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Arbitrary Dimensional Majorana Dualities and Network Architectures for Topological Matter
Mesoscale and Nanoscale Physics · Submitted 14 Mar 2012
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We prove that (a) these Majorana systems, (b) quantum Ising gauge theories, and (c) transverse-field Ising models with annealed bimodal disorder are all dual to one another on general graphs.
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The abstract, sentence by sentence
Motivated by the prospect of attaining Majorana modes at the ends of nanowires, we analyze interacting Majorana systems on general networks and lattices in an arbitrary number of dimensions, and derive various universal spin duals. Such general complex Majorana architectures (other than those of simple square or other crystalline arrangements) might be of empirical relevance. As these systems display low-dimensional symmetries, they are candidates for realizing topological quantum order. We prove that (a) these Majorana systems, (b) quantum Ising gauge theories, and (c) transverse-field Ising models with annealed bimodal disorder are all dual to one another on general graphs. As any Dirac fermion (including electronic) operator can be expressed as a linear combination of two Majorana fermion operators, our results further lead to dualities between interacting Dirac fermionic systems. The spin duals allow us to predict the feasibility of various standard transitions as well as spin-glass type behavior in {\it interacting} Majorana fermion or electronic systems. Several new systems that can be simulated by arrays of Majorana wires are further introduced and investigated: (1) the {\it XXZ honeycomb compass} model (intermediate between the classical Ising model on the honeycomb lattice and Kitaev's honeycomb model), (2) a checkerboard lattice realization of the model of Xu and Moore for superconducting $(p+ip)$ arrays, and a (3) compass type two-flavor Hubbard model with both pairing and hopping terms. By the use of dualities, we show that all of these systems lie in the 3D Ising universality class. We discuss how the existence of topological orders and bounds on autocorrelation times can be inferred by the use of symmetries and also propose to engineer {\it quantum simulators} out of these Majorana networks.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Motivated by the prospect of attaining Majorana modes at the ends of nanowires, we analyze interacting Majorana systems on general networks and lattices in an arbitrary number of dimensions, and derive various universal spin duals. | contribution 0.98 | 0.96 | 0.09 | asserted 0.98 |
| s1 | Such general complex Majorana architectures (other than those of simple square or other crystalline arrangements) might be of empirical relevance. | interpretation 0.73 | 0.10 | 0.03 | speculative 0.98 |
| s2 | As these systems display low-dimensional symmetries, they are candidates for realizing topological quantum order. | interpretation 0.81 | 0.26 | 0.05 | hedged possibility 0.85 |
| s3 | We prove that (a) these Majorana systems, (b) quantum Ising gauge theories, and (c) transverse-field Ising models with annealed bimodal disorder are all dual to one another on general graphs. | result 0.97 | 0.94 | 0.51 | asserted 1.00 |
| s4 | As any Dirac fermion (including electronic) operator can be expressed as a linear combination of two Majorana fermion operators, our results further lead to dualities between interacting Dirac fermionic systems. | result 0.60 | 0.77 | 0.11 | asserted 0.98 |
| s5 | The spin duals allow us to predict the feasibility of various standard transitions as well as spin-glass type behavior in {\it interacting} Majorana fermion or electronic systems. | interpretation 0.85 | 0.81 | 0.14 | asserted 0.67 |
| s6 | Several new systems that can be simulated by arrays of Majorana wires are further introduced and investigated: (1) the {\it XXZ honeycomb compass} model (intermediate between the classical Ising model on the honeycomb lattice and Kitaev's honeycomb model), (2) a checkerboard lattice realization of the model of Xu and Moore for superconducting $(p+ip)$ arrays, and a (3) compass type two-flavor Hubbard model with both pairing and hopping terms. | contribution 1.00 | 0.90 | 0.07 | asserted 0.99 |
| s7 | By the use of dualities, we show that all of these systems lie in the 3D Ising universality class. | result 1.00 | 0.94 | 0.26 | asserted 1.00 |
| s8 | We discuss how the existence of topological orders and bounds on autocorrelation times can be inferred by the use of symmetries and also propose to engineer {\it quantum simulators} out of these Majorana networks. | future work 0.81 | 0.75 | 0.13 | hedged possibility 0.43 |
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The paper
Original abstract
Motivated by the prospect of attaining Majorana modes at the ends of nanowires, we analyze interacting Majorana systems on general networks and lattices in an arbitrary number of dimensions, and derive various universal spin duals. Such general complex Majorana architectures (other than those of simple square or other crystalline arrangements) might be of empirical relevance. As these systems display low-dimensional symmetries, they are candidates for realizing topological quantum order. We prove that (a) these Majorana systems, (b) quantum Ising gauge theories, and (c) transverse-field Ising models with annealed bimodal disorder are all dual to one another on general graphs. As any Dirac fermion (including electronic) operator can be expressed as a linear combination of two Majorana fermion operators, our results further lead to dualities between interacting Dirac fermionic systems. The spin duals allow us to predict the feasibility of various standard transitions as well as spin-glass type behavior in {\it interacting} Majorana fermion or electronic systems. Several new systems that can be simulated by arrays of Majorana wires are further introduced and investigated: (1) the {\it XXZ honeycomb compass} model (intermediate between the classical Ising model on the honeycomb lattice and Kitaev's honeycomb model), (2) a checkerboard lattice realization of the model of Xu and Moore for superconducting $(p+ip)$ arrays, and a (3) compass type two-flavor Hubbard model with both pairing and hopping terms. By the use of dualities, we show that all of these systems lie in the 3D Ising universality class. We discuss how the existence of topological orders and bounds on autocorrelation times can be inferred by the use of symmetries and also propose to engineer {\it quantum simulators} out of these Majorana networks.
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References
How the literature uses it
5 arXiv papers cite this in 9 sentences
Papers citing it · 6 sentences without a confident label
Uses its method or tool 1 paper
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This is done by the transformation [equation] using the bond algebraic approach [ref] (see Appendix [ref] for details).
Topological ordering in the Majorana Toric Code · The model · 1.00
Cites it as background 1 paper
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Since its discovery, the Jordan-Wigner transformation has been generalized to an entire family of exact free-fermion solutions [ref] , yielding new understanding for a wide class of spin models.
A Unified Graph-Theoretic Framework for Free-Fermion Solvability · Introduction · 0.87
What it cites · 14 sentences without a confident label
Builds on 1 paper
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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.
From Majorana Fermions to Topological Order · Introduction · 1.00
Uses the method or tool of 2 papers
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[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.
Effective and exact holographies from symmetries and dualities · Introduction · 0.96
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[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.
From Majorana Fermions to Topological Order · Introduction · 0.96
Compares with 1 paper
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This latter result was derived, using methods very different to ours, in Ref. [ref] .
From Majorana Fermions to Topological Order · Duality to annealed transverse-field Ising models · 0.81
Cites as background 2 papers
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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.
From Majorana Fermions to Topological Order · Networks of Superconducting Grains and Nanowires · 0.86
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More recently, there has been a flurry of activity in the study of Majorana fermions in candidate condensed matter realizations, [ref] including lattice [ref] and other [ref] systems inspired by the prospect of topological quantum computing.
Search for Majorana fermions in superconductors · Introduction · 0.88
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