Fermionic influence superoperator for transport through Majorana zero modes

TL;DR

This paper tackles open quantum transport through Majorana zero modes by constructing a rigorous fermionic influence superoperator and its differential HEOM formulation, enabling non-Markovian system–bath dynamics to be treated exactly. It builds a canonical algebra using graded tensor products, a partial-trace framework, and a fermionic Wick theorem to derive an influence functional that encodes bath effects via two-point correlators and exponential decompositions. The authors then formulate the hierarchical equations of motion (HEOM) with auxiliary density operators to capture all bath-induced correlations, and provide a functional-derivative scheme to link transport observables to the first-tier ADOs. Numerical demonstrations compare the Majorana impurity to a regular fermionic impurity, revealing distinctive signatures such as nonzero steady-state current under certain biases and a Landauer-like peak structure at εM = 0, validating the approach as a numerically exact tool for Majorana transport physics with broad applicability.

Abstract

In recent years, the study of Majorana signatures in quantum transport has become a central focus in condensed matter physics. Here, we present a rigorous and systematic derivation of the fermionic superoperator describing the open quantum dynamics of electron transport through Majorana zero modes, building on the techniques introduced in Phys. Rev. B 105, 035121 (2022). The numerical implementation of this superoperator is to construct its differential equivalence, the hierarchical equations of motion (HEOM). The HEOM approach describes the system-bath correlated dynamics. Furthermore, we also develop a functional derivative scheme that provides exact expressions for the transport observables in terms of the auxiliary density operators introduced in the HEOM formulation. The superoperator formalism establishes a solid theoretical foundation for analyzing key transport signatures that may uncover the unique characteristics of Majorana physics in mesoscopic systems.

Figures (2)

• Figure 1: Transient dynamics of transport current for Majorana impurity [(a) and (b)] and regular fermion [(c) and (d)]. We set $\varepsilon_{\rm M} = \varepsilon_{\rm F} = 10\Delta$ and $\beta_{\rm L} = \beta_{\rm R} = 1000\Delta^{-1}$. The values of external bias voltages are given as $\varphi_{\rm L} = 0, 5\Delta, 10\Delta$ and $\varphi_{\rm R} = 0$, respectively. Remarkably, the Majorana impurity model exhibits a non-vanishing steady-state current when $\varphi_{\rm L}$ is absent. And the FBO model does not have such a feature, since its total particle number is conserved at the steady-state.
• Figure 2: The steady-state differential conductance as a function of bias voltage at various values ($0,5\Delta,10\Delta$) of system energy level. The parameters are set to $\beta_{\rm L} = \beta_{\rm R} = 1000\Delta^{-1}$ and $\varphi_{\rm R} = 0$.