Kitaev链中的Majorana波函数振荡、费米子宇称开关和无序
Majorana wavefunction oscillations, fermion parity switches, and disorder in Kitaev chains
摘要：
研究了一维拓扑超导晶格系统中Majorana费米子波函数和基态费米子宇称的衰减和振荡。使用马约拉纳传输矩阵方法，我们发现，马约拉纳波函数的属性被编码在相关的李雅普诺夫指数，这反过来又是两个独立的组件的总和：一个“超导组件”的间隙诱导衰变的特点，和“正常的组件”，它确定的振荡和化学势配置的响应。拓扑相变分离阶段与马约拉纳结束模式被认为是这两个组件的取消。我们表明，马约拉纳波函数振荡是完全由一个潜在的非超导紧束缚模型，并完全负责GS费米子奇偶开关在有限大小的系统。这些观察使我们能够解析地绘制出波函数振荡，由此产生的GS奇偶配置作为一个函数的参数空间在均匀的电线，和特殊的奇偶开关点，退化的零能量马约拉纳模式恢复，尽管有限的尺寸效应。对于无序的线，我们发现，带振荡被完全洗掉，导致第二本地化长度的马约拉纳模式和剩余的振荡随机化，在正常系统中的安德森局域化物理。我们的转移矩阵方法还允许我们i）再现关于中间带隙马约拉纳态的缩放的已知结果，并证明其对数正态分布的起源，ii）识别偶数与奇数个晶格位点的情况下的无序依赖的GS奇偶开关的对比行为，以及iii）绘制GS奇偶配置和相关奇偶开关点作为无序强度的函数。
英文摘要：
We study the decay and oscillations of Majorana fermion wavefunctions and ground state (GS) fermion parity in one-dimensional topological superconducting lattice systems. Using a Majorana transfer matrix method, we find that Majorana wavefunction properties are encoded in the associated Lyapunov exponent, which in turn is the sum of two independent components: a `superconducting component' which characterizes the gap induced decay, and the `normal component', which determines the oscillations and response to chemical potential configurations. The topological phase transition separating phases with and without Majorana end modes is seen to be a cancellation of these two components. We show that Majorana wavefunction oscillations are completely determined by an underlying non-superconducting tight-binding model and are solely responsible for GS fermion parity switches in finite-sized systems. These observations enable us to analytically chart out wavefunction oscillations, the resultant GS parity configuration as a function of parameter space in uniform wires, and special parity switch points where degenerate zero energy Majorana modes are restored in spite of finite size effects. For disordered wires, we find that band oscillations are completely washed out leading to a second localization length for the Majorana mode and the remnant oscillations are randomized as per Anderson localization physics in normal systems. Our transfer matrix method further allows us to i) reproduce known results on the scaling of mid-gap Majorana states and demonstrate the origin of its log-normal distribution, ii) identify contrasting behavior of disorder-dependent GS parity switches for the cases of even versus odd number of lattice sites, and iii) chart out the GS parity configuration and associated parity switch points as a function of disorder strength.
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