Zohar Nussinov, Gerardo Ortiz, Emilio Cobanera
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. As these systems display low-dimensional symmetries, they are candidates for topological quantum order. We prove that (a) these Majorana systems, (b) quantum Ising gauge theories, and (c) annealed, bimodal transverse-field Ising models are all dual to one another on general graphs. This leads to an interesting connection between heavily disordered annealed Ising systems and uniform Ising theories. As any Dirac fermion can be expressed as a linear combination of two Majorana fermions, our results further lead to dualities between interacting fermionic systems on rather general graphs and spin 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 systems on general networks. Several new systems that can be simulated by arrays of Majorana wires are further introduced and investigated: (1) the {\it XXZ Kitaev} model (intermediate between the classical Ising model on the honeycomb lattice and Kitaev's honeycomb model), and (2) a checkerboard lattice realization of the model of Xu and Moore for superconducting $(p+ip)$ arrays. By the use of dualities, we show that both of these systems lie in the 3D Ising universality class. We discuss how the existence of topological orders and bounds on auto-correlation times can be inferred by the use of symmetries.
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http://arxiv.org/abs/1203.2983
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