Fuka Watanabe1, Keijiro Hirata1, Tatsuki Tojo1, Kyozaburo Takeda1
1. Waseda University, Tokyo, Japan
Fuka Watanabe1, Keijiro Hirata1, Tatsuki Tojo1, Kyozaburo Takeda1
1. Waseda University, Tokyo, Japan
Physicomathematics can create new frontiers in condensed-matter physics. The application of topology proposes a novel concept, topological materials, that can deepen our understanding regarding quantum phenomena, particularly those related to spin and spin–orbit interactions (SOIs), such as the spin current, spin Hall effects, and spin-Zeeman effect [1]. In these topological materials, the SOI around the degenerate point is key in inducing a characteristic effective magnetic field (EMF). The degenerate state determines its own orthogonal eigenvectors but is not unique because of the “phase” indefiniteness. The Dirac corn is a prime example and results in a topological singularity originating from degeneracy. Spatial asymmetry can generate singularities in the EMF distribution, such as vortices, sources, and sinks, via the SOI around the degenerate point; subsequently, the spin is polarized characteristically therein, particularly when the system exhibits bulk inversion asymmetry (BIA) and/or structure inversion asymmetry (SIA).
In this study, we investigate the manner by which multiple degenerate points in the Brillouin zone (BZ) modify the topological features. We focus on a semiconductor system with a two-dimensional quantum well structure and investigate topological modifications via numerical calculations of the EMF, Berry connection, and curvature. Next, we reveal the energy dependence of the Berry phase modulated by the interaction between the degenerate points. We employ the Hamiltonian, in which the spatial asymmetry of the SIA and/or BIA is considered [1]. The effect of the competition between the SIA and BIA is investigated using tunable parameters a and b. To provide multiple degenerate points in the BZ, we locate the degenerate state at point Γ and then transfer it to a peculiar direction by ±ki. We vary the locations of the degenerate points by appropriately substituting a value for ki and examine the directional dependence of the synthesized EMF distribution. We further introduce anisotropy by changing the directional effective mass.
In the SIA system, each degenerate point presents a SIA-like singularity and the concentric EMF distribution results. The synthesis of these two concentric distributions generates a source and a sink at the middle position between the two prepared degenerate points when the points (±ki) are located in the <110> direction. Accordingly, the EMF is distributed hyperbolically, and the BIA-like singularity newly results. By contrast, the topological singularity in the BIA system distributes the EMF hyperbolically around each degenerate point. The synthesis of these two hyperbolic distributions results in a new concentric EMF distribution at the middle position. These newborn singularities in the EMF distribution results in accidental degeneracy at the middle position. The sign of the Berry curvature around the newly generated point is opposite to that at the prepared degenerate points.
Finally, we discuss the energy dependence of the Berry phase based on the calculated Berry connection and curvature. Since each degenerate point results in the Berry phase of ±π, the multiple degenerate points quantize the Berry phase by π and result in a plateau profile against energy. We further demonstrate that this π quantization can be understood by counting the number of degenerate points in the BZ in addition to identifying the sign of the Berry curvature [2].
[1] S.D. Ganichev, L.E. Golub, physica status solidi (b) 251, 1801 (2014).
[2] T. Tojo, K. Takeda, Phys. Lett. A 389, 127091 (2021).