160 research outputs found
Surface Magnetization of Aperiodic Ising Quantum Chains
We study the surface magnetization of aperiodic Ising quantum chains. Using
fermion techniques, exact results are obtained in the critical region for
quasiperiodic sequences generated through an irrational number as well as for
the automatic binary Thue-Morse sequence and its generalizations modulo p. The
surface magnetization exponent keeps its Ising value, beta_s=1/2, for all the
sequences studied. The critical amplitude of the surface magnetization depends
on the strength of the modulation and also on the starting point of the chain
along the aperiodic sequence.Comment: 11 pages, 6 eps-figures, Plain TeX, eps
Using the Hadamard and related transforms for simplifying the spectrum of the quantum baker's map
We rationalize the somewhat surprising efficacy of the Hadamard transform in
simplifying the eigenstates of the quantum baker's map, a paradigmatic model of
quantum chaos. This allows us to construct closely related, but new, transforms
that do significantly better, thus nearly solving for many states of the
quantum baker's map. These new transforms, which combine the standard Fourier
and Hadamard transforms in an interesting manner, are constructed from
eigenvectors of the shift permutation operator that are also simultaneous
eigenvectors of bit-flip (parity) and possess bit-reversal (time-reversal)
symmetry.Comment: Version to appear in J. Phys. A. Added discussions; modified title;
corrected minor error
Remarks on the Spectral Properties of Tight Binding and Kronig-Penney Models with Substitution Sequences
We comment on some recent investigations on the electronic properties of
models associated to the Thue-Morse chain and point out that their conclusions
are in contradiction with rigorously proven theorems and indicate some of the
sources of these misinterpretations. We briefly review and explain the current
status of mathematical results in this field and discuss some conjectures and
open problems.Comment: 15,CPT-94/P.3003,tex,
Doppler Tolerance, Complementary Code Sets and the Generalized Thue-Morse Sequence
We generalize the construction of Doppler-tolerant Golay complementary
waveforms by Pezeshki-Calderbank-Moran-Howard to complementary code sets having
more than two codes. This is accomplished by exploiting number-theoretic
results involving the sum-of-digits function, equal sums of like powers, and a
generalization to more than two symbols of the classical two-symbol
Prouhet-Thue-Morse sequence.Comment: 12 page
Resonant Photonic Quasicrystalline and Aperiodic Structures
We have theoretically studied propagation of exciton-polaritons in
deterministic aperiodic multiple-quantum-well structures, particularly, in the
Fibonacci and Thue-Morse chains. The attention is concentrated on the
structures tuned to the resonant Bragg condition with two-dimensional
quantum-well exciton. The superradiant or photonic-quasicrystal regimes are
realized in these structures depending on the number of the wells. The
developed theory based on the two-wave approximation allows one to describe
analytically the exact transfer-matrix computations for transmittance and
reflectance spectra in the whole frequency range except for a narrow region
near the exciton resonance. In this region the optical spectra and the
exciton-polariton dispersion demonstrate scaling invariance and self-similarity
which can be interpreted in terms of the ``band-edge'' cycle of the trace map,
in the case of Fibonacci structures, and in terms of zero reflection
frequencies, in the case of Thue-Morse structures.Comment: 13 pages, 9 figures, submitted to Phys. Rev.
Spectral Approximation for Quasiperiodic Jacobi Operators
Quasiperiodic Jacobi operators arise as mathematical models of quasicrystals
and in more general studies of structures exhibiting aperiodic order. The
spectra of these self-adjoint operators can be quite exotic, such as Cantor
sets, and their fine properties yield insight into associated dynamical
systems. Quasiperiodic operators can be approximated by periodic ones, the
spectra of which can be computed via two finite dimensional eigenvalue
problems. Since long periods are necessary to get detailed approximations, both
computational efficiency and numerical accuracy become a concern. We describe a
simple method for numerically computing the spectrum of a period- Jacobi
operator in operations, and use it to investigate the spectra of
Schr\"odinger operators with Fibonacci, period doubling, and Thue-Morse
potentials
Similarity density of the Thue-Morse word with overlap-free infinite binary words
We consider a measure of similarity for infinite words that generalizes the
notion of asymptotic or natural density of subsets of natural numbers from
number theory. We show that every overlap-free infinite binary word, other than
the Thue-Morse word t and its complement t bar, has this measure of similarity
with t between 1/4 and 3/4. This is a partial generalization of a classical
1927 result of Mahler.Comment: In Proceedings AFL 2014, arXiv:1405.527
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