Bulk-Edge Correspondence for Nonlinear Eigenvalue Problems

Although topological phenomena attract growing interest not only in linear systems but also in nonlinear systems, the bulk-edge correspondence under the nonlinearity of eigenvalues has not been established so far. We address this issue by introducing auxiliary eigenvalues. We reveal that the topological edge states of auxiliary eigenstates are topologically inherited as physical edge states when the nonlinearity is weak but finite (i.e., auxiliary eigenvalues are monotonic as for the physical one). This result leads to the bulk-edge correspondence with the nonlinearity of eigenvalues.

Figure 1

Schematic picture of the auxiliary bands $\lambda (k,\omega )$ for two-dimensional cylindrical system. The system is periodic in $y$ direction and with boundaries for $x$ direction. Bulk states are illustrated in green, while edge states are represented in red for each omega. The eigenstates that cross the $\lambda =0$ plane are of particular interest in the context of physics. Physical bands of $\omega (k)$ for each $k$ is shown in the inset. The gapless edge states of the auxiliary bands $\lambda$ are inherited to the physical nonlinear bands of $\omega$. Namely, red, blue, green, and white dots on the gray plane of $\lambda =0$, respectively, correspond to the physical edge states. For example, they can be dots in the inset.

Figure 2

Band structure of the auxiliary eigenvalue $\lambda$ of the two-dimensional model. Edge (bulk) states are plotted in red (gray). (a1)–(c1) [(a2)–(c2)]: Plot of $\lambda$ for each $\omega$ [$k_x$] with $M_0=-1$, $M_0=0.35$, and $M_0=1$, respectively. The parameter $k_x$ ($\omega$) is fixed in $k_x=0$ ($\omega =1$). Parameter $k_x$ ($\omega$) is chosen in $k_x=0$ ($\omega =1$). Edge states emerge when $M_0<0.35$.

Figure 3

Band structure of $\omega$ of the two-dimensional model. Edge (bulk) states are plotted in red (gray). (a)–(c) Plot of $\omega$ for each $k_x$ and the Chern number of the bands of $\lambda$ with $M_0=-1$, $M_0=0.35$, and $M_0=1$, respectively. The band gap is fully opening in the orange-colored regions. (d) Plot of the Chern number calculated from the eigenstates of ${\lambda }_1$ for each $M_0$ and $\omega$. The Chern number takes a nonzero (zero) value in the blue-colored region. The black line represents $\omega =1$. The band structures in Figs. 3 emerge on the black, red, and green dots. The region where the Chern number takes nonzero values corresponds to the region where the edge states emerge.

Figure 4

Plot of the Chern number calculated by the eigenstates of $\lambda$ for each $k_z$ and $\omega$. The Chern number takes 1 (0) in the blue (white) region. The black line represents $\omega =1$. Band structures of Figs. 3 emerge on the black, red, and green dots. The boundaries between the blue and the white regions correspond to the Weyl nodes.