18 research outputs found
Tridiagonal substitution Hamiltonians
We consider a family of discrete Jacobi operators on the one-dimensional
integer lattice with Laplacian and potential terms modulated by a primitive
invertible two-letter substitution. We investigate the spectrum and the
spectral type, the fractal structure and fractal dimensions of the spectrum,
exact dimensionality of the integrated density of states, and the gap
structure. We present a review of previous results, some applications, and open
problems. Our investigation is based largely on the dynamics of trace maps.
This work is an extension of similar results on Schroedinger operators,
although some of the results that we obtain differ qualitatively and
quantitatively from those for the Schoedinger operators. The nontrivialities of
this extension lie in the dynamics of the associated trace map as one attempts
to extend the trace map formalism from the Schroedinger cocycle to the Jacobi
one. In fact, the Jacobi operators considered here are, in a sense, a test
item, as many other models can be attacked via the same techniques, and we
present an extensive discussion on this.Comment: 41 pages, 5 figures, 81 reference
New Anomalous Lieb-Robinson Bounds in Quasi-Periodic XY Chains
We announce and sketch the rigorous proof of a new kind of anomalous (or
sub-ballistic) Lieb-Robinson bound for an isotropic XY chain in a
quasi-periodic transversal magnetic field. By "anomalous", we mean that the
usual effective light cone defined by is replaced by the region
for some . In fact, we can characterize
exactly the values of for which this holds as those exceeding the
upper transport exponent of an appropriate one-body discrete
Schr\"odinger operator. Previous study has produced a good amount of
quantitative information on .
The result is obtained by mapping to free fermions, obtaining good dynamical
bounds on the one-body level by adapting techniques developed by Damanik,
Gorodetski, Tcheremchantsev, and Yessen and then "pulling back" these bounds
through the non-local Jordan-Wigner transformation, following an idea of Hamza,
Sims, and Stolz. To our knowledge, this is the first rigorous derivation of
anomalous many-body transport.
We also explain why our method does not extend to yield anomalous LR bounds
of power-law type if one replaces the quasi-periodic field by a random dimer
field.Comment: 5 pages, to appear in Phys. Rev. Let
On Anomalous Lieb-Robinson Bounds for the Fibonacci XY Chain
We rigorously prove a new kind of anomalous (or sub-ballistic) Lieb-Robinson
bound for the isotropic XY chain with Fibonacci external magnetic field at
arbitrary coupling. It is anomalous in that the usual exponential decay in
is replaced by exponential decay in with . In
fact, we can characterize the values of for which such a bound holds
as those exceeding , the upper transport exponent of the one-body
Fibonacci Hamiltonian. Following the approach of \cite{HSS11}, we relate
Lieb-Robinson bounds to dynamical bounds for the one-body Hamiltonian
corresponding to the XY chain via the Jordan-Wigner transformation; in our case
the one-body Hamiltonian with Fibonacci potential. We can bound its dynamics by
adapting techniques developed in \cite{DT07, DT08, D05, DGY} to our purposes.
We also explain why our method does not extend to yield anomalous
Lieb-Robinson bounds of power-law type for the random dimer model.Comment: 21 pages, Final version to appear in J. Spectr. Theor
Characterizations of Uniform Hyperbolicity and Spectra of CMV Matrices
We provide an elementary proof of the equivalence of various notions of
uniform hyperbolicity for a class of cocycles and
establish a Johnson-type theorem for extended CMV matrices, relating the
spectrum to the points on the unit circle for which the associated Szeg\H{o}
cocycle is not uniformly hyperbolic.Comment: 15 pages; based on the appendices of arXiv:1404.7065, which were
removed in passing from arXiv:1404.7065v1 to arXiv:1404.7065v2, the latter
being the version accepted for publication in IMR