Arbitrary Dimensional Majorana Dualities and Network Architectures for Topological Matter

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.

v3,19 pages, 18 figures, submitted to Physical Review B. 11 new figures, new section on simulating the Hubbard model with nanowire systems, and two new appendices

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• Topological phases in two-dimensional arrays of parafermionic zero modes
• Interacting Majorana Fermions
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• Physical Implementation of a Majorana Fermion Surface Code for Fault-Tolerant Quantum Computation
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• Geometric Criterion for Solvability of Lattice Spin Systems
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• Simulating topological tensor networks with Majorana qubits
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• Quantum phase transitions of the Majorana toric code in the presence of finite Cooper-pair tunneling
• Compass and Kitaev models -- Theory and Physical Motivations
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• Anatomy of Z2 fluxes in anyon Fermi liquids and Bose condensates
• Induced topological order at the boundary of 3D topological superconductors
• The non-Abelian Duality Problem
• Topological ordering in the Majorana Toric Code
• Charge response of the Majorana toric code
• A Theorem on Extensive Spectral Degeneracy for Systems with Higher Symmetries in General Dimensions
• The Hilbert-space structure of free fermions in disguise
• Designing ${\mathbb{Z}}_2$ and ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ topological orders in networks of Majorana bound states
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