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arXiv:2303.03837 · New Journal of Physics

On the conclusive detection of Majorana zero modes: conductance spectroscopy, disconnected entanglement entropy and the fermion parity noise

Arnav Arora, Abhishek Kejriwal, Bhaskaran Muralidharan

Published Jan 29, 2024• 5 citations
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Abstract

Abstract Semiconducting nanowires with strong Rashba spin–orbit coupling in the proximity with a superconductor and under a strong Zeeman field can potentially manifest Majorana zero modes (MZMs) at their edges and are a topical candidate for topological superconductivity. However, protocols for their detection based on the local and the non-local conductance spectroscopy have been subject to intense scrutiny. In this work, by taking current experimental setups into account, we detail mathematical ideas related to the entanglement entropy and the fermion parity fluctuations to faithfully distinguish between true MZMs and trivial quasi-MZMs. We demonstrate that the disconnected entanglement entropy, derived from the von Neumann entanglement entropy, provides a distinct and robust signature of the topological phase transition which is immune to system parameters, size and disorders. In order to understand the entanglement entropy of the Rashba nanowire system, we establish its connection to a model of interacting spinfull Kitaev chains. Moreover, we relate the entanglement entropy to the fermionic parity fluctuation, and show that it behaves concordantly with entanglement entropy, hence making it a suitable metric for the detection of MZMs. In connection with the topological gap protocol that is based on the conductance spectra, the aforesaid metrics can reliably point toward the topological transitions even in realistic setups.

DOI
10.1088/1367-2630/ad23a2

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Abstract Semiconducting nanowires with strong Rashba spin–orbit coupling in the proximity with a superconductor and under a strong Zeeman field can potentially manifest Majorana zero modes (MZMs) at their edges and are a topical candidate for topological superconductivity. However, protocols for their detection based on the local and the non-local conductance spectroscopy have been subject to intense scrutiny. In this work, by taking current experimental setups into account, we detail mathematical ideas related to the entanglement entropy and the fermion parity fluctuations to faithfully distinguish between true MZMs and trivial quasi-MZMs. We demonstrate that the disconnected entanglement entropy, derived from the von Neumann entanglement entropy, provides a distinct and robust signature of the topological phase transition which is immune to system parameters, size and disorders. In order to understand the entanglement entropy of the Rashba nanowire system, we establish its connection to a model of interacting spinfull Kitaev chains. Moreover, we relate the entanglement entropy to the fermionic parity fluctuation, and show that it behaves concordantly with entanglement entropy, hence making it a suitable metric for the detection of MZMs. In connection with the topological gap protocol that is based on the conductance spectra, the aforesaid metrics can reliably point toward the topological transitions even in realistic setups.
The authors employ entanglement entropy as the means to reliably predict the presence of Majorana zero modes (MZMs) in experimentally relevant setups. Connecting with quantum transport calculations of the conductance spectra, they present ways to reliably signal the presence of topological MZMs that possess true nonlocal correlations, thereby distinguishing topological phase transitions from trivial ones. The detection of MZMs is a contentious issue, and the models presented here set the stage for looking beyond conductance spectra for their reliable identification.
Zero-energy Majorana fermion states (Majoranas) can arise at the ends of a semiconducting wire in proximity with a superconductor. A first generation of experiments has detected a zero bias conductance peak in these systems that strongly suggests these Majoranas do exist; however, a definitive demonstration of the long-ranged entanglement that is crucial for potential applications in quantum computing has yet to be carried out. This work discusses two possible measurement schemes to detect this long-ranged entanglement in a wire system with two coupled pairs of Majoranas, by varying the coupling between one pair while measuring the fermion parity of the second pair. First, in a system with two coupled pairs of Majoranas, we discuss how varying the coupling of one pair in time, while measuring temporal fermion parity correlations of the second pair, allows for an experimental probe of long-ranged Majorana entanglement. Second, we show that the power spectrum of the charge noise (fermion parity noise) of one pair contains signatures of these correlations, as well as allowing one to infer the parity relaxation rate.
Majorana zero modes are the hallmark of topological superconductivity. In one-dimensional systems, these zero modes are usually introduced in the context of gapped, mean-field models that do not conserve particle number, such as the Kitaev chain. By non-locally encoding a conventional fermion across spatially separated Majorana zero modes, these systems become inherently immune to local decoherence. In this work, we show that signatures of Majorana edge physics persist in a number-conserving, gapless Luttinger liquid of spinless fermions with short-range attractive interactions. We identify the two-point correlator as a sharp diagnostic, revealing an edge-to-edge revival whose sign depends on the fermion-number parity. This revival is robust in the thermodynamic limit, and persists in the excited states of the system and at different fillings. A simple particle-hole ansatz for the ground state of the system with an odd number of fermions captures the physics of the system for a wide range of interaction strengths, interpolating between the free-fermion limit and the strongly interacting Majorana regime. Finally, we propose a concrete protocol to realize this model with ultracold dipolar molecules or atoms in an optical lattice, and to detect the revival via beam-splitter interferometry, opening an experimental route to Majorana physics beyond the conventional gapped-superconductor paradigm.
Z2\mathbb{Z}_2Z2​ Pfaffian Invariant and Entanglement Spectrum Signatures in Disordered Majorana Wires
In a one-dimensional (1D) disordered superconducting wire (Kitaev chain), the topological phase hosts Majorana zero modes (MZMs) at its boundaries, while the trivial phase hosts ordinary fermionic states. We identify the Z2\mathbb{Z}_2Z2​ Pfaffian invariant, defined as the sign of the Pfaffian of the real-space Majorana Hamiltonian matrix, as the precise topological invariant distinguishing these phases in the presence of strong disorder. We demonstrate that this invariant is exactly mirrored in the ground-state entanglement spectrum (ES). By bipartitioning the chain, the reduced density matrix of the ground state can be expressed via an entanglement Hamiltonian determined by the restricted single-particle correlation matrix. The Z2\mathbb{Z}_2Z2​ invariant is computed as the sign of the Pfaffian of the restricted correlation matrix, tracking the parity of its negative eigenvalues. In the trivial phase, the ES is non-degenerate; in the topological phase, the ES exhibits a universal two-fold degeneracy. The sign flip of this Pfaffian provides a sharp, falsifiable signature of the topological phase transition, robust to local disorder. ## 1. Introduction The Kitaev chain, a one-dimensional (1D) p-wave superconducting wire, provides a paradigmatic model for topological superconductivity. In its topological phase, the wire hosts Majorana zero modes (MZMs)—operators that are their own antiparticles—at its boundaries. In the presence of strong disorder, the distinction between the topological phase and the trivial phase hosting ordinary fermionic states becomes subtle, as momentum-space topological invariants are ill-defined. This raises the question of what precise topological invariant distinguishes these phases in real space, and whether this invariant can be extracted from the ground-state entanglement spectrum (ES) to provide a falsifiable, non-local signature of the topological phase transition. Here, we show that the Z2\mathbb{Z}_2Z2​ Pfaffian invariant of the real-space M
In this work, we perform a systematic investigation of the correlated topological superconductors (TSCs), especially their non-trivial Majorana zero modes (MZMs). Compared to the non-interacting MZMs, the emerged correlated MZMs become projected MZMs. To prove this, we study the topological superconducting nanowire under the Hubbard and Hatsugai-Kohmoto interactions. Both of them become correlated TSCs under a magnetic field. Their topological properties are numerically computed by the Wilson loop and entanglement spectrum. We successfully extract the projected MZMs connecting the ground state and excited state through exact diagonalization. We also extend our results to the spinful Kitaev chain. Our results can provide a new perspective and understanding of the correlated MZMs.
We analyze the entanglement entropy properties of a 2D p-wave superconductor with Rashba spin-orbit coupling, which displays a rich phase-space that supports non-trivial topological phases, as the chemical potential and the Zeeman term are varied. We show that the entanglement entropy and its derivatives clearly signal the topological transitions and we find numerical evidence that for this model the derivative with respect to the magnetization provides a sensible signature of each topological phase. Following the area law for the entanglement entropy, we systematically analyze the contributions that are proportional to or independent of the perimeter of the system, as a function of the Hamiltonian coupling constants and the geometry of the finite subsystem. For this model, we show that even though the topological entanglement entropy vanishes, it signals the topological phase transitions in a finite system. We also observe a relationship between a topological contribution to the entanglement entropy in a half-cylinder geometry and the number of edge states, and that the entanglement spectrum has robust modes associated with each edge state, as in other topological systems.