Related papers: The Spectrum of the Almost Mathieu Operator
In this paper we use results on reducibility, localization and duality for the Almost Mathieu operator, \[ (H_{b,\phi} x)_n= x_{n+1} +x_{n-1} + b \cos(2 \pi n \omega + \phi)x_n \] on $l^2(\mathbb{Z})$ and its associated eigenvalue equation…
We prove the conjecture (known as the ``Ten Martini Problem'' after Kac and Simon) that the spectrum of the almost Mathieu operator is a Cantor set for all non-zero values of the coupling and all irrational frequencies.
We develop a quantitative version of Aubry duality and use it to obtain several sharp estimates for the dynamics of Schr\"odinger cocycles associated to a non-perturbatively small analytic potential and Diophantine frequency. In particular,…
It is known that the spectral type of the almost Mathieu operator depends in a fundamental way on both the strength of the coupling constant and the arithmetic properties of the frequency. We study the competition between those factors and…
It is shown that for any irrational rotation number and any admissible gap labelling number the almost Mathieu operator (also known as Harper's operator) has a gap in its spectrum with that labelling number. This answers the strong version…
We solve the Dry Ten Martini Problem for the unitary almost Mathieu operator with Diophantine frequencies in the non-critical regime.
We apply recently developed methods for the construction of quasi-periodic transfer matrices to the Dry Ten Martini problem for the critical almost-Mathieu Operator, also known as the Aubry-Andre-Harper (AAH) model.
We introduce a unitary almost-Mathieu operator, which is obtained from a two-dimensional quantum walk in a uniform magnetic field. We exhibit a version of Aubry--Andr\'{e} duality for this model, which partitions the parameter space into…
For the almost Mathieu operator with a small coupling constant, for a series of spectral gaps, we describe the asymptotic locations of the gaps and get lower bounds for their lengths. The results are obtained by analysing a monodromy…
I review a recent progress towards solution of the Almost Mathieu equation (A.G. Abanov, J.C. Talstra, P.B. Wiegmann, Nucl. Phys. B 525, 571, 1998), known also as Harper's equation or Azbel-Hofstadter problem. The spectrum of this equation…
We study how the Mott metal-insulator transition (MIT) is affected when we have to deal with electrons with different angular momentum quantum numbers. For that purpose we apply ab-initio quantum-chemical methods to lithium rings in order…
Recent advances in many-body physics have made it possible to study correlated electron systems at the two-particle level. In Dynamical Mean-Field theory, it has been shown that the metal-insulator phase diagram is closely related to the…
Experimental results on the metal-insulator transition and related phenomena in strongly interacting two-dimensional electron systems are discussed. Special attention is given to recent results for the strongly enhanced spin susceptibility,…
This paper establishes several sharp spectral results for analytic quasiperiodic Schrodinger operators. Key contributions include: (1) exact exponential decay rates for spectral gaps of the almost Mathieu operator, addressing a question…
We reexamine the problem of delocalization of two-dimensional electrons in the presence of random magnetic field. By introducing spatial correlations among random fluxes, a well-defined metal-insulator transition characterized by a…
We prove that the spectrum of the almost Mathieu operator is absolutely continuous if and only if the coupling is subcritical. This settles Problem 6 of Barry Simon's list of Schr\"odinger operator problems for the twenty-first century.
The lecture introduces a reader to the relatively young field of physics of strongly interacting and disordered 2D electron system, in particular, to the phenomena of the metallic conduction and the apparent metal-insulator transition in…
A high-resolution investigation of the electron spectra close to the metal-to-insulator transition in dynamic mean-field theory is presented. An all-numerical, consistent confirmation of a smooth transition at zero temperature is provided.…
In this overview we provide a general introduction to metal-insulator transitions, with focus on specific mechanisms that can localize the electrons in absence of magnetic or charge ordering, and produce well defined quantum critical…
Experimental results on the metal-insulator transitions and the anomalous properties of strongly interacting two-dimensional electron systems are reviewed and critically analyzed. Special attention is given to recent results for the…