for the 15th mode (red line in Fig. 4g). If we extract the normal mode near f = 158 Hz from the experimental result (where FFT shows high intensity in Fig. 4c), we see the equivalent localization phenomenon as the blue solid line in Fig. 4g shows. Recall that our eigenanalysis in the previous sections identified the mode inside the BG1 as topological mode regardless of the twist angle. This implies that the experimentally observed normal mode in Fig. 4g  We employ loading protocol that initiates and terminates smoothly, to avoid the excitation of other modes. The boundary angle profile ϕ b is expressed as, where T is the total loading time, and the superscripts (0) and (1)   This is in agreement with the previous analysis in Fig. 3 (also reprinted in the inset panel of end of the lattice, respectively, which decays exponentially as presented in Fig. 5c and e. As noted before, during the transition between these two angles, the normal mode can exhibit sinusoidal-like bulk mode profile (Fig. 5d) when quasi-static torsion is absent (ϕ b = 0 • ). This normal mode is reminiscent of those shown through linear eigenanalysis in Fig. 3f, and it is the mode that plays as a vehicle to transfer the energy through the lattice between the two localized modes. This can be confirmed by the eigenmode transition curve in the sub-panel of Fig. 5b, where we evidently see the change of the color-from red to blue, representing the migration of the localization index LI-along the curve. (See also Supplementary Video 3 for the comparison between BG1 and BG2 cases.) To quantify and confirm the state transfer from one edge to another, we define the fidelity F via the dot product [32,49], where u = (u 1 , v 1 , ϕ 1 , ϑ 1 , · · · , u N , v N , ϕ N , ϑ N ) T is the normal mode vector based on the envelope function extracted through Hilbert transform (see Supplementary Note 9). The subscript target refers to the analytically predicted normal mode at ϕ b = 240 • , which corresponds to the excited eigenfrequency. Note that when the numerically solved normal mode vector u is similar to the analytical prediction u target , we obtain F → 1. Figure 5f shows the fidelity as a function of time for the cases shown in Fig. 5a  Wave dispersion relationships of the dimer Kresling lattice are also shown. c All four branches shown with the band gap (BG1) as blue shaded area. d Enlarged view of the lower two branches.
The color map of the solid lines represents the intensity of the axial component described by the polarization factor P u = m|u k | 2 + m|v k | 2 m|u k | 2 + m|v k | 2 + j|ϕ k | 2 + j|ϑ k | 2 . e The linear wave dispersion relationships for negatively twisted lattice (left), natural condition (center, reprinted from panel