From Majorana Fermions to Topological Order

Barbara M. Terhal; David P. DiVincenzo; Fabian Hassler

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From Majorana Fermions to Topological Order

Barbara M. Terhal, Fabian Hassler, David P. DiVincenzo

Quantum Physics · Submitted 18 Jan 2012

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We show that the fermionic Hamiltonian modeling this system is topologically-ordered in a region of parameter space.

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.16 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.27 · compares with prior results 0.17 · mentions code or data availability 0.02.

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arXiv listing abstract v4. Results and the main claim are underlined; show every role.

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. In particular 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 (FM or AFM) are determined by additional gauge bits. Our mapping allows an understanding of the non-perturbative regime and the phase transition to a non-topological 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.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We consider a system consisting of a 2D network of links between Majorana fermions on superconducting islands.	contribution 0.54	0.33	0.03	asserted 0.75
s1	We show that the fermionic Hamiltonian modeling this system is topologically-ordered in a region of parameter space.	result 0.97	0.95	0.76	asserted 1.00
s2	In particular we show that Kitaev's toric code emerges in fourth-order perturbation theory.	result 0.99	0.91	0.23	asserted 1.00
s3	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 (FM or AFM) are determined by additional gauge bits.	method 0.97	0.89	0.06	asserted 0.88
s4	Our mapping allows an understanding of the non-perturbative regime and the phase transition to a non-topological phase.	interpretation 0.64	0.86	0.12	asserted 0.95
s5	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.	interpretation 0.81	0.60	0.20	hedged possibility 0.96

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

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arXiv listing abstract v4

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full text
figures and tables
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T body sentences
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original 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. In particular 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 (FM or AFM) are determined by additional gauge bits. Our mapping allows an understanding of the non-perturbative regime and the phase transition to a non-topological 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.

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How the literature uses it

4 arXiv papers cite this in 22 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 10 sentences without a confident label

Builds on it 1 paper

To illustrate the basic premise as it may be applied to architectures with Majorana fermions, we will advance and study a generalization of a model introduced in Ref. [ref] to describe a square lattice array of Josephson-coupled nanowires on superconducting grains.

Arbitrary Dimensional Majorana Dualities and Network Architectures for Topological Matter · Introduction · 1.00

Uses its method or tool 2 papers

[ref] The bond-algebraic approach, [ref] that we employ to study general exact dualities and fermionization, [ref] allows for the derivation of earlier known dualities as well as a plethora of many new others for rather general networks (or graphs) in arbitrary dimensions and boundary conditions.

Arbitrary Dimensional Majorana Dualities and Network Architectures for Topological Matter · Introduction · 0.96
The coupling of independent slips is then calculated as the tunnelling amplitude in the cosine potential using semi-classical WKB approximation [ref] , leading to $lnΔ ∝-√(E_J / E_C)$.

Topological ordering in the Majorana Toric Code · Dominant Josephson tunneling rate · 0.95

Compares with it 2 papers

This latter result was derived, using methods very different to ours, in Ref. [ref] .

Arbitrary Dimensional Majorana Dualities and Network Architectures for Topological Matter · Duality to annealed transverse-field Ising models · 0.81
These architectures display some similarities to the surface-code setups introduced recently in Refs. [ref] , but seek to leverage the topological quantum information processing afforded by MZMs, rather than pursuing active error correction.

Scalable Designs for Quasiparticle-Poisoning-Protected Topological Quantum Computation with Majorana Zero Modes · Conclusions and Near-term Directions · 1.00

Cites it as background 3 papers

The phase-diagram (see Fig. [ref]) of the system is rich and has been analyzed [ref] .

Topological ordering in the Majorana Toric Code · The model · 0.92
[ref] In particular, the possibility of realizing them as zero-energy states localized at the end of one-dimensional systems [so-called Majorana end states (MES)] has been the subject of many recent investigations.

Majorana states in inhomogeneous spin ladders · Introduction · 0.97
In the Introduction, we succinctly reviewed the effective Hamiltonian for the square lattice array, [ref] depicted in Fig. [ref], of Josephson-coupled granular superconductors carrying each two nanowires.

Arbitrary Dimensional Majorana Dualities and Network Architectures for Topological Matter · Networks of Superconducting Grains and Nanowires · 0.86

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