Department of Physics, Sharif University of Technology, Tehran 11155-9161, Iran.
Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, Iran.
Sci Rep. 2016 Sep 6;6:32720. doi: 10.1038/srep32720.
We present an exactly solvable extension of the quantum XY chain with longer range multi-spin interactions. Topological phase transitions of the model are classified in terms of the number of Majorana zero modes, nM which are in turn related to an integer winding number, nW. The present class of exactly solvable models belong to the BDI class in the Altland-Zirnbauer classification of topological superconductors. We show that time reversal symmetry of the spin variables translates into a sliding particle-hole (PH) transformation in the language of Jordan-Wigner fermions - a PH transformation followed by a π shift in the wave vector which we call it the πPH. Presence of πPH symmetry restricts the nW (nM) of time-reversal symmetric extensions of XY to odd (even) integers. The πPH operator may serve in further detailed classification of topological superconductors in higher dimensions as well.
我们提出了一个具有更长范围多自旋相互作用的量子 XY 链的精确可解扩展。该模型的拓扑相变根据马约拉纳零模的数量 nM 进行分类，nM 又与整数缠绕数 nW 相关。目前这一类精确可解模型属于 Altland-Zirnbauer 拓扑超导体分类中的 BDI 类。我们表明，自旋变量的时间反演对称性在 Jordan-Wigner 费米子的语言中转化为滑动粒子空穴 (PH) 变换——PH 变换后波矢发生 π 位移，我们称之为 πPH。πPH 对称性的存在将 XY 的时间反演对称扩展的 nW（nM）限制为奇数（偶数）整数。πPH 算子也可以用于更高维度的拓扑超导体的进一步详细分类。
