2,502 research outputs found
Lattice generalization of the Dirac equation to general spin and the role of the flat band
We provide a novel setup for generalizing the two-dimensional pseudospin
S=1/2 Dirac equation, arising in graphene's honeycomb lattice, to general
pseudospin-S. We engineer these band structures as a nearest-neighbor hopping
Hamiltonian involving stacked triangular lattices. We obtain multi-layered low
energy excitations around half-filling described by a two-dimensional Dirac
equation of the form H=v_F S\cdot p, where S represents an arbitrary spin-S
(integer or half-integer). For integer-S, a flat band appears, whose presence
modifies qualitatively the response of the system. Among physical observables,
the density of states, the optical conductivity and the peculiarities of Klein
tunneling are investigated. We also study Chern numbers as well as the
zero-energy Landau level degeneracy. By changing the stacking pattern, the
topological properties are altered significantly, with no obvious analogue in
multilayer graphene stacks.Comment: 14 pages, 6 figures, 1 table, revised version with a new section on
experimental possibilitie
A constant-time algorithm for middle levels Gray codes
For any integer a middle levels Gray code is a cyclic listing of
all -element and -element subsets of such that
any two consecutive subsets differ in adding or removing a single element. The
question whether such a Gray code exists for any has been the subject
of intensive research during the last 30 years, and has been answered
affirmatively only recently [T. M\"utze. Proof of the middle levels conjecture.
Proc. London Math. Soc., 112(4):677--713, 2016]. In a follow-up paper [T.
M\"utze and J. Nummenpalo. An efficient algorithm for computing a middle levels
Gray code. To appear in ACM Transactions on Algorithms, 2018] this existence
proof was turned into an algorithm that computes each new set in the Gray code
in time on average. In this work we present an algorithm for
computing a middle levels Gray code in optimal time and space: each new set is
generated in time on average, and the required space is
Coordinated Robot Navigation via Hierarchical Clustering
We introduce the use of hierarchical clustering for relaxed, deterministic
coordination and control of multiple robots. Traditionally an unsupervised
learning method, hierarchical clustering offers a formalism for identifying and
representing spatially cohesive and segregated robot groups at different
resolutions by relating the continuous space of configurations to the
combinatorial space of trees. We formalize and exploit this relation,
developing computationally effective reactive algorithms for navigating through
the combinatorial space in concert with geometric realizations for a particular
choice of hierarchical clustering method. These constructions yield
computationally effective vector field planners for both hierarchically
invariant as well as transitional navigation in the configuration space. We
apply these methods to the centralized coordination and control of
perfectly sensed and actuated Euclidean spheres in a -dimensional ambient
space (for arbitrary and ). Given a desired configuration supporting a
desired hierarchy, we construct a hybrid controller which is quadratic in
and algebraic in and prove that its execution brings all but a measure zero
set of initial configurations to the desired goal with the guarantee of no
collisions along the way.Comment: 29 pages, 13 figures, 8 tables, extended version of a paper in
preparation for submission to a journa
Reduced Chern-Simons Quiver Theories and Cohomological 3-Algebra Models
We study the BPS spectrum and vacuum moduli spaces in dimensional reductions
of Chern-Simons-matter theories with N>=2 supersymmetry to zero dimensions. Our
main example is a matrix model version of the ABJM theory which we relate
explicitly to certain reduced 3-algebra models. We find the explicit maps from
Chern-Simons quiver matrix models to dual IKKT matrix models. We address the
problem of topologically twisting the ABJM matrix model, and along the way
construct a new twist of the IKKT model. We construct a cohomological matrix
model whose partition function localizes onto a moduli space specified by
3-algebra relations which live in the double of the conifold quiver. It
computes an equivariant index enumerating framed BPS states with specified
R-charges which can be expressed as a combinatorial sum over certain filtered
pyramid partitions.Comment: 47 page
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