Spatiotemporal Chaos in the Interface Growth of Topological Insulators

TL;DR

The work reveals an intrinsic interfacial instability in topological insulators arising from boundary states that generate negative surface stiffness, leading to spatiotemporal chaos in interface growth. By deriving an interface-growth equation from a thermodynamic free-energy functional, it shows that the stiffness term becomes a constant $\tilde{\gamma}=\alpha$, with $\alpha<0$ in the topological phase, causing the Kuramoto--Sivashinsky dynamics to emerge in the negative-stiffness regime. The authors validate the mechanism with tight-binding models for Chern insulators and the BHZ model, demonstrating negative stiffness in the topological phases and positive stiffness in trivial phases, and quantify surface-energy contributions via $\gamma$ and $e_b$. They further argue for a generalization to 3D TI and connect their findings to observed interfacial roughness in materials like Bi$_2$Se$_3$ and Te$_2$Se$_3$, offering an intrinsic origin of interface chaos linked to topological boundary states.

Abstract

We demonstrate that topological insulators exhibit an intrinsic interfacial instability that amplifies small interface fluctuations, resulting in chaotic behavior during interface growth. This mechanism is fundamentally different from conventional interfacial instabilities in crystal growth that are driven by external non-uniformities such as surface diffusion, and instead arises from intrinsic electronic properties of topological materials. We find that the boundary states of topological insulators have a pronounced impact on the surface stiffness, which quantifies how strongly a surface resists changes in its shape or orientation. While trivial insulators possess positive stiffness that smooths out surface roughness, topological insulators exhibit negative stiffness that amplifies small shape fluctuations. We derive an effective equation of the interface growth with this negative stiffness and demonstrate that the interface dynamics is governed by the Kuramoto--Sivashinsky equation, a prototypical nonlinear equation exhibiting spatiotemporal chaos.

Figures (9)

• Figure 1: Schematic illustration of interface growth with positive and negative stiffness.
• Figure 2: (a) Schematic illustration of a square lattice in strip geometry. The blue shaded region represents the unit cell in this geometry, and $A$ is the surface length of the unit cell. (b) The transition from decoupled 1D topological insulators to a Chern insulator. (c) and (d) The band structures of the boundary states along the $y$ direction (left panel) and parity eigenvalues at the TRIM (right panel) (c) for the decoupled 1D topological insulators and (d) for the Chern insulator. The red lines in the band structures indicate the occupied boundary states.
• Figure 3: (a) Schematic illustration of the tight-binding model in Eq. \ref{['eq:Chern_model']}. The hopping amplitude in the $y$ direction is proportional to the real parameter $\lambda$. In the absence of hopping in the $y$ direction, this model reduces to a set of decoupled SSH chains. (b) and (c) The surface energy $\gamma$ for the (10) surface for (b) $m=0.8$ and (c) $m=4$ in the model [Eq. \ref{['eq:Chern_model']}]. The parameter is set to $v=0.7$, and the system size is $L_x=L_y=40$.
• Figure 4: (a) Dangling bonds on the surface with the Miller index $(k,l)$. Each plaquette is a unit cell of a bulk. (b,c) Surface energy $\gamma$ [Eq. \ref{['eq:surf_ene']}] of the model [Eq. \ref{['eq:Chern_model']}] (b) in the trivial phase ($m=4$) and (c) in the topological phase ($m=0.8$). "Tight-binding model" indicates the direct calculation of the model. "Fitting" indicates the fitting with Eq. \ref{['eq:broken_bond_ec']}. (d) Band structures of the tight-binding model ($m=0.8$) under the OBC in the $x$ direction and the PBC in the $y$ direction. The color bar indicates the inverse participation ratio (IPR) $\sum_{x}|\psi_{x}(k_y)|^4/(\sum_{x}|\psi_{x}(k_y)|^2)^2$ for the boundary states, which characterizes the localization of the boundary states along the two edges in the strip geometry. The parameters are the same as Fig. \ref{['fig:energy_vs_lambda']}, and the width of the strip [$L$ in (a)] is 40.
• Figure 5: The dynamics of the interface $h$ described by Eq. \ref{['eq:interface_growth']} with the system size $L=100$ under the PBC for (a) the topological insulator ($\tilde{\gamma}<0$) and (b) the trivial insulator ($\tilde{\gamma}>0$). The lower panels show the spatiotemporal evolution of the interface fluctuation $h-\bar{h}$. Here, $\bar{h}$ denotes the spatial average of the interface height $h$ at each time step [$\nu=-1$ in (a) and $\nu=1$ in (b). $0\leq t\leq 100$, $\kappa=1$, and $\eta=1$ in both (a) and (b)].