31,924 research outputs found
Fixed Point and Aperiodic Tilings
An aperiodic tile set was first constructed by R.Berger while proving the
undecidability of the domino problem. It turned out that aperiodic tile sets
appear in many topics ranging from logic (the Entscheidungsproblem) to physics
(quasicrystals) We present a new construction of an aperiodic tile set that is
based on Kleene's fixed-point construction instead of geometric arguments. This
construction is similar to J. von Neumann self-reproducing automata; similar
ideas were also used by P. Gacs in the context of error-correcting
computations. The flexibility of this construction allows us to construct a
"robust" aperiodic tile set that does not have periodic (or close to periodic)
tilings even if we allow some (sparse enough) tiling errors. This property was
not known for any of the existing aperiodic tile sets.Comment: v5: technical revision (positions of figures are shifted
Complex patterns on the plane: different types of basin fractalization in a two-dimensional mapping
Basins generated by a noninvertible mapping formed by two symmetrically
coupled logistic maps are studied when the only parameter \lambda of the system
is modified. Complex patterns on the plane are visualised as a consequence of
basins' bifurcations. According to the already established nomenclature in the
literature, we present the relevant phenomenology organised in different
scenarios: fractal islands disaggregation, finite disaggregation, infinitely
disconnected basin, infinitely many converging sequences of lakes, countable
self-similar disaggregation and sharp fractal boundary. By use of critical
curves, we determine the influence of zones with different number of first rank
preimages in the mechanisms of basin fractalization.Comment: 19 pages, 11 figure
Exact Spectral Form Factor in a Minimal Model of Many-Body Quantum Chaos
The most general and versatile defining feature of quantum chaotic systems is
that they possess an energy spectrum with correlations universally described by
random matrix theory (RMT). This feature can be exhibited by systems with a
well defined classical limit as well as by systems with no classical
correspondence, such as locally interacting spins or fermions. Despite great
phenomenological success, a general mechanism explaining the emergence of RMT
without reference to semiclassical concepts is still missing. Here we provide
the example of a quantum many-body system with no semiclassical limit (no large
parameter) where the emergence of RMT spectral correlations is proven exactly.
Specifically, we consider a periodically driven Ising model and write the
Fourier transform of spectral density's two-point function, the spectral form
factor, in terms of a partition function of a two-dimensional classical Ising
model featuring a space-time duality. We show that the self-dual cases provide
a minimal model of many-body quantum chaos, where the spectral form factor is
demonstrated to match RMT for all values of the integer time variable in
the thermodynamic limit. In particular, we rigorously prove RMT form factor for
odd , while we formulate a precise conjecture for even . The results
imply ergodicity for any finite amount of disorder in the longitudinal field,
rigorously excluding the possibility of many-body localization. Our method
provides a novel route for obtaining exact nonperturbative results in
non-integrable systems.Comment: 6 + 22 pages, 3 figures; v2: improved presentation of the proofs in
the appendices; v3: as appears in Physical Review Letter
Recent advances in open billiards with some open problems
Much recent interest has focused on "open" dynamical systems, in which a
classical map or flow is considered only until the trajectory reaches a "hole",
at which the dynamics is no longer considered. Here we consider questions
pertaining to the survival probability as a function of time, given an initial
measure on phase space. We focus on the case of billiard dynamics, namely that
of a point particle moving with constant velocity except for mirror-like
reflections at the boundary, and give a number of recent results, physical
applications and open problems.Comment: 16 pages, 1 figure in six parts. To appear in Frontiers in the study
of chaotic dynamical systems with open problems (Ed. Z. Elhadj and J. C.
Sprott, World Scientific
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