Self-induced topological transition in phononic crystals by nonlinearity management

A design paradigm of topology has recently emerged to manipulate the flow of phonons. At its heart lies a topological transition to a nontrivial state with exotic properties. This framework has been limited to linear lattice dynamics so far. Here we show a topological transition in a nonlinear regime and its implication in emerging nonlinear solutions. We employ nonlinearity management such that the system consists of masses connected with two types of nonlinear springs, “stiffening” and “softening” types, alternating along the length. We show, analytically and numerically, that the lattice makes a topological transition simply by changing the excitation amplitude and invoking nonlinear dynamics. Consequently, we witness the emergence of a family of finite-frequency edge modes, not observed in linear phononic systems. We also report the existence of kink solitons at the topological transition point. These correspond to heteroclinic orbits that form a closed curve in the phase portrait separating the two topologically distinct regimes. These findings suggest that nonlinearity can be used as a strategic tuning knob to alter topological characteristics of phononic crystals. These also provide fresh perspectives towards understanding a different family of nonlinear solutions in light of topology.

Figure 1

(a) A schematic of the “self-induced” topological transition in the system. For small amplitude excitation, the system has a trivial band gap, with no edge or bulk mode at the midgap frequency ${\mathrm{\Omega }}_m$. However, for large amplitude, nonlinearity kicks in and forces the system to a nontrivial state. This leads to the emergence of an edge mode. (b) A chain of point masses and nonlinear springs (stiffening in red and softening in blue) is designed to induce the aforementioned effect. The box highlights the unit cell.

Figure 2

(a) Dispersion relation of the unit cell with small strain amplitude $\mathrm{\Delta }A$, a linear regime. It highlights a band gap between acoustical (lower) and optical (upper) branches. (b) Increase in strain amplitude $\mathrm{\Delta }A$ changes the band gap, by closing it first and then opening again to a nontrivial state.

Figure 3

(a) Phase portrait of the bulk showing steady-state solutions at the midgap frequency. It highlights nine fixed points in black dots. The arrows in red, blue, and green indicate the presence of multiple evanescent and soliton solutions. (b) Soliton profile that corresponds to the heteroclinic orbits in green in the phase portrait. (c) Effective stiffness for all the springs along the chain, calculated for the aforementioned mode profile. This indicates the closure of “local band gap.” (d),(e) For small amplitude regimes, a profile and its effective stiffness, respectively, extracted from one of the evanescent solutions shown in red in the phase portrait. Note the amplitude forms a plateau towards the left edge. (f),(g) For large amplitude regimes, a profile and its effective stiffness, respectively, extracted from one of the evanescent solutions shown in blue in the phase portrait. Note the amplitude forms a plateau towards the right edge. (h) A schematic to show topologically distinct regions in the phase portrait that are separated by the circular trajectory of solitons.

Figure 4

Full numerical simulations verifying which bulk solutions exist in a finite chain with specific boundary condition. (a)–(c) Solutions for the boundary condition that preserves the lattice symmetry. These show space-time evolution of the absolute value of strain $\left|\mathrm{\Delta }\xi \right|$ with the initial conditions obtained from the bulk solutions in Figs. 3, 3, and 3, respectively. (d)–(f) The same but for the boundary condition that breaks the lattice symmetry. ${\tau }_m=2\pi /{\mathrm{\Omega }}_m$ represents the time period of oscillations at the midgap frequency. Spring index 1 denotes the spring attached to the wall. Note that we are not interested in seeing the effect of boundary on the soliton solutions in (b) and (e), hence those are kept the same.

Figure 5

(a) Effective stiffness of nonlinear springs as a function of strain amplitude $\mathrm{\Delta }A$ when we change nonlinearity type from cubic to saturating. (b) Resulting profile of the edge mode. (c) Effective stiffness of springs along the chain for this mode. (d) Full numerical simulation showing space-time evolution of the absolute value of strain $\left|\mathrm{\Delta }\xi \right|$ when we take the edge mode profile as the initial condition in a finite chain with the symmetry-preserving boundary.

Figure 6

(a) Space-time evolution of strain $\mathrm{\Delta }\xi$ when we excite the left wall with a Gaussian-modulated profile of displacement amplitude $A_w=0.01$. (b) Spectrum calculated from the time history of spring strains for the case above. (c),(d) The same with displacement amplitude $A_w=1.4$. (e),(f) The same with displacement amplitude $A_w=3$. Spectrum plots in (b), (d), and (f) indicate the presence of a band gap, its closure, and the emergence of an edge mode, respectively. Blue dashed lines mark the lower and upper band edges for the initial (linear) configuration. ${\mathrm{\Omega }}_c$ denotes the upper cutoff frequency of the optical branch.

Figure 7

(a) Normalized modal amplitude of the first spring attached to the wall as a function of excitation amplitude $A_w$. This shows that the emergence of an edge mode is a result of band-gap closure. (b) Modal amplitude of the first spring as a function of excitation amplitude at the midgap frequency. The critical strain amplitude $\mathrm{\Delta }{A}_0$ is marked to indicate the point of band closure and steep increase in modal amplitude thereafter.

Figure 8

(a) Phase portrait for a linear system ($\mathrm{\Gamma }=0$) showing steady-state solutions at the midgap frequency. There is a fixed point indicated by the black dot at the origin. Red arrows show the evanescent solutions in the bulk. (b) An evanescent mode solution decaying from the left corresponds to the solutions on $u=0$ axis. (c) Space-time evolution of the absolute value of strain when we use the aforementioned mode profile as an initial condition for a finite chain with the symmetry-preserving left boundary.