3 research outputs found
Superselection sectors in the 3d Toric Code
We rigorously define superselection sectors in the 3d (spatial dimensions)
Toric Code Model on the infinite lattice . We begin by
constructing automorphisms that correspond to infinite flux strings, a
phenomenon that's only possible in open manifolds. We then classify all ground
state superselection sectors containing infinite flux strings, and find a rich
structure that depends on the geometry and number of strings in the
configuration. In particular, for a single infinite flux string configuration
to be a ground state, it must be monotonic. For configurations containing
multiple infinite flux strings, we define "infinity directions" and use that to
establish a necessary and sufficient condition for a state to be in a ground
state superselection sector. Notably, we also find that if a state contains
more than 3 infinite flux strings, then it is not in a ground state
superselection sector.Comment: 33 pages, 16 figure
Classification of the anyon sectors of Kitaev's quantum double model
We give a complete classification of the anyon sectors of Kitaev's quantum
double model on the infinite triangular lattice and for finite gauge group ,
including the non-abelian case. As conjectured, the anyon sectors of the model
correspond precisely to the irreducible representations of the quantum double
algebra of . Our proof consists of two main parts. In the first part, we
construct for each irreducible representation of the quantum double algebra a
pure state and show that the GNS representations of these pure states are
pairwise disjoint anyon sectors. In the second part we show that any anyon
sector is unitarily equivalent to one of the anyon sectors constructed in the
first part. Purity of the states constructed in the first part is shown by
characterising these states as the unique states that satisfy appropriate local
constraints. These constraints are of two types, namely flux constraints and
gauge constraints. The flux constraints single out certain string-net states,
while the gauge constraints fix the way in which these string-nets condense. At
the core of the proof is the fact that certain groups of local gauge
transformations act freely and transitively on collections of local
string-nets. The proof that the GNS representations of these states are anyon
sectors relies on showing that they are unitarily equivalent to amplimorphism
representations which are much easier to compare to the ground state
representation. For the second part, we show that any anyon sector contains a
pure state that satisfies all but a finite number of the constraints
characterising the pure states of the first part. Using known techniques we can
then construct a pure state in the anyon sector that satisfies all but one of
these constraints. Finally, we show that any such state must be a vector state
in one of the anyon sectors constructed in the first part.Comment: Errors of version 1 fixed, presentation throughout the manuscript
improved. 83 pages, 16 figure
Dynamical abelian anyons with bound states and scattering states
We introduce a family of quantum spin Hamiltonians on that can
be regarded as perturbations of Kitaev's abelian quantum double models that
preserve the gauge and duality symmetries of these models. We analyze in detail
the sector with one electric charge and one magnetic flux and show that the
spectrum in this sector consists of both bound states and scattering states of
abelian anyons. Concretely, we have defined a family of lattice models in which
abelian anyons arise naturally as finite-size quasi-particles with non-trivial
dynamics that consist of a charge-flux pair. In particular, the anyons exhibit
a non-trivial holonomy with a quantized phase, consistent with the gauge and
duality symmetries of the Hamiltonian.Comment: 21 pages, 10 figure