We introduce a scheme for preparation, manipulation, and read out of Majorana zero modes in semiconducting wires with mesoscopic superconducting islands. Our approach synthesizes recent advances in materials growth with tools commonly used in quantum-dot experiments, including gate control of tunnel barriers and Coulomb effects, charge sensing, and charge pumping. We outline a sequence of milestones interpolating between zero-mode detection and quantum computing that includes (1) detection of fusion rules for non-Abelian anyons using either proximal charge sensors or pumped current, (2) validation of a prototype topological qubit, and (3) demonstration of non-Abelian statistics by braiding in a branched geometry. The first two milestones require only a single wire with two islands, and additionally enable sensitive measurements of the system's excitation gap, quasiparticle poisoning rates, residual Majorana zero-mode splittings, and topological-qubit coherence times. These pre-braiding experiments can be adapted to other manipulation and read out schemes as well.

In this paper and its sequel, we construct topologically invariant defects in two-dimensional classical lattice models and quantum spin chains. We show how defect lines commute with the transfer matrix/Hamiltonian when they obey the defect commutation relations, cousins of the Yang-Baxter equation. These relations and their solutions can be extended to allow defect lines to branch and fuse, again with properties depending only on topology. In this part I, we focus on the simplest example, the Ising model. We define lattice spin-flip and duality defects and their branching, and prove they are topological. One useful consequence is a simple implementation of Kramers-Wannier duality on the torus and higher genus surfaces by using the fusion of duality defects. We use these topological defects to do simple calculations that yield exact properties of the conformal field theory describing the continuum limit. For example, the shift in momentum quantization with duality-twisted boundary conditions yields the conformal spin 1/16 of the chiral spin field. Even more strikingly, we derive the modular transformation matrices explicitly and exactly.

Foliated fracton order is a qualitatively new kind of phase of matter. It is similar to topological order, but with the fundamental difference that a layered structure, referred to as a foliation, plays an essential role and determines the mobility restrictions of the topological excitations. In this work, we introduce a new kind of field theory to describe these phases: a foliated field theory. We also introduce a new lattice model and string-membrane-net condensation picture of these phases, which is analogous to the string-net condensation picture of topological order.Contents arXiv:1812.01613v3 [cond-mat.str-el]

We study fermionic topological phases using the technique of fermion condensation. We give a prescription for performing fermion condensation in bosonic topological phases which contain a fermion. Our approach to fermion condensation can roughly be understood as coupling the parent bosonic topological phase to a phase of physical fermions, and condensing pairs of physical and emergent fermions. There are two distinct types of objects in the resulting fermionic fusion categories, which we call "m-type" and "q-type" objects. The endomorphism algebras of q-type objects are complex Clifford algebras, and they have no analogues in bosonic theories. We construct a fermionic generalization of the tube category, which allows us to compute the quasiparticle excitations arising from the condensed theories. We prove a series of results relating data in fermionic theories to data in their parent bosonic theories; for example, if C is a modular tensor category containing a fermion, then the tube category constructed from the condensed theory satisfies Tube(C/ψ) ∼ = C × (C/ψ). We also study how modular transformations, fusion rules, and coherence relations are modified in the fermionic setting, prove a fermionic version of the Verlinde dimension formula, construct a commuting projector lattice Hamiltonian for fermionic theories, and write down a fermionic version of the Turaev-Viro-Barrett-Westbury state sum. A large portion of this work is devoted to three detailed examples of performing fermion condensation to produce fermionic topological phases: we condense fermions in the Ising theory, the SO(3) 6 theory, and the 1 2 E 6 theory, and compute the quasiparticle excitation spectrum in each of the condensed theories. arXiv:1709.01941v2 [cond-mat.str-el]

We show that particle detectors, such as two-level atoms, in noninertial motion (or in gravitational fields) could be used to build quantum gates for the processing of quantum information. Concretely, we show that through suitably chosen noninertial trajectories of the detectors the interaction Hamiltonian's time dependence can be modulated to yield arbitrary rotations in the Bloch sphere due to relativistic quantum effects.

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Recent experiments have produced mounting evidence of Majorana zero modes in nanowiresuperconductor hybrids. Signatures of an expected topological phase transition accompanying the onset of these modes nevertheless remain elusive. We investigate a fundamental question concerning this issue: Do well-formed Majorana modes necessarily entail a sharp phase transition in these setups? Assuming reasonable parameters, we argue that finite-size effects can dramatically smooth this putative transition into a crossover, even in systems large enough to support well-localized Majorana modes. We propose overcoming such finite-size effects by examining the behavior of low-lying excited states through tunneling spectroscopy. In particular, the excited-state energies exhibit characteristic field and density dependence, and scaling with system size, that expose an approaching topological phase transition. We suggest several experiments for extracting the predicted behavior. As a useful byproduct, the protocols also allow one to measure the wire's spin-orbit coupling directly in its superconducting environment.

Recent thermal-conductivity measurements evidence a magnetic-field-induced non-Abelian spin-liquid phase in the Kitaev material α-RuCl 3. Although the platform is a good Mott insulator, we propose experiments that electrically probe the spin liquid's hallmark chiral Majorana edge state and bulk anyons, including their exotic exchange statistics. We specifically introduce circuits that exploit interfaces between electrically active systems and Kitaev materials to "perfectly" convert electrons from the former into emergent fermions in the latter-thereby enabling variations of transport probes invented for topological superconductors and fractional quantum-Hall states. Along the way, we resolve puzzles in the literature concerning interacting Majorana fermions, and also develop an anyon-interferometry framework that incorporates nontrivial energy-partitioning effects. Our results illuminate a partial pathway toward topological quantum computation with Kitaev materials.