1,677 research outputs found
Microscopic models for Kitaev's sixteenfold way of anyon theories
In two dimensions, the topological order described by gauge
theory coupled to free or weakly interacting fermions with a nonzero spectral
Chern number is classified by as predicted by
Kitaev [Ann. Phys. 321, 2 (2006)]. Here we provide a systematic and complete
construction of microscopic models realizing this so-called sixteenfold way of
anyon theories. These models are defined by matrices satisfying the
Clifford algebra, enjoy a global symmetry, and live on
either square or honeycomb lattices depending on the parity of . We show
that all these models are exactly solvable by using a Majorana representation
and characterize the topological order by calculating the topological spin of
an anyonic quasiparticle and the ground-state degeneracy. The possible
relevance of the and models to materials with
Kugel-Khomskii-type spin-orbital interactions is discussed.Comment: 6+9 pages, 2+1 figures, published versio
Detection of Symmetry Enriched Topological Phases
Topologically ordered systems in the presence of symmetries can exhibit new
structures which are referred to as symmetry enriched topological (SET) phases.
We introduce simple methods to detect the SET order directly from a complete
set of topologically degenerate ground state wave functions. In particular, we
first show how to directly determine the characteristic symmetry
fractionalization of the quasiparticles from the reduced density matrix of the
minimally entangled states. Second, we show how a simple generalization of a
non-local order parameter can be measured to detect SETs. The usefulness of the
proposed approached is demonstrated by examining two concrete model states
which exhibit SET: (i) a spin-1 model on the honeycomb lattice and (ii) the
resonating valence bond state on a kagome lattice. We conclude that the spin-1
model and the RVB state are in the same SET phases
Geometry of the momentum space: From wire networks to quivers and monopoles
A new nano--material in the form of a double gyroid has motivated us to study
(non-commutative geometry of periodic wire networks and the associated
graph Hamiltonians. Here we present the general abstract framework, which is
given by certain quiver representations, with special attention to the original
case of the gyroid as well as related cases, such as graphene. In these
geometric situations, the non- commutativity is introduced by a constant
magnetic field and the theory splits into two pieces: commutative and
non-commutative, both of which are governed by a geometry.
In the non-commutative case, we can use tools such as K-theory to make
statements about the band structure. In the commutative case, we give geometric
and algebraic methods to study band intersections; these methods come from
singularity theory and representation theory. We also provide new tools in the
study, using -theory and Chern classes. The latter can be computed using
Berry connection in the momentum space. This brings monopole charges and issues
of topological stability into the picture.Comment: 31 pages, 4 figure
Symmetry protected fractional Chern insulators and fractional topological insulators
In this paper we construct fully symmetric wavefunctions for the
spin-polarized fractional Chern insulators (FCI) and time-reversal-invariant
fractional topological insulators (FTI) in two dimensions using the parton
approach. We show that the lattice symmetry gives rise to many different FCI
and FTI phases even with the same filling fraction (and the same
quantized Hall conductance in FCI case). They have different
symmetry-protected topological orders, which are characterized by different
projective symmetry groups. We mainly focus on FCI phases which are realized in
a partially filled band with Chern number one. The low-energy gauge groups of a
generic FCI wavefunctions can be either or
the discrete group , and in the latter case the associated low-energy
physics are described by Chern-Simons-Higgs theories. We use our construction
to compute the ground state degeneracy. Examples of FCI/FTI wavefunctions on
honeycomb lattice and checkerboard lattice are explicitly given. Possible
non-Abelian FCI phases which may be realized in a partially filled band with
Chern number two are discussed. Generic FTI wavefunctions in the absence of
spin conservation are also presented whose low-energy gauge groups can be
either or . The constructed wavefunctions
also set up the framework for future variational Monte Carlo simulations.Comment: 24 pages, 13 figures, published versio
Double Semion Phase in an Exactly Solvable Quantum Dimer Model on the Kagome Lattice
Quantum dimer models typically arise in various low energy theories like
those of frustrated antiferromagnets. We introduce a quantum dimer model on the
kagome lattice which stabilizes an alternative topological
order, namely the so-called "double semion" order. For a particular set of
parameters, the model is exactly solvable, allowing us to access the ground
state as well as the excited states. We show that the double semion phase is
stable over a wide range of parameters using numerical exact diagonalization.
Furthermore, we propose a simple microscopic spin Hamiltonian for which the
low-energy physics is described by the derived quantum dimer model.Comment: 7 pages, 5 figure
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