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arXiv:2510.04959 · Chinese Journal of Chemical Physics

Fermionic influence superoperator for transport through majorana zero modes

J. Pan, Zi-Fan Zhu, Shixuan Chen, Yu Su, Yao Wang

Published Dec 1, 2025• 1 citation
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Abstract

In recent years, the study of Majorana signatures in quantum transport has become acentral 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 super-operator 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.

DOI
10.1063/1674-0068/cjcp2509147

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In recent years, the study of Majorana signatures in quantum transport has become acentral 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 super-operator 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.
In this review we give a comprehensive account of a hierarchical equations of motion (HEOM) approach to the characterization of stationary and dynamic properties of open quantum systems. This approach is rooted at the Feynman-Vernon influence functional path integral formalism, but much more implementable numerically and operationally for the study of various complex molecular dynamics and quantum transport in strongly correlated electronic systems. By construction, HEOM resolves nonperturbatively the combined effects of many-particle interaction, system-bath coupling,and non-Markovian memory. Finally the practicality of HEOM to address physical and chemical problems is exemplified with a model simulation of coherent two-dimensional spectroscopy signals of a biological light-harvesting system and a time-dependent quantum transport system involving dynamic Kondo transition.級聯方程已成為研究量子開放系統的穩態性質和動力學過程的重要方法。本文旨在系統綜述量子耗散和量子輸運的級聯方程組方法的建立、發展以及在理論、算法和應用方面的一些最新進展。級聯方程形式理論的建立以影響泛函路徑積分為基礎,并具有數值上的高效性和應用上的靈活性,可用于研究分子體系的復雜動力學過程以及強關聯電子體系中的量子輸運。其級聯耦合結構以非微擾的方式揭示了多體相互作用、體系-環境耦合、非馬爾可夫記憶等的綜合效應。作為應用示例,我們采用級聯方程模擬了生物光富集體系的二維相干動力學光譜以及含時電子輸運過程中的動態近藤效應。
The hierarchical equations of motion (HEOM) formalism developed recently is becoming an important method for investigating the properties and dynamical processes of quantum open systems. A comprehensive account on its establishment and recent development was given, covering theoretical formulations, numerical algorithm and practical applications. By construction HEOM resolves in a nonperturbative manner the combined effects of many-body interaction, system-environment coupling and non-Markovian memory. While it is mathematically equivalent to an exact path integral formalism, HEOM is also very convenient in computation and versatile in various systems of study. Moreover, it is shown recently that HEOM is actually a correlated system-environment dynamics theory. The practicality of HEOM to address physical and chemical problems was exemplified with examples relevant to quantum dissipation and quantum transport.
Investigations of strongly correlated quantum impurity systems (QIS), which exhibit diversified novel and intriguing quantum phenomena, have become a highly concerning subject in recent years. The hierarchical equations of motion (HEOM) method is one of the most popular numerical methods to characterize QIS linearly coupled to the environment. This review provides a comprehensive account of a formally rigorous and numerical convergent HEOM method, including a modeling description of the QIS and an overview of the fermionic HEOM formalism. Moreover, a variety of spectrum decomposition schemes and hierarchal terminators have been proposed and developed, which significantly improve the accuracy and efficiency of the HEOM method, especially in cryogenic temperature regimes. The practicality and usefulness of the HEOM method to tackle strongly correlated issues are exemplified by numerical simulations for the characterization of nonequilibrium quantum transport and strongly correlated Kondo states as well as the investigation of nonequilibrium quantum thermodynamics.
A quantum dissipation theory is formulated in terms of hierarchically coupled equations of motion for an arbitrary fermionic system coupled with grand canonical fermionic bath ensembles.1 The theoretical construction starts with the second-quantization influence functional in path integral formalism, in which the fermionic creation and annihilation operators are represented by Grassmann variables. Temporal derivatives on influence functionals are then performed in a hierarchical manner, on the basis of calculus-on-path-integral algorithm. Both the multiple-frequency-dispersion and the non-Markovian reservoir parametrization schemes are considered for the desired hierarchy construction. The resulting formalism is in principle exact, applicable to interacting systems, with arbitrary time-dependent external fields. It renders an exact tool to evaluate various transient and stationary quantum transport properties of many-electron systems. At the second-tier truncation level the present theory recovers the real-time diagrammatic formalism developed by Schön and coworkers.2 For a single-particle system, the hierarchical formalism terminates at the second tier exactly, and the Landuer-Büttiker's transport current expression is readily recovered. Numerical studies will be presented to highlight the richness of transient current through both interacting and noninteracting model systems.
We study electron quantum transport through a strongly interacting Anderson quantum dot at finite bias voltage and magnetic field at zero temperature using the real-time renormalization group (RT-RG) in the framework of a kinetic (generalized master) equation for the reduced density operator. To this end, we further develop the general, finite-temperature real-time transport formalism by introducing field superoperators that obey fermionic statistics. This direct second quantization in Liouville Fock space strongly simplifies the construction of operators and superoperators that transform irreducibly under the Anderson-model symmetry transformations. The fermionic field superoperators naturally arise from the univalence (fermion-parity) superselection rule of quantum mechanics for the total system of quantum dot plus reservoirs. Expressed in these field superoperators, the causal structure of the perturbation theory for the effective time-evolution superoperator kernel becomes explicit. Using the constraints of the causal structure, we construct a parametrization of the exact effective time-evolution kernel for which we analytically find the eigenvectors and eigenvalues in terms of a minimal set of only 30 independent coefficients. The causal structure also implies the existence of a fermion-parity protected eigenvector of the exact Liouvillian, explaining a recently reported result on adiabatic driving [Contreras-Pulido et al., Phys. Rev. B 85, 075301 (2012)] and generalizing it to arbitrary order in the tunnel coupling $\ensuremath{\Gamma}$. Furthermore, in the wide-band limit, the causal representation exponentially reduces the number of diagrams for the time-evolution kernel. The remaining diagrams can be identified simply by their topology and are manifestly independent of the energy cutoff term by term. By an exact reformulation of this series, we integrate out all infinite-temperature effects, obtaining an expansion targeting only the nontrivial, finite-temperature corrections, and the exactly conserved transport current follows directly from the time-evolution kernel. From this new series, the previously formulated RT-RG equations are obtained naturally. We perform a complete one-plus-two-loop RG analysis at finite voltage and magnetic field, while systematically accounting for the dependence of all renormalized quantities on both the quantum dot and reservoir frequencies. Using the second quantization in Liouville space and symmetry restrictions, we obtain analytical RT-RG equations, which can be solved numerically in an efficient way, and we extensively study the model parameter space, excluding the Kondo regime where the one-plus-two-loop approach is obviously invalid. The incorporated renormalization effects result in an enhancement of the inelastic cotunneling peak, even at a voltage $\ensuremath{\sim}$magnetic field $\ensuremath{\sim}$tunnel coupling $\ensuremath{\Gamma}$. Moreover, we find a tunnel-induced nonlinearity of the stability diagrams (Coulomb diamonds) at finite voltage, both in the single-electron tunneling and inelastic cotunneling regime.