Size Effects on the Conditions of Edge State Formation in 1D Systems
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ONIC PROPERTIES OF SOLID
Size Effects on the Conditions of Edge State Formation in 1D Systems A. D. Fedoseev Kirensky Institute of Physics, Federal Research Center Krasnoyarsk Science Center, Siberian Branch, Russian Academy of Sciences, Krasnoyarsk, 660036 Russia e-mail: [email protected] Received May 21, 2018; revised June 21, 2018; accepted June 27, 2018
Abstract—A criterion for revealing edge states in the case when the size of a system is comparable with the localization length of these states has been proposed. The application of the algorithm for determining edge states in short systems has been demonstrated on examples of the Bernevig–Hughes–Zhang (BHZ) model in the cylindrical geometry, the Kitaev model, and a chain with the spin–orbit interaction and induced superconductivity. It has been shown that for finite-length 1D systems, there exist ranges of parameters in which the edge states are not formed, although the topological index is nontrivial; conversely, the emergence of the Majorana modes in regions with a trivial topological index has been demonstrated. DOI: 10.1134/S106377611812004X
1. INTRODUCTION The properties of topologically nontrivial systems have attracted attention of researchers in recent decades [1–7]. Such systems are distinguished by the existence of topologically protected states in the dielectric gap, which ensure, among other things, the motion of a fermion without scattering by nonmagnetic impurities. The properties of such edge states are studied commonly using semi-infinite models with a single boundary, and the introduction of limited systems involves computation difficulties in most cases. The results obtained for semi-infinite systems are extended in this case to finite-size systems, and vice versa. Additional interest in systems in which edge states are realized was induced by Kitaev’s prediction of the existence of zero-energy edge (Majorana) modes in 1D systems with superconducting pairing [8]. The ranges of parameters ensuring the emergence of the Majorana modes in open systems are detected in most cases via the search of topologically nontrivial phases taking into account the periodic boundary conditions. The classification of such phases for noninteracting electrons was performed in [9, 10]. The systems were considered to be large enough for the size effects of the system to be disregarded. The limits of this approach were extended in [11, 12], where the emergence of lines of parameters corresponding to the formation of the Majorana modes separating the ground-state regions with different parities was demonstrated for 1D finite-size models.
Despite the huge number of features devoted to analysis of the properties of edge states in systems with a single boundary, the peculiarities in the realization of edge states in short chains have not been studied comprehensively [13–17]. All these publications were aimed at revealing the peculiar properties of edge states, which are associated with the finite size of such systems, while the size effects on the conditions for the emer
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