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arXiv:1412.8183

Majorana Edge States and Braiding in an Exactly Solvable One-dimensional Spin Model

Zhao-Yang Dong, Jianxin Li

Published Dec 28, 2014• 0 citations
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Abstract

We derive an exactly solvable one-dimensional (1D) spin model from the three-band Hubbard model with a strong spin-orbit coupling by introducing U(1) gauge fields to the isospin states. We find that it has a topological nontrivial phase characterized by Majorana end modes which are protected by a new Z_2 topological invariant related to the parity of the lattice sites (odd or even number of sites) in the spin chain. With the protection of this Z_2 topological invariant, a novel braiding of two Majorana edge states in this strictly geometric 1D chain is realized. We also discuss the possible realization of the gauge fields.

DOI
10.48550/arxiv.1412.8183

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Evolution of Majorona zero-energy edge states in a T2=−1T^2 = -1T2=−1 symmetry protected 1D topological superconductor with dominant spin-orbit coupling
We consider a 1D topological superconductor (TSC) constructed by coupling a pair of Kitaev's Majorana chains with opposite spin configurations. Such a 1D lattice model is known to be protected by a T2=−1T^2 = -1T2=−1 time-reversal symmetry. Furthermore, we consider a modeled Rashba spin-orbit coupling on such a system of T2=−1T^2=-1T2=−1 time-reversal symmetric TSC. The Rashba spin-orbit coupling together with the chemical potential engineered the phase transitions of the edge states in the system and consequently the number of Majorona's zero-energy edge modes (MZM's) emerging at the edge of the coupled chains. Correspondingly, the topological nature of the system is described by a phase diagram consisting of three different phases. The three phases are characterized by a topological winding number, W=1\mathcal{W}=1W=1, 222 (with one and two MZM's: topological phases) and W=0\mathcal{W}=0W=0 (devoid of any MZM: trivial insulating phase).
A family of two-dimensional (2D) spin-1/2 models have been constructed to realize Kitaev’s sixteen-fold way of anyon theories. Defining a one-dimensional (1D) path through all the lattice sites, and performing the Jordan-Wigner transformation with the help of the 1D path, we find that such a spin-1/2 model is equivalent to a model with \nu ν species of Majorana fermions coupled to a static \mathbb{Z}_2 ℤ 2 gauge field. Here each species of Majorana fermions gives rise to an energy band that carries a Chern number \mathcal{C}=1 𝒞 = 1 , yielding a total Chern number \mathcal{C}=\nu 𝒞 = ν . It has been shown that the ground states are three (four)-fold topologically degenerate on a torus, when \nu ν is an odd (even) number. These exactly solvable models can be achieved by quantum simulations.
The Kitaev chain model exhibits topological order that manifests as topological degeneracy, Majorana edge modes and Z2 topological invariant of the bulk spectrum. This model can be obtained from a transverse field Ising model(TFIM) using the Jordan–Wigner transformation. TFIM has neither topological degeneracy nor any edge modes. Topological degeneracy associated with topological order is central to topological quantum computation. In this paper, we explore topological protection of the ground state manifold in the case of Majorana fermion models which exhibit Z2 topological order. We show that there are at least two different ways to understand this topological protection of Majorana fermion qubits: one way is based on fermionic mode operators and the other is based on anti-commuting symmetry operators. We also show how these two different ways are related to each other. We provide a very general approach to understanding the topological protection of Majorana fermion qubits in the case of lattice Hamiltonians. We then show how in topological phases in Majorana fermion models gives rise to new braid group representations. So, we give a unifying and broad perspective of topological phases in Majorana fermion models based on anti-commuting symmetry operators and braid group representations of Majorana fermions as anyons.
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