相关论文: Bloch electrons on honeycomb lattice and toric Cal…
We revisit the spectral problem for Bloch electrons in a two-dimensional bipartite honeycomb lattice under a uniform magnetic field. It is well-known that such a honeycomb structure is realized in graphene. We present a systematic framework…
The B-model approach of topological string theory leads to difference equations by quantizing algebraic mirror curves. It is known that these quantum mechanical systems are solved by the refined topological strings. Recently, it was pointed…
We investigate Bloch electrons in two dimensions subject to constant electric and magnetic fields. The model that results from our pursuit is governed by a finite difference equation with a quasienergy spectrum that interpolates between a…
Flux-energy and angle-energy diagrams for an exact three-dimensional Hamiltonian of the Bloch electron in a uniform magnetic field are analyzed. The dependence of the structure of the diagrams on the direction of the field, the geometry of…
Two-dimensional Bloch electrons in a uniform magnetic field exhibit complex energy spectrum. When static electric and magnetic modulations with a checkerboard pattern are superimposed on the uniform magnetic field, more structures and…
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…
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…
In a previous paper, we presented a matrix model reproducing the topological string partition function on an arbitrary given toric Calabi-Yau manifold. Here, we study the spectral curve of our matrix model and thus derive, upon imposing…
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…
The tight-binding model of quantum particles on a honeycomb lattice is investigated in the presence of homogeneous magnetic field. Provided the magnetic flux per unit hexagon is rational of the elementary flux, the one-particle Hamiltonian…
The Bethe ansatz (BA) equations for the two-dimensional Bloch electrons in a uniform magnetic field are treated in the weak field limit. We have calculated energies near the lower boundary of the energy spectrum up to the first nontrivial…
Electron hopping models on the honeycomb lattice are studied. The lattice consists of two triangular sublattices, and it is non-Bravais. The dual space has non-trivial topology. The gauge fields of Bloch electrons have the U(1) symmetry and…
We develop a self-consistent approach to study the spectral properties of a class of quantum mechanical operators by using the knowledge about monodromies of $2\times 2$ linear systems (Riemann-Hilbert correspondence). Our technique applies…
A connection of a variety of tight-binding models of noninteracting electrons on a rectangular lattice in a magnetic field with theta functions is established. A new spectrum generating symmetry is discovered which essentialy reduces the…
We investigate the spectrum and eigenstates of ultracold fermionic atoms in the bilayer honeycomb optical lattice. In the low energy approximation, the dispersion relation has parabolic form and the quasiparticles are chiral. In the…
We propose a solution to the problem of Bloch electrons in a homogeneous magnetic field by including the quantum fluctuations of the photon field. A generalized quantum electrodynamical (QED) Bloch theory from first principles is presented.…
We review the Landau problem of an electron in a constant uniform magnetic field. The magnetic translations are the invariant transformations of the free Hamiltonian. A K\"ahler polarization of the plane has been used for the geometric…
We address the spectral problem of the normal quantum mechanical operator associated to the quantized mirror curve of the toric (almost) del Pezzo Calabi--Yau threefold called local $\mathbb{P}^2$ in the case of complex values of Planck's…
The problem of Bloch electrons in two dimensions subject to magnetic and intense electric fields is investigated. Magnetic translations, electric evolution and energy translation operators are used to specify the solutions of the…
Inspired by recent connections between spectral theory and topological string theory, we propose exact quantization conditions for the relativistic Toda lattice of N particles. These conditions involve the Nekrasov-Shatashvili free energy,…