1,001 research outputs found
Abelian powers in paper-folding words
We show that paper folding words contain arbitrarily large abelian powers
The fully residually F quotients of F*<x,y>
We describe the fully residually F; or limit groups relative to F; (where F
is a free group) that arise from systems of equations in two variables over F
that have coefficients in F.Comment: 64 pages, 2 figures. Following recommendations from a referee, the
paper has been completely reorganized and many small mistakes have been
corrected. There were also a few gaps in the earlier version of the paper
that have been fixed. In particular much of the content of Section 8 in the
previous version had to be replaced. This paper is to appear in Groups. Geom.
Dy
A Multiple Commutator Formula for the Sum of Feynman Diagrams
In the presence of a large parameter, such as mass or energy, leading
behavior of individual Feynman diagrams often get cancelled in the sum. This is
known to happen in large- QCD in the presence of a baryon, and also in the
case of high-energy electron-electron as well as quark-quark scatterings. We
present an exact combinatorial formula, involving multiple commutators of the
vertices, which can be used to compute such cancellations. It is a non-abelian
generalization of the eikonal formula, and will be applied in subsequent
publications to study the consistency of large- QCD involving baryons, as
well as high-energy quark-quark scattering in ordinary QCD.Comment: uu-encoded latex file with two postscript figure
Knapsack Problems in Groups
We generalize the classical knapsack and subset sum problems to arbitrary
groups and study the computational complexity of these new problems. We show
that these problems, as well as the bounded submonoid membership problem, are
P-time decidable in hyperbolic groups and give various examples of finitely
presented groups where the subset sum problem is NP-complete.Comment: 28 pages, 12 figure
Emergent Many-Body Translational Symmetries of Abelian and Non-Abelian Fractionally Filled Topological Insulators
The energy and entanglement spectrum of fractionally filled interacting
topological insulators exhibit a peculiar manifold of low energy states
separated by a gap from a high energy set of spurious states. In the current
manuscript, we show that in the case of fractionally filled Chern insulators,
the topological information of the many-body state developing in the system
resides in this low-energy manifold. We identify an emergent many-body
translational symmetry which allows us to separate the states in
quasi-degenerate center of mass momentum sectors. Within one center of mass
sector, the states can be further classified as eigenstates of an emergent (in
the thermodynamic limit) set of many-body relative translation operators. We
analytically establish a mapping between the two-dimensional Brillouin zone for
the Fractional Quantum Hall effect on the torus and the one for the fractional
Chern insulator. We show that the counting of quasi-degenerate levels below the
gap for the Fractional Chern Insulator should arise from a folding of the
states in the Fractional Quantum Hall system at identical filling factor. We
show how to count and separate the excitations of the Laughlin, Moore-Read and
Read-Rezayi series in the Fractional Quantum Hall effect into two-dimensional
Brillouin zone momentum sectors, and then how to map these into the momentum
sectors of the Fractional Chern Insulator. We numerically check our results by
showing the emergent symmetry at work for Laughlin, Moore-Read and Read-Rezayi
states on the checkerboard model of a Chern insulator, thereby also showing, as
a proof of principle, that non-Abelian Fractional Chern Insulators exist.Comment: 32 pages, 9 figure
- …