5,363 research outputs found
On string topology of classifying spaces
Let G be a compact Lie group. By work of Chataur and Menichi, the homology of
the space of free loops in the classifying space of G is known to be the value
on the circle in a homological conformal field theory. This means in particular
that it admits operations parameterized by homology classes of classifying
spaces of diffeomorphism groups of surfaces. Here we present a radical
extension of this result, giving a new construction in which diffeomorphisms
are replaced with homotopy equivalences, and surfaces with boundary are
replaced with arbitrary spaces homotopy equivalent to finite graphs. The result
is a novel kind of field theory which is related to both the diffeomorphism
groups of surfaces and the automorphism groups of free groups with boundaries.
Our work shows that the algebraic structures in string topology of classifying
spaces can be brought into line with, and in fact far exceed, those available
in string topology of manifolds. For simplicity, we restrict to the
characteristic 2 case. The generalization to arbitrary characteristic will be
addressed in a subsequent paper.Comment: 93 pages; v4: minor changes; to appear in Advances in Mathematic
Kitaev spin models from topological nanowire networks
We show that networks of topological nanowires can realize the physics of
exactly solvable Kitaev spin models with two-body interactions. This connection
arises from the description of the low-energy theory of both systems in terms
of a tight-binding model of Majorana modes. In Kitaev spin models the Majorana
description provides a convenient representation to solve the model, whereas in
an array of topological nanowires it arises, because the physical Majorana
modes localized at wire ends permit tunnelling between wire ends and across
different Josephson junctions. We explicitly show that an array of junctions of
three wires -- a setup relevant to topological quantum computing with nanowires
-- can realize the Yao-Kivelson model, a variant of Kitaev spin models on a
decorated honeycomb lattice. Translating the results from the latter, we show
that the network can be constructed to give rise to collective states
characterized by Chern numbers \nu = 0, +/-1 and +/-2, and that defects in an
array can be associated with vortex-like quasi-particle excitations. Finally,
we analyze the stability of the collective states as well as that of the
network as a quantum information processor. We show that decoherence inducing
instabilities, be them due to disorder or phase fluctuations, can be understood
in terms of proliferation of the vortex-like quasi-particles.Comment: 15 pages, 9 figure
A hierarchy of exactly solvable spin-1/2 chains with so(N)_1 critical points
We construct a hierarchy of exactly solvable spin-1/2 chains with so(N)_1
critical points. Our construction is based on the framework of
condensate-induced transitions between topological phases. We employ this
framework to construct a Hamiltonian term that couples N transverse field Ising
chains such that the resulting theory is critical and described by the so(N)_1
conformal field theory. By employing spin duality transformations, we then cast
these spin chains for arbitrary N into translationally invariant forms that all
allow exact solution by the means of a Jordan-Wigner transformation. For odd N
our models generalize the phase diagram of the transverse field Ising chain,
the simplest model in our hierarchy. For even N the models can be viewed as
longer ranger generalizations of the XY chain, the next model in the hierarchy.
We also demonstrate that our method of constructing spin chains with given
critical points goes beyond exactly solvable models. Applying the same strategy
to the Blume-Capel model, a spin-1 generalization of the Ising chain in a
generic magnetic field, we construct another critical spin-1 chain with the
predicted CFT describing the criticality.Comment: 24 pages, 5 figures; v2: minor changes and added reference
Scaling of random spreading in small world networks
In this study we have carried out computer simulations of random walks on
Watts-Strogatz-type small world networks and measured the mean number of
visited sites and the return probabilities. These quantities were found to obey
scaling behavior with intuitively reasoned exponents as long as the probability
of having a long range bond was sufficiently low.Comment: 3 pages, 4 figure
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