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arXiv:1201.3757 · Physical Review Letters

From Majorana fermions to topological order

Barbara M. Terhal, Fabian Hassler, David Peter Divincenzo

Published Jun 28, 2012• 81 citations
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Abstract

We consider a system consisting of a 2D network of links between Majorana fermions on superconducting islands. We show that the fermionic Hamiltonian modeling this system is topologically ordered in a region of parameter space: we show that Kitaev's toric code emerges in fourth-order perturbation theory. By using a Jordan-Wigner transformation we can map the model onto a family of signed 2D Ising models in a transverse field where the signs, ferromagnetic or antiferromagnetic, are determined by additional gauge bits. Our mapping allows an understanding of the nonperturbative regime and the phase transition to a nontopological phase. We discuss the physics behind a possible implementation of this model and argue how it can be used for topological quantum computation by adiabatic changes in the Hamiltonian.

DOI
10.1103/physrevlett.108.260504

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We consider a system consisting of a 2D network of links between Majorana fermions on superconducting islands. We show that the fermionic Hamiltonian modeling this system is topologically ordered in a region of parameter space: we show that Kitaev's toric code emerges in fourth-order perturbation theory. By using a Jordan-Wigner transformation we can map the model onto a family of signed 2D Ising models in a transverse field where the signs, ferromagnetic or antiferromagnetic, are determined by additional gauge bits. Our mapping allows an understanding of the nonperturbative regime and the phase transition to a nontopological phase. We discuss the physics behind a possible implementation of this model and argue how it can be used for topological quantum computation by adiabatic changes in the Hamiltonian.
The Kitaev chain model exhibits topological order that manifests as topological degeneracy, Majorana edge modes and Z2Z_{2}Z2​ topological invariance of the abulk 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 will explore topological protection of the ground state manifold in the case of Majorana fermion models which exhibit Z2Z_{2}Z2​ topological order. We will 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 will also show how these two different ways are related to each other. We provide a very general approach of understanding the topological protection of Majorana fermion qubits in the case of lattice Hamiltonians.
At zero temperature, a two-dimensional lattice of Majorana zero modes on mesoscopic superconducting islands exhibits a topologically ordered toric code phase. Recently, a Landau field theory was used to describe the different phases of the aforementioned system and the phase transitions separating them. While the field theory provides details on the properties of the system close to the phase transitions, signatures of topological ordering in the different phases have not been computed. This is the primary goal of this work. We describe a lattice gauge theory of the Majorana toric code in terms of U(1)\mathrm{U}(1)U(1) matter fields coupled to an emergent Z2{\mathbb{Z}}_{2}Z2​ gauge field. Subsequently, we use a generalized Wilson-loop order parameter, the equal-time Fredenhagen-Marcu order parameter, to characterize the topological ordering in the different phases. Our computation provides evidence of the toric code phase both in the Mott-insulator and the charge-2e2e2e superconductor phases, while showing that the toric code phase disappears in the charge-eee superconductor phase. In addition, we perturbatively analyze the influence of Cooper pair tunneling on the topological gap of the toric code in the limit of strong charging energy and show that the toric code phase is, in fact, stabilized by the Cooper pair tunneling. Our results are relevant for experimental realizations of the Majorana toric code.
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.
Applying the Jordan-Wigner transformation to spin-1/2 operators on special quasi-one dimensional paths, we show that the two-dimensional Kitaev model can be exactly mapped to a free Majorana fermion model without any redundant degrees of freedom. Via duality transformation, it can be further shown that the quantum phase transitions of the Kitaev model are described by non-local topological order parameters, which become Landau type local order parameters in the dual space. A closed relationship between conventional and topological quantum phase transitions is revealed, and the validity of conventional Landau phase transition theory with spontaneous symmetry breaking and local order parameters is also extended.
We have introduced a novel Majorana representation of S=1/2 spins using the Jordan-Wigner transformation and have shown that a generalized spin model of Kitaev defined on a brick-wall lattice is equivalent to a model of noninteracting Majorana fermions with Z2 gauge fields without redundant degrees of freedom. The quantum phase transitions of the system at zero temperature are found to be of topological type and can be characterized by nonlocal string order parameters (SOP). In appropriate dual representations, these SOP become local order parameters and the basic concept of Landau theory of continuous phase transition can be applied.
In this paper, we prove, extend and review possible mappings between the two-dimensional (2D) cluster state, Wen's model, the 2D Ising chain and Kitaev's toric code model. We introduce a 2D duality transformation to map the 2D lattice cluster state into the topologically ordered Wen model. Then, we investigate how this mapping could be achieved physically, which allows us to discuss the rate at which a topologically ordered system can be achieved. Next, using a lattice fermionization method, Wen's model is mapped into a series of 1D Ising interactions. Considering the boundary terms with this mapping then reveals how the Ising chains interact with one another. The duality of these models can be taken as a starting point to address questions as to how their gate operations in different quantum computational models can be related to each other.