48,158 research outputs found
Localization of Two-Dimensional Quantum Walks
The Grover walk, which is related to the Grover's search algorithm on a
quantum computer, is one of the typical discrete time quantum walks. However, a
localization of the two-dimensional Grover walk starting from a fixed point is
striking different from other types of quantum walks. The present paper
explains the reason why the walker who moves according to the degree-four
Grover's operator can remain at the starting point with a high probability. It
is shown that the key factor for the localization is due to the degeneration of
eigenvalues of the time evolution operator. In fact, the global time evolution
of the quantum walk on a large lattice is mainly determined by the degree of
degeneration. The dependence of the localization on the initial state is also
considered by calculating the wave function analytically.Comment: 21 pages RevTeX, 4 figures ep
Linear semigroups with coarsely dense orbits
Let be a finitely generated abelian semigroup of invertible linear
operators on a finite dimensional real or complex vector space . We show
that every coarsely dense orbit of is actually dense in . More
generally, if the orbit contains a coarsely dense subset of some open cone
in then the closure of the orbit contains the closure of . In the
complex case the orbit is then actually dense in . For the real case we give
precise information about the possible cases for the closure of the orbit.Comment: We added comments and remarks at various places. 14 page
Topologically Driven Swelling of a Polymer Loop
Numerical studies of the average size of trivially knotted polymer loops with
no excluded volume are undertaken. Topology is identified by Alexander and
Vassiliev degree 2 invariants. Probability of a trivial knot, average gyration
radius, and probability density distributions as functions of gyration radius
are generated for loops of up to N=3000 segments. Gyration radii of trivially
knotted loops are found to follow a power law similar to that of self avoiding
walks consistent with earlier theoretical predictions.Comment: 6 pages, 4 figures, submitted to PNAS (USA) in Feb 200
Finite Cluster Typical Medium Theory for Disordered Electronic Systems
We use the recently developed typical medium dynamical cluster (TMDCA)
approach~[Ekuma \etal,~\textit{Phys. Rev. B \textbf{89}, 081107 (2014)}] to
perform a detailed study of the Anderson localization transition in three
dimensions for the Box, Gaussian, Lorentzian, and Binary disorder
distributions, and benchmark them with exact numerical results. Utilizing the
nonlocal hybridization function and the momentum resolved typical spectra to
characterize the localization transition in three dimensions, we demonstrate
the importance of both spatial correlations and a typical environment for the
proper characterization of the localization transition in all the disorder
distributions studied. As a function of increasing cluster size, the TMDCA
systematically recovers the re-entrance behavior of the mobility edge for
disorder distributions with finite variance, obtaining the correct critical
disorder strengths, and shows that the order parameter critical exponent for
the Anderson localization transition is universal. The TMDCA is computationally
efficient, requiring only a small cluster to obtain qualitative and
quantitative data in good agreement with numerical exact results at a fraction
of the computational cost. Our results demonstrate that the TMDCA provides a
consistent and systematic description of the Anderson localization transition.Comment: 20 Pages, 19 Figures, 3 Table
Elliptic Genera and 3d Gravity
We describe general constraints on the elliptic genus of a 2d supersymmetric
conformal field theory which has a gravity dual with large radius in Planck
units. We give examples of theories which do and do not satisfy the bounds we
derive, by describing the elliptic genera of symmetric product orbifolds of
, product manifolds, certain simple families of Calabi-Yau hypersurfaces,
and symmetric products of the "Monster CFT." We discuss the distinction between
theories with supergravity duals and those whose duals have strings at the
scale set by the AdS curvature. Under natural assumptions we attempt to
quantify the fraction of (2,2) supersymmetric conformal theories which admit a
weakly curved gravity description, at large central charge.Comment: 50 pages, 9 figures, v2: minor corrections to section
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