6,959 research outputs found
Open Spin Chains and Complexity in the High Energy Limit
In the high energy limit of scattering amplitudes in Quantum Chromodynamics
and supersymmetric theories the dominant Feynman diagrams are characterized by
a hidden integrability. A well-known example is that of Odderon exchange, which
can be described as a bound state of three reggeized gluons and corresponds to
a closed spin chain with periodic boundary conditions. In the
supersymmetric Yang-Mills theory a similar spin chain arises in the multi-Regge
asymptotics of the eight-point amplitude in the planar limit. We investigate
the associated open spin chain in transverse momentum and rapidity variables
solving the corresponding effective Feynman diagrams. We introduce the concept
of complexity in the high energy effective field theory and study its emerging
scaling laws.Comment: 24 pages, 13 figures; references adde
Simulation of braiding anyons using Matrix Product States
Anyons exist as point like particles in two dimensions and carry braid
statistics which enable interactions that are independent of the distance
between the particles. Except for a relatively few number of models which are
analytically tractable, much of the physics of anyons remain still unexplored.
In this paper, we show how U(1)-symmetry can be combined with the previously
proposed anyonic Matrix Product States to simulate ground states and dynamics
of anyonic systems on a lattice at any rational particle number density. We
provide proof of principle by studying itinerant anyons on a one dimensional
chain where no natural notion of braiding arises and also on a two-leg ladder
where the anyons hop between sites and possibly braid. We compare the result of
the ground state energies of Fibonacci anyons against hardcore bosons and
spinless fermions. In addition, we report the entanglement entropies of the
ground states of interacting Fibonacci anyons on a fully filled two-leg ladder
at different interaction strength, identifying gapped or gapless points in the
parameter space. As an outlook, our approach can also prove useful in studying
the time dynamics of a finite number of nonabelian anyons on a finite
two-dimensional lattice.Comment: Revised version: 20 pages, 14 captioned figures, 2 new tables. We
have moved a significant amount of material concerning symmetric tensors for
anyons --- which can be found in prior works --- to Appendices in order to
streamline our exposition of the modified Anyonic-U(1) ansat
Maximal Entropy Random Walk: solvable cases of dynamics
We focus on the study of dynamics of two kinds of random walk: generic random
walk (GRW) and maximal entropy random walk (MERW) on two model networks: Cayley
trees and ladder graphs. The stationary probability distribution for MERW is
given by the squared components of the eigenvector associated with the largest
eigenvalue \lambda_0 of the adjacency matrix of a graph, while the dynamics of
the probability distribution approaching to the stationary state depends on the
second largest eigenvalue \lambda_1.
Firstly, we give analytic solutions for Cayley trees with arbitrary branching
number, root degree, and number of generations. We determine three regimes of a
tree structure that result in different statics and dynamics of MERW, which are
due to strongly, critically, and weakly branched roots. We show how the
relaxation times, generically shorter for MERW than for GRW, scale with the
graph size.
Secondly, we give numerical results for ladder graphs with symmetric defects.
MERW shows a clear exponential growth of the relaxation time with the size of
defective regions, which indicates trapping of a particle within highly
entropic intact region and its escaping that resembles quantum tunneling
through a potential barrier. GRW shows standard diffusive dependence
irrespective of the defects.Comment: 13 pages, 6 figures, 24th Marian Smoluchowski Symposium on
Statistical Physics (Zakopane, Poland, September 17-22, 2011
Extremal Infinite Graph Theory
We survey various aspects of infinite extremal graph theory and prove several
new results. The lead role play the parameters connectivity and degree. This
includes the end degree. Many open problems are suggested.Comment: 41 pages, 16 figure
Rectangular Layouts and Contact Graphs
Contact graphs of isothetic rectangles unify many concepts from applications
including VLSI and architectural design, computational geometry, and GIS.
Minimizing the area of their corresponding {\em rectangular layouts} is a key
problem. We study the area-optimization problem and show that it is NP-hard to
find a minimum-area rectangular layout of a given contact graph. We present
O(n)-time algorithms that construct -area rectangular layouts for
general contact graphs and -area rectangular layouts for trees.
(For trees, this is an -approximation algorithm.) We also present an
infinite family of graphs (rsp., trees) that require (rsp.,
) area.
We derive these results by presenting a new characterization of graphs that
admit rectangular layouts using the related concept of {\em rectangular duals}.
A corollary to our results relates the class of graphs that admit rectangular
layouts to {\em rectangle of influence drawings}.Comment: 28 pages, 13 figures, 55 references, 1 appendi
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