相关论文: Bloch and Bethe ansatze for the Harper model: A bu…

Following an approach developed by Gieseker, Kn\"orrer and Trubowitz for discretized Schr\"odinger operators, we study the spectral theory of Harper operators in dimension two and one, as a discretized model of magnetic Laplacians, from the…

We revisit the Hofstadter butterfly for a subset of topologically trivial Bloch bands arising from a continuum free electron Hamiltonian in a periodic lattice potential. We employ the recently developed procedure -- which was previously…

The problem of two-dimensional, independent electrons subject to a periodic potential and a uniform perpendicular magnetic field unveils surprisingly rich physics, as epitomized by the fractal energy spectrum known as Hofstadter's…

A method for finding the exact analytical solutions for the bulk and edge energy levels and corresponding eigenstates for all commensurate Aubry-Andr\'e/Harper single-particle models under open boundary conditions is presented here, both…

While the Bloch spectrum of translationally invariant noninteracting lattice models is trivially obtained by a Fourier transformation, diagonalizing the same problem in the presence of open boundary conditions is typically only possible…

Motivated by recent experimental breakthroughs in realizing hyperbolic lattices in superconducting waveguides and electric circuits, we compute the Hofstadter butterfly on regular hyperbolic tilings. By utilizing large hyperbolic lattices…

We present a generalization of Bloch's theorem to finite-range lattice systems of independent fermions, in which translation symmetry is broken only by arbitrary boundary conditions, by providing exact, analytic expressions for all energy…

We study the topological edge states of the Haldane's graphene model with zigzag/armchair lattice edges. The Harper equation for solving the energies of the edge states is derived. The results show that there are two edge states in the bulk…

We investigate the edge state of a two-dimensional topological insulator based on the Kane-Mele model. Using complex wave numbers of the Bloch wave function, we derive an analytical expression for the edge state localized near the edge of a…

The Hofstadter model, well-known for its fractal butterfly spectrum, describes two-dimensional electrons under a perpendicular magnetic field, which gives rise to the integer quantum hall effect. Inspired by the real-space building blocks…

Two-dimensional electron systems in both magnetic fields and periodic potentials are described by Hofstadter butterfly, a fundamental problem of solid-state physics. While moir\'e systems provide a powerful method to realize this spectrum,…

The trans-series completion of perturbative series of a wide class of quantum mechanical systems can be determined by combining the resurgence program and extra input coming from exact WKB analysis. In this paper, we reexamine the…

Based on a thorough numerical analysis of the spectrum of Harper's operator, which describes, e.g., an electron on a two-dimensional lattice subjected to a magnetic field perpendicular to the lattice plane, we make the following conjecture:…

For a wide class of noninteracting tight-binding models in one dimension we present an analytical solution for all scattering and edge states on a half-infinite system. Without assuming any symmetry constraints we consider models with…

In this chapter, we investigate the energy spectra as well as the bulk and surface states in a two-dimensional system composed of a coupled stack of one-dimensional dimerized chains in the presence of an external magnetic field.…

We consider Bloch electrons on the honeycomb lattice under a uniform magnetic field with $2 \pi p/q$ flux per cell. It is shown that the problem factorizes to two triangular lattices. Treating magnetic translations as Heisenberg-Weyl group…

We study edge states of a random Schroedinger operator for an electron submitted to a magnetic field in a finite macroscopic two dimensional system of linear dimensions equal to L. The y direction is L-periodic and in the x direction the…

We point out that the recent conjectural solution to the spectral problem for the Hamiltonian $H=e^{x}+e^{-x}+e^{p}+e^{-p}$ in terms of the refined topological invariants of a local Calabi-Yau geometry has an intimate relation with…

We analyze the driven Harper model, which appears in the problem of tight-binding electrons in the Hall configuration (normal to the lattice plane magnetic field plus in-plane electric field). The presence of an electric field extends the…

We investigate the electronic properties in the Bloch electron on a square lattice with vacancies in the uniform magnetic field. We show that a single vacancy site introduced to the system creates a defect energy level in every single…