Quantum theory for edge current and noise in 2D topological superconductors

TL;DR

The paper develops a Green-function and T-matrix framework to calculate edge spectral functions, edge current, and edge noise in two-dimensional topological superconductors. It shows that edge current vanishes for non-chiral edges while edge noise persists, and that edge noise tracks the Chern number, with bulk-noise peaks signaling topological transitions. Applications to the Qi-Wu-Zhang and twisted bilayer models demonstrate that the edge current–Chern number relationship is robust and that bulk noise behaves as a topological susceptibility. The work links topological invariants to measurable edge and bulk fluctuations, suggesting experimental routes to detect edge currents and noise in topological superconductors.

Abstract

We calculate the edge current and its fluctuations, i.e. noise, in a 2D topological superconductor using the T-matrix and the Green function techniques. We show that the current is zero for non-chiral edge states and non-zero for chiral edge states, while the edge noise is non-zero whatever the chirality of the edge states. By applying our results to toy models with chiral edge states, we find that the noise is closely related to the Chern number. The edge noise is non-zero only when the Chern number is non-zero, and the bulk noise exhibits a peak each time the Chern number varies, meaning that there is strong current fluctuations when a topological phase transition occurs. Our results suggest that the bulk noise could be seen as a topological susceptibility.

Figures (11)

• Figure 1: Schematically picture of the 2D system, modeled as a square lattice, with an impurity line at $y=0$ with significant strong potential $U_0$, and an edge at $y=a$, where $a$ is the inter-atomic distance. The translation invariance symmetry is broken along $y$-axis while it remains preserved along $x$-axis.
• Figure 2: Bulk and edge DOS at (a) $\mu=0$, (b) $\mu=t$, (c) $\mu=2t$, (d) and Chern number $\mathcal{C}$ as a function of $\mu$ in the QWZ model. The parameters are $\Delta=t$, $\eta= 0.02t$ and $U_0=1000t$. In panel (d), the vertical gray lines indicate the values of $\mu$ for which the Chern number value changes, i.e. $\mu=0$ and $\mu=\pm 2\,t$.
• Figure 3: Edge spectral function $A_{k_xa}(\varepsilon)$ in the QWZ model for various values of $\mu$, with $\Delta=t$, $\eta= 0.02t$ and $U_0=1000t$.
• Figure 4: (a) Current $\langle I_y\rangle$ and (b) zero-frequency noise $\mathcal{S}_{yy}(\omega=0)$ in the QWZ model as a function of $y$, for various values of $\mu$, at $k_BT=0.1t$, $\Delta=t$, $\eta= 0.02t$ and $U_0=1000t$.
• Figure 5: (a) Edge current $\langle I_a\rangle$ and bulk current $\langle I_{y\gg a}\rangle$, (b) derivative of edge current according to $\mu$, (c) edge noise and (d) bulk noise in the QWZ model as a function of $\mu$, at $k_BT=0.1t$, $\Delta=t$, $\eta= 0.02t$ and $U_0=1000t$. In panels (c) and (d), the noise is plotted for various values of frequency $\omega$. The vertical gray lines indicate the $\mu$ values for which the Chern number changes.