Related papers: The Spectrum of the Almost Mathieu Operator
In this paper we estimate the rest of the approximation of a stationary process by a martingale in terms of the projections of partial sums. Then, based on this estimate, we obtain almost sure approximation of partial sums by a martingale…
We consider quantum wave propagation in one-dimensional quasiperiodic lattices. We propose an iterative construction of quasiperiodic potentials from sequences of potentials with increasing spatial period. At each finite iteration step the…
These are lecture notes compiled for a short lecture series at the 2023 Condensed Matter Summer School at the University of Minnesota. They are designed to be conversational and fun, and not to take the place of review articles that do a…
We discuss a recent mapping of the Anderson-Mott metal-insulator transition onto a random field magnet problem. The most important new idea introduced is to describe the metal-insulator transition in terms of an order parameter expansion…
A two-dimensional gas of non-interacting quasiparticles in a nearly periodic potential is considered at zero temperature. The potential is a superposition of a periodic potential, induced by the charge density wave of a Wigner crystal, and…
This script is based on the notes the author prepared to give a set of six lectures at the Les Houches School "Integrability in Atomic and Condensed Matter Physics" in the summer of 2018. The school had its focus on the application of…
We have studied numerically the spectral correlations in a metallic phase and at the metal-insulator transition. We have calculated directly the two-point correlation function of the density of states $R(s,s')$. In the metallic phase, it is…
We present a model for the metal-insulator transition in 2D, observed in the recent years. Our starting point consists of two ingredients only, which are ubiquitous in the experiments: Coulomb interactions and weak disorder spin-orbit…
We solve the Dry Ten Martini Problem in the non-critical case, i.e., all possible spectral gaps are open for almost Mathieu operators with $\lambda\ne \pm 1$.
Thermodynamic and dynamical properties of filling-control metal-insulator transition (MIT) in the Hubbard model are studied by the operator projection method, especially in two dimensions. This is a non-perturbative analytic approach to…
Landau suggested that the low-temperature properties of metals can be understood in terms of long-lived quasiparticles with all complex interactions included in Fermi-liquid parameters, such as the effective mass $m^{\star}$. Despite its…
We study the unitary almost Mathieu operator (UAMO), a one-dimensional quasi-periodic unitary operator arising from a two-dimensional discrete-time quantum walk on $\mathbb Z^2$ in a homogeneous magnetic field. In the positive Lyapunov…
The goal of this paper is to highlight several issues which are most crucial for the understanding of the ``metal-insulator transition'' in two dimensions. We discuss some common problems in interpreting experimental results on high…
The type IIA string theory on a non-compact Calabi-Yau geometry known as the local $\mathbb{P}^{1} \times \mathbb{P}^{1}$ gives rise to five-dimensional N =1 supersymmetric SU(2) gauge theory compactified on a circle, known as geometric…
Kicked Harper operators and on-resonance double kicked rotor operators model quantum systems whose semiclassical limits exhibit chaotic dynamics. Recent computational studies indicate a striking resemblance between the spectrums of these…
We show that the integrated density of states of the almost Mathieu operator is absolutely continuous if and only if the coupling is non-critical. We deduce for subcritical coupling that the spectrum is purely absolutely continuous for…
We study the phase transion line of the almost Mathieu operator, that separates arithmetic regions corresponding to singular continuous and a.e. pure point regimes, and prove that both purely singular continuous and a.e. pure point spectrum…
We have studied corrections to conductivity due to the coherent backscattering in low-disordered two-dimensional electron systems in silicon for a range of electron densities including the vicinity of the metal-insulator transition, where…
It is shown that recent experiments indicating a metal-insulator transition in 2D electron systems can be interpreted in terms of a simple model, in which the resistivity is controlled by scattering at charged hole traps located in the…
In this article, we study quasi-isospectral operators as a generalization of isospectral operators. The paper contains both expository material and original results. We begin by reviewing known results on isospectral potentials on compact…