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arXiv:1209.5460 · Physical Review B

Majorana edge states for Z2 topological orders of the Wen plaquette and toric code models

Jing Yu, Xing-Hai Zhang, Su-Peng Kou

Published May 3, 2013• 12 citations
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Abstract

In this paper, we study the symmetry protected Majorana edge states for the ${\mathbb{Z}}_{2}$ topological order of the Wen plaquette and toric code models and calculate the dispersion of the Majorana edge states. For the system with translational symmetry, the Majorana edge states are gapless and have the nodal points at $k=0$ and $\ensuremath{\pi}$. For the edge states of the toric code model without translational symmetry, the edge modes become gapped.

DOI
10.1103/physrevb.87.184402

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In this paper, we study the symmetry protected Majorana edge states for the Z2{\mathbb{Z}}_{2}Z2​ topological order of the Wen plaquette and toric code models and calculate the dispersion of the Majorana edge states. For the system with translational symmetry, the Majorana edge states are gapless and have the nodal points at k=0k=0k=0 and \ensuremathπ\ensuremath{\pi}\ensuremathπ. For the edge states of the toric code model without translational symmetry, the edge modes become gapped.
We study possible gapless edge states of translation-symmetric Z2{\mathbb{Z}}_{2}Z2​ spin liquids. The gapless edge states emerge from dangling Majorana fermions at the boundary. We construct a series of mean-field Hamiltonians of Z2{\mathbb{Z}}_{2}Z2​ spin liquids on the square lattice; these models can be obtained by generalization of Wen's exactly solvable plaquette model. We also study the details of the edge theory of these Z2{\mathbb{Z}}_{2}Z2​ spin liquids and find their effective background field (BF) theory descriptions. The effective BF theories are shown to describe the crystal momenta of the ground states and their degeneracies and to predict the edge theories of these Z2{\mathbb{Z}}_{2}Z2​ spin liquids. As a by-product, we obtained a way to classify the BF theories reflecting the lattice symmetries. We discuss in closing three-dimensional Z2{\mathbb{Z}}_{2}Z2​ spin liquids with gapless surface states on the cubic lattice.
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.
We reveal the connection between two-dimensional subsystem symmetry-protected topological (SSPT) states and two-dimensional topological orders via a self-dual frustrated toric code model. This model, an enrichment of the toric code (TC) with its dual interactions, can be mapped to a model defined on the dual lattice with subsystem symmetries and subextensive ground state degeneracy. The map connects exactly the frustrated TC to two copies of the topological plaquette Ising model (TPIM), as a strong SSPT model with linear subsystem symmetries. The membrane order parameter of the TPIM is exactly mapped to dual TC stabilizers as the order parameter of the frustrated TC model, SSPT gapless edge states of the TPIM are mapped to zero-energy dangling operators under open boundaries, and the transition from the SSPT-ordered TPIM to the trivial paramagnetic phase is mapped to the transition between two distinct topological orders. We also demonstrate that this mapping can be used to elucidate the structure of other SSPT models, reflecting the subtle linkage between SSPT order and topological order in two dimensions.
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.
Stabilizer codes provide exact lattice realizations of bosonic topological orders. In contrast, systematic stabilizer descriptions of intrinsically fermionic topological phases remain much less developed. In this work, we introduce Majorana-Pauli stabilizer codes, a class of exactly solvable fermionic lattice models whose stabilizers are built from both generalized Pauli operators and Majorana operators. As a main example, we construct an exactly solvable stabilizer realization of the fermionic toric code: an intrinsically fermionic Z2\mathbb Z_2Z2​ topological order in (2+1)(2{+}1)(2+1) dimensions, using Z8\mathbb Z_8Z8​ Pauli operators coupled to Majorana modes. Within this stabilizer framework, the anyons, string operators, fusion rules, and braiding statistics all follow naturally from the stabilizer algebra. More broadly, we show that the fermionic toric code belongs to a duality web generated by anyon condensation and by gauging or ungauging bosonic or fermionic symmetries. This web connects bosonic topological orders, symmetry-enriched topological (SET) phases, and both bosonic and fermionic symmetry-protected topological (SPT) phases, all within a common stabilizer description. We extend this construction to all Abelian fermionic topological orders with gapped boundaries and to fermionic SPT phases whose fully gauged theories are Abelian. Going beyond Majorana operators, we introduce fermionic versions of the clock and shift operators and use them to construct an exact bosonization map for ZDF\mathbb Z_D^FZDF​ symmetries for DDD even. Using this, we realize a stabilizer model for a nontrivial Z8F\mathbb Z_8^FZ8F​ fermionic SPT phase with no free-fermion analog. Altogether, these results extend the stabilizer-code paradigm to a broad class of intrinsically fermionic phases bridging fermionic quantum many-body physics to quantum error correction.
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.