130 research outputs found
Fermion condensation and super pivotal categories
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
fermionic theories, which we call "m-type" and "q-type" particles. The
endomorphism algebras of q-type particles 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 in fermionic topological phases. We then prove a
series of results relating data in condensed theories to data in their parent
theories; for example, if is a modular tensor category containing
a fermion, then the tube category of the condensed theory satisfies
.
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
theory, and the theory, and compute the
quasiparticle excitation spectrum in each of these examples.Comment: 161 pages; v2: corrected typos (including 18 instances of "the the")
and added some reference
Foliated Field Theory and String-Membrane-Net Condensation Picture of Fracton Order
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.Comment: 22+15 pages, 8 figures; v3 added a summary of our model near the end
of the introductio
Topological Defects on the Lattice I: The Ising model
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.Comment: 45 pages, 9 figure
Interaction effects in superconductor/quantum spin Hall devices: universal transport signatures and fractional Coulomb blockade
Interfacing s-wave superconductors and quantum spin Hall edges produces
time-reversal-invariant topological superconductivity of a type that can not
arise in strictly 1D systems. With the aim of establishing sharp fingerprints
of this novel phase, we use renormalization group methods to extract universal
transport characteristics of superconductor/quantum spin Hall heterostructures
where the native edge states serve as leads. We determine scaling forms for the
conductance through a grounded superconductor and show that the results depend
sensitively on the interaction strength in the leads, the size of the
superconducting region, and the presence or absence of time-reversal-breaking
perturbations. We also study transport across a floating superconducting island
isolated by magnetic barriers. Here we predict e-periodic Coulomb-blockade
peaks, as recently observed in nanowire devices [Albrecht et al., Nature 531,
206 (2016)], with the added feature that the island can support fractional
charge tunable via the relative orientation of the barrier magnetizations. As
an interesting corollary, when the magnetic barriers arise from strong
interactions at the edge that spontaneously break time-reversal symmetry, the
Coulomb-blockade periodicity changes from e to e/2. These findings suggest
several future experiments that probe unique characteristics of topological
superconductivity at the quantum spin Hall edge.Comment: 18 pages, 7 figure
Approaching a topological phase transition in Majorana nanowires
Recent experiments have produced mounting evidence of Majorana zero modes in
nanowire-superconductor 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.Comment: 13 pages, 8 figure
Torsorial actions on G-crossed braided tensor categories
We develop a method for generating the complete set of basic data under the
torsorial actions of and on
a -crossed braided tensor category , where
is the set of invertible simple objects in the braided tensor
category . When is a modular tensor category, the
and torsorial action gives a
complete generation of possible -crossed extensions, and hence provides a
classification. This torsorial classification can be (partially) collapsed by
relabeling equivalences that appear when computing the set of -crossed
braided extensions of . The torsor method presented here reduces
these redundancies by systematizing relabelings by -valued
-cochains. We also use our methods to compute the composition rule of these
torsor functors.Comment: 34 pages, several figures; v2: added Sec V, VI, and minor correction
Super Pivotal Categories, Fermion Condensation, and Fermionic Topological Phases
We describe a systematic way of producing fermionic topological phases using the technique of fermion condensation. We give a prescription for performing fermion condensation in bosonic topological phases which contain an emergent 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 fermionic theories, which we call “m-type” and “q-type” particles. The endomorphism algebras of q-type particles 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 in fermionic topological phases. We then prove a series of results relating data in condensed theories to data in their parent theories; for example, if C is a modular tensor category containing a fermion, then the tube category of 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.</p
Church networks, peacebuilding and women’s participation in Eastern DRC and the Great Lakes Region – a mapping study
This report (i) map the religious civil society networks, (ii) assess their role in local and regional peace processes and (iii) address the organisation of women and the promotion of issues of women, peace and security within these networks. The focus in this report is on the linkages between provincial centres and churches at village level in the vast rural areas of the Kivus where civilians continue to suffer from war crimes. The legitimacy of the church in peacebuilding at local and national levels hinges on the assumption that (i) the church leadership has a mandate based in their constituency on the ground, and that (ii) church coordinating structures in the province or at the national level have the capacity to coordinate church activities at lower levels of the church hierarchy.publishedVersio
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