Analysis of the bulk-boundary correspondence in nonlinear eigenvalue problems by the auxiliary eigenvalues

Topological band theory has been extensively studied in these fifteen years [1,2]. So far, the platforms of the topological band theory have been extended including classical systems [3]. Furthermore, recent studies have been attempting to extend the topological band theory to systems that involve nonlinearity of eigenvectors [4]. This is because the nonlinearity is common in nature and those extension is significant for the development of the topological band theory. On the other hand, few studies focusing on the interplay between the topology and the nonlinearity of eigenvalues, which is another type of the nonlinearity.

In this talk, we address the bulk-boundary correspondence (BBC) [5] in systems with the nonlinearity of eigenvalues. Specifically, we introduce auxiliary eigenvalues of the matrix pencil and establish the BBC between the bulk topology of auxiliary eigenstates and physical boundary states [6]. We demonstrate the existence of the BBC in insulators and semimetals by analyzing models with nonlinearities of eigenvalues.



[1] M. Z. Hasan and C. L. Kane, RMP 82, 3045 (2010).

[2] X. -L. Qi and S. -C. Zhang, RMP 83, 1057 (2011).

[3] F. D. M. Haldane and S. Raghu, PRL 100, 013904 (2008).

[4] K. Sone, Y. Ashida, and T. Sagawa, PRR 4, 023211 (2022).

[5] Y. Hatsugai, PRL 71, 3697 (1993).

[6] T. Isobe, T. Yoshida, and Y. Hatsugai, arXiv:2310.12577 (2023).