Quench dynamics and bulk-edge correspondence in nonlinear mechanical systems

We study a topological physics in a one-dimensional nonlinear system by taking an instance of a mechanical rotator model with alternating spring constants. This nonlinear model is smoothly connected to an acoustic model described by the Su-Schrieffer-Heeger model in the linear limit. We numerically show that quench dynamics of the kinetic and potential energies for the nonlinear model is well understood in terms of the topological and trivial phases defined in the associated linearized model. It indicates phenomenologically the emergence of the edge state in the topological phase even for the nonlinear system, which may be the bulk-edge correspondence in nonlinear system.

Topological physics is mostly studied in linear systems. There are only a few studies on it in nonlinear systems [53][54][55][56] because the study of the topological properties is not straightforward. One of the reasons is that it is a formidable problem to obtain a band structure and a topological number in a generic nonlinear system. Recently, it is proposed to account for the topological properties in nonlinear systems phenomenologically based on the bulk-edge correspondence well established in the linear theory 56 . It seems to work provided that the nonlinear theory is continuously connected to a linear theory where the topological number is well defined. One may say that the topological properties are inherited from a linear theory to a nonlinear theory. It is an interesting problem to explore other systems which share similar properties.
A good signal to detect the topological phase transition is to study quench dynamics starting from one of the edges in the case of one dimension based on the bulk-edge correspondence 57 . There is almost no time evolution and the state remains almost localized at the edge for a topological phase. It is because the initial state is almost given by the topological localized edge state, which has no dynamics. On the other hand, the state rapidly spreads into the bulk for a trivial phase because the initial state is composed of bulk eigen functions. Although the usefulness of quench dynamics has been established in a linear system, it is also useful in a nonlinear system 56 .
In this paper, we investigate a one-dimensional mechanical rotator model with alternating spring constants: See Fig.1. It is a nonlinear generalization of the Su-Schrieffer-Heeger (SSH) model. We solve the quench dynamics of a mechanical rotator model as an initial condition problem, where only one rotator at the left-most edge is excited initially. The potential and kinetic energies exhibit distinct behaviors depending on whether the system is in the topological or trivial phase defined in the linearized model. Namely, after enough time, they are well localized in the topological phase, while they are spread over the bulk in the trivial phase. These phenomena are understood in terms of the emergence of the topological edge state in the topological phase. It would represent the bulkedge correspondence in nonlinear system.

II. MECHANICAL ROTATOR MODEL
We consider a mechanical rotator model 13 as illustrated in Fig.1. We place rotators indexed by the number j, j = 1, 2, · · · , N . Each rotator has the radius 0 , and its center is fixed at the position (jL 0 , 0) on the x axis, with L 0 the distance between two adjacent centers. The dynamical variables are angles θ j for rotators j.
The Hamiltonian of the system consists of the kinetic energy U K , the potential energy of the springs U spring and the gravitational energy of the U g , They are given by with the mass m,  10), respectively. This illustration is argued to hold even in nonlinear system described by Eq.(9).
with the gravitational constant g, and where L j (θ j , θ j+1 ) is the length of the spring between j and j + 1 nodes, The equation of motion is given by The spring constant is assumed to be alternating, where the dimerization is controlled by λ with |λ| ≤ 1. For λ < 0, the spring constant κ j with odd (even) j is strong (weak). On the other hand, for λ > 0, the spring constant κ j with odd (even) j is weak (strong): See Fig.2.

III. LINEARIZED MODEL
Provided the angle θ j is small enough, the potential energy is well approximated by the harmonic potential, and the equation of motion is obtained as which is a linearized model. Eq. (12) is rewritten in the form of where is identical to the SSH Hamiltonian. After a Fourier transformation, we have It is known 13,14 that the system is topological for λ < 0 and trivial for λ > 0. There are two isolated edge states in the limit λ −1, while all of the states are dimerized in the limit λ 1: See Fig.2.
The topological number associated with the SSH Hamiltonian is the chiral index defined by where K is given by Eq. (15). We obtain Γ = 1 for λ < 0 and Γ = 0 for λ > 0.
Our interest is the effect of the nonlinearity based on Eq.(9) instead of the linear equation of motion (12).

IV. NONLINEAR QUENCH DYNAMICS AND BULK-EDGE CORRESPONDENCE
The quench dynamics starting from a state localized at one edge well captures the topological phase transition in linear systems 57 , where the eigenfunctions are easily obtained and a topological number is well defined. In the topological phase, there are topological edge states well localized at edges, which is known as the bulk-edge correspondence. If we excite only one edge site, the most component is dominated by a topological edge state. The topological edge state remains as it is after time evolution. On the other hand, there is no such localized edge state in the trivial phase. Hence, all of the components of the initial state are bulk eigenfunctions. They are spread into the bulk after the time evolution. Thus, it is possible to distinguish topological and trivial phases by checking whether the state is localized or spread into the bulk.
The above observation is also applicable even for nonlinear systems 56 . The existence of the localized topological edge state is obscure in nonlinear systems because it is not possible to diagonalize the Hamiltonian and obtain eigenfunctions. Nevertheless, the quench dynamics shows distinct behaviors between the topological and trivial phases as in the case of the linear system. Thus, the quench dynamics starting from one of the edges is a strong signal to detect a topological phase transition.
We show the time evolution of sin θ j , the potential energy U spring j and the kinetic energy U K j in Fig.3. The quench dynamics of sin θ j is significantly different between the topological and trivial phases. The amplitude of sin θ 1 at the left edge site remains finite in the topological phase. On the other hand, the amplitude of sin θ 1 decreases in the trivial phase. The potential energy U spring j remains well localized at the left edge in the topological phase although the absolute value of sin θ j spreads into the bulk as shown in Fig.3(b). On the other hand, it moves as if it were a soliton in the trivial phase as shown in Fig.3(e). If θ 0 = π/2, there remains a finite com- ponent in U spring j as well as a soliton-like propagation. This will be because that sin θ j takes a maximum for θ 0 = π/2, where the dynamics is most enhanced and it takes longer time to reach a steady state.
We also show the time evolution of the kinetic energy U K j in Fig.3(c) and (f). The behavior of U K j is quite similar to that of U spring j .
The dynamics between θ 0 = π/4 and 3π/4 are very similar as shown in Fig.3. It will be due to the fact that sin π/4 = sin 3π/4 although we have cos π/4 = cos 3π/4. These results indicate a symmetry between the angle θ 1 and θ 2 satisfying the condition sin θ 1 = sin θ 2 , where the dynamics is similar.
The potential and kinetic energies after enough time present a good signal for the topological phase transition comparing to the dynamics of sin θ j . We show the kinetic energy U K j and the potential energy U spring j after enough time as a function of λ in Fig.4. It is finite for the topological phase with λ < 0, while it is almost zero for the trivial phase with λ > 0. These features are common for all initial conditions θ 0 = π/10, π/4, π/2 and 3π/4. It indicates the bulk-edge correspondence in nonlinear systems. It suggests the validity of the topological number (16) even in strong nonlinear regime.
A comment is in order with respect to finite components around λ 1 in Fig.4. They are interpreted as follows. In the limit λ 1, the system is almost dimerized as shown in Fig.2, where there are no isolated sites at the both edges. In this limit, the energy is not well transferred to the bulk because the energy is localized in the dimer located at the edge to which the energy is injected, resulting in finite components around λ 1.

V. DISCUSSION
We have shown that the topological and trivial phases are well differentiated in the mechanical rotator model even in a strong nonlinear regime based on the bulk-edge correspondence. Our finding is that the topological properties are inherited to the nonlinear model from the associated linearized model provided they are smoothly connected. Then, it would be possible to use a topological number defined in the linearized model. This phenomenon is quite similar to the one in the Toda lattice 56 , which is a typical exactly solvable model containing a soliton.
The author is very much grateful to N. Nagaosa for helpful discussions on the subject. This work is supported by the Grants-in-Aid for Scientific Research from MEXT KAK-ENHI (Grants No. JP17K05490 and No. JP18H03676). This work is also supported by CREST, JST (JPMJCR16F1 and JPMJCR20T2).