Hexagonal Warping Control of Exceptional Points in Topological Insulator--Ferromagnetic Heterojunctions

TL;DR

The paper addresses how hexagonal warping of topological_insulator surface states influences non-Hermitian exceptional points at TI–FM interfaces and shows that an in-plane magnetic field can tune both the number and positions of these EPs. By deriving an effective NH Hamiltonian that includes lead-induced self-energy and expressing it as $\tilde H = \epsilon_0 + \mathbf{d}\cdot\boldsymbol{\sigma}$, the authors derive EP conditions from $\mathbf{d}_R^2 = \mathbf{d}_I^2$ and $\mathbf{d}_R\cdot\mathbf{d}_I = 0$, obtaining six EPs arranged in a hexagonal pattern whose locations are independent of the warping strength $\lambda$. The magnetic field drives EPs, annihilating four at $B_c = \sqrt{\frac{4}{3}}\gamma$ and leaving two along $k_x = 0$ for larger fields, with hexagonal symmetry broken by the field. When $\lambda=0$ an exceptional ring forms, but finite $\lambda$ fragments this ring into six robust EPs at hexagon vertices, with stronger warping sharpening the phase rigidity features. This work establishes TI–FM heterostructures with hexagonally warped TI surface states as a realistic and tunable platform for exploring non-Hermitian degeneracies in condensed matter systems, with Bi$_2$Te$_3$-based realizations and gating/doping providing experimental handles.

Abstract

Exceptional points (EPs) are non-Hermitian degeneracies, where both eigenvalues and eigenvectors coalesce, which are fundamentally distinct from their Hermitian counterparts. In this study, we investigate the influence of hexagonal warping on EPs emerging at the interfaces between topological insulators and ferromagnets. We demonstrate that the presence of the warping term plays a crucial role in determining the locations of the EPs. Furthermore, we show that the number as well as the positions of EPs emerging at such junctions can be tuned by an applied magnetic field. Our results establish a realistic and experimentally accessible platform for exploring non-Hermitian physics in topological insulator-ferromagnet junctions.

Figures (4)

• Figure 1: Schematic of the proposed topological insulator-ferromagnet junction. The junction is formed at $z=0$, while the region $z<0$ represents the FM lead which is in contact with a TI film (extending for $z>0$). The red dots in the inset denote the emergent EPs, which follow the symmetry imposed by the hexagonal warping term. The thick black arrow denotes the direction of the externally applied magnetic field, which serves as a tuning parameter for controlling the position and evolution of the EPs.
• Figure 2: Complex band diagram of the junction. The (a) real and (b) imaginary parts of the energy for the TI-FM junction. The two bands are presented in green and blue colors. Note that both the real and the imaginary parts of the eigenvalues coalesce at the red points, indicating the presence of six EPs which exhibit a hexagonal symmetry. Here, we choose $\lambda=1$, $\alpha=1$,$\gamma=1$, $B_x=0$.
• Figure 3: Magnetic-field tuning of exceptional points. Evolution of the EP positions for different magnetic field values (a) $B_x=0.0B_c$, (b) $B_x=0.5B_c$, (c) $B_x=0.7B_c$, and (d) $B_x=1.1B_c$. The EPs are shown by red dots, while the phase rigidity is plotted in color (darker colors representing lower values of phase rigidity). As the magnetic field increases, the EPs move in momentum space, and above the critical field, $B_c$, four EPs annihilate, leaving behind two remanent EPs. The applied field also breaks the underlying hexagonal symmetry arising from the warping term. Here, we choose $\lambda=1$, $\alpha=1$,$\gamma=1$.
• Figure 4: Role of hexagonal warping in shaping exceptional degeneracies. The phase rigidity in the $k_x-k_y$ plane with (a) $\lambda = 0.0$, (b) $\lambda= 0.1$, (c) $\lambda = 0.5$, and (d) $\lambda = 1.0$. At zero hexagonal warping, an exceptional ring is formed centered at the origin. Increasing $\lambda$ drives the splitting of the exceptional ring into discrete EPs, reflecting the underlying hexagonal symmetry of the warping term. The EPs are robust to the strength of the hexagonal warping. Here, we choose $\alpha=1$,$\gamma=1$, $B_x=0$.