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Zero modes arising from a planar Majorana equation in the presence of $N$ vortices require an $mathcal{N}$-dimensional state-space, where $mathcal{N} = 2^{N/2}$ for $N$ even and $mathcal{N} = 2^{(N + 1)/2}$ for $N$ odd. The mode operators form a restricted $mathcal{N}$-dimensional Clifford algebra.
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We describe the occurrence and physical role of zero-energy modes in the Dirac equation with a topologically non-trivial background.
We provide a current perspective on the rapidly developing field of Majorana zero modes in solid state systems. We emphasize the theoretical prediction, experimental realization, and potential use of Majorana zero modes in future information processing devices through braiding-based topological quantum computation. Well-separated Majorana zero modes should manifest non-Abelian braiding statistics suitable for unitary gate operations for topological quantum computation. Recent experimental work, following earlier theoretical predictions, has shown specific signatures consistent with the existence of Majorana modes localized at the ends of semiconductor nanowires in the presence of superconducting proximity effect. We discuss the experimental findings and their theoretical analyses, and provide a perspective on the extent to which the observations indicate the existence of anyonic Majorana zero modes in solid state systems. We also discuss fractional quantum Hall systems (the 5/2 state) in this context. We describe proposed schemes for carrying out braiding with Majorana zero modes as well as the necessary steps for implementing topological quantum computation.
We introduce and study a class of anyon models that are a natural generalization of Ising anyons and Majorana fermion zero modes. These models combine an Ising anyon sector with a sector associated with $SO(m)_2$ Chern-Simons theory. We show how they can arise in a simple scenario for electron fractionalization and give a complete account of their quasiparticles types, fusion rules, and braiding. We show that the image of the braid group is finite for a collection of $2n$ fundamental quasiparticles and is a proper subgroup of the metaplectic representation of $Sp(2n-2,mathbb{F}_m)ltimes H(2n-2,mathbb{F}_m)$, where $Sp(2n-2,mathbb{F}_m)$ is the symplectic group over the finite field $mathbb{F}_m$ and $H(2n-2,mathbb{F}_m)$ is the extra special group (also called the $(2n-1)$-dimensional Heisenberg group) over $mathbb{F}_m$. Moreover, the braiding of fundamental quasiparticles can be efficiently simulated classically. However, computing the result of braiding a certain type of composite quasiparticle is $# P$-hard, although it is not universal for quantum computation because it has a finite braid group image. This a rare example of a topological phase that is not universal for quantum computation through braiding but nevertheless has $# P$-hard link invariants. We argue that our models are closely related to recent analyses finding non-Abelian anyonic properties for defects in quantum Hall systems, generalizing Majorana zero modes in quasi-1D systems.
We investigate the decoherence patterns of topological qubits in contact with the environment by a novel way of deriving the open system dynamics other than the Feynman-Vernon. Each topological qubit is made of two Majorana modes of a 1D Kitaevs chain. These two Majorana modes interact with the environment in an incoherent way which yields peculiar decoherence patterns of the topological qubit. More specifically, we consider the open system dynamics of the topological qubits which are weakly coupled to the fermionic/bosonic Ohmic-like environments. We find atypical patterns of quantum decoherence. In contrast to the cases of non-topological qubits for which they always decohere completely in all Ohmic-like environments, the topological qubits decohere completely in the Ohmic and sub-Ohmic environments but not in the super-Ohmic ones. Moreover, we find that the fermion parities of the topological qubits though cannot prevent the qubit states from decoherence in the sub-Ohmic environments, can prevent from thermalization turning into Gibbs state. We also study the cases in which each Majorana mode can couple to different Ohmic-like environments and the time dependence of concurrence for two topological qubits.
We investigate the topological properties of spin polarized fermionic polar molecules loaded in a multi-layer structure with the electric dipole moment polarized to the normal direction. When polar molecules are paired by attractive inter-layer interaction, unpaired Majorana fermions can be macroscopically generated in the top and bottom layers in dilute density regime. We show that the resulting topological state is effectively composed by a bundle of 1D Kitaev ladders labeled by in-plane momenta k and -k, and hence belongs to BDI class characterized by the winding number Z, protected by the time reversal symmetry. The Majorana surface modes exhibit a flatband at zero energy, fully gapped from Bogoliubov excitations in the bulk, and hence becomes an idea system to investigate the interaction effects on quantum degenerate Majorana fermions. We further show that additional interference fringes can be identified as a signature of such 2D Majorana surface modes in the time-of-flight experiment.
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