FINDING: Anyon topological qubits are realized via braiding operations in modular tensor categories, with interferometric readout achievable through Josephson vortex dynamics rather than Abrikosov cores. | MATH: Fibonacci anyon braiding gates: single-qubit gate set compiled from braid words in the Fibonacci anyon model, where the fusion space dimension grows as Fibonacci numbers Fn (golden ratio ϕ=(1+5)/2≈1.618); anyon condensation described by modular tensor category inclusion C→D with a commutative algebra object A∈C; Chern–Simons term SCS=4πk∫A∧dA with level k determining anyonic statistics θ=π/k; flux qubit readout maps anyonic braid to Josephson phase φ with quantum interference condition Δφ=2πn. | CONNECTION: Fibonacci anyons directly encode the golden ratio ϕ=1.618 and its inverse 1/ϕ=0.618 in the fusion rules \(\tau \otimes \tau = 1 \op Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Abstract In a physical system undergoing a continuous quantum phase transition, spontaneous symmetry breaking occurs when certain symmetries of the Hamiltonian fail to be preserved in the ground state. In the traditional Landau theory, a symmetry group can break down to any subgroup. However, this no longer holds across a continuous phase transition driven by anyon condensation in symmetry enriched topological orders (SETOs). For a SETO described by aG-crossed braided extension C⊆CG× C⊆CG× , we show that physical considerations require that a connected étale algebraA∈ C C admit aG-equivariant algebra structure for symmetry to be preserved under condensation ofA. Given any categorical actionG→EqBr( C C ) such thatg(A) ≅Afor allg∈G, we show there is a short exact sequence whose splittings correspond toG-equivariant algebra structures. The non-splitting of this sequence forces spontaneous symmetry breaking under condensation ofA, while inequivalent splittings of the sequence correspond to different SETOs resulting from the anyon-condensation transition. Furthermore, we show that if symmetry is preserved, there is a canonically associated SETO of CAloc CAloc , and gauging this symmetry commutes with anyon condensation.
Instead of studying anyon condensation in various concrete models, we take a bootstrap approach by considering an abstract situation, in which an anyon condensation happens in a 2-d topological phase with anyonic excitations given by a modular tensor category C; and the anyons in the condensed phase are given by another modular tensor category D. By a bootstrap analysis, we derive a relation between anyons in D-phase and anyons in C-phase from natural physical requirements. It turns out that the vacuum (or the tensor unit) A in D-phase is necessary to be a connected commutative separable algebra in C, and the category D is equivalent to the category of local A-modules as modular tensor categories. This condensation also produces a gapped domain wall with wall excitations given by the category of A-modules in C. A more general situation is also studied in this paper. We will also show how to determine such algebra A from the initial and final data. Multi-condensations and 1-d condensations will also be briefly discussed. Examples will be given in the toric code model, Kitaev quantum double models, Levin–Wen types of lattice models and some chiral topological phases.