The localization of electron states due to disorder, known as Anderson localization, has been actively studied. In recent years, there has been an interest on studying single-particle states with unique characteristics. Flat band systems are one of such examples, and indeed, there have been several reports of their distinctive behaviors to disorder [1, 2]. In our previous study [3], it also was suggested that flat band states in a model called molecular-orbital (MO) model [4] exhibit some distinct features. However, we find that the characterization methods used for single-particle wave functions, such as multifractal analysis, are ineffective for the flat band state due to the macroscopic degeneracy of the flat band.

In this work, to characterize the flat band state of the MO model, we employ a concept of hyperuniformity [5]. Hyperuniformity is a concept that quantifies the spatial fluctuation of distributions, and has recently been used for condensed matter systems, such as quasiperiodic systems [6]. In this presentation, we will show the results for quantitatively characterizing the features of the flat band states in a MO model with randomness using hyperuniformity [7].

[1] M. Goda, et al., PRL 96, 126401 (2006).

[2] J. T. Chalker, et al., PRB 82, 104209 (2010).

[3] T. Kuroda, et al., JPSJ 91, 044703 (2022).

[4] Y. Hatsugai, Annals of Physics 168453 (2021).

[5] S. Torquato, et al., PRE 68, 041113 (2003).

[6] S. Sakai, et al. PRB 105, 054202 (2022).

[7] T. Kuroda, T. Mizoguchi, Y. Hatsugai, in preparation.

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).