# Semiclassical asymptotics in magnetic Bloch bands

@article{Dimassi2002SemiclassicalAI, title={Semiclassical asymptotics in magnetic Bloch bands}, author={Mouez Dimassi and Jean-Claude Guillot and James Ralston}, journal={Journal of Physics A}, year={2002}, volume={35}, pages={7597-7605} }

This paper gives a simple construction of wave packets localized near semiclassical trajectories for an electron subject to external electric and magnetic fields. We assume that the magnetic and electric potentials are slowly varying perturbations of the potential of a constant magnetic field and a periodic lattice potential, respectively.

## 33 Citations

### Semiclassical Asymptotics for Weakly Nonlinear Bloch Waves

- Mathematics, PhysicsJournal of Statistical Physics
- 2004

We study the simultaneous semi-classical and adiabatic asymptotics for a class of (weakly) nonlinear Schrödinger equations with a fast periodic potential and a slowly varying confinement potential. A…

### Gaussian Beam Construction for Adiabatic Perturbations

- Physics
- 2006

We construct wave packets concentrated near a single space-time trajectory of the semi-classical Hamiltonian for a Bloch electron in a crystal lattice subject to slowly varying external electric and…

### On Effective Hamiltonians for Adiabatic Perturbations of Magnetic Schrodinger Operators

- Physics, Mathematics
- 2004

We construct almost invariant subspaces and the corresponding effective Hamiltonian for magnetic Bloch bands. We also discuss the question of the dynamics related to the effective Hamiltonian. We…

### Space-adiabatic perturbation theory for Dirac-Bloch electrons

- Physics
- 2006

In this thesis, the quantum mechanical dynamics of an electron in a crystal as governed by the Dirac equation is studied. The electron is subject to the periodic potential generated by the crystal…

### Motion of Electrons in Adiabatically Perturbed Periodic Structures

- Physics
- 2006

We study the motion of electrons in a periodic background potential (usually resulting from a crystalline solid). For small velocities one would use either the non-magnetic or the magnetic Bloch…

### Resonances for Perturbed Periodic Schrödinger Operator

- Mathematics, Physics
- 2012

In the semiclassical regime, we obtain a lower bound for the counting function of resonances corresponding to the perturbed periodic Schrodinger operator . Here is a periodic potential, a decreasing…

### Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice II. impurities, confinement and Bloch oscillations

- Physics, Mathematics
- 2004

### Asymptotic analysis of quantum dynamics in crystals: the Bloch-Wigner transform, Bloch dynamics and Berry phase

- Physics, Mathematics
- 2013

We study the semi-classical limit of the Schrödinger equation in a crystal in the presence of an external potential and magnetic field. We first introduce the Bloch-Wigner transform and derive the…

### Effective Dynamics for Bloch Electrons: Peierls Substitution and Beyond

- Physics
- 2002

AbstractWe consider an electron moving in a periodic potential and subject to an additional slowly varying external electrostatic potential, φ(ɛx), and vector potential A(ɛx), with xℝd and ɛ≪1. We…

### Resonances for Perturbed Periodic Schrödinger Operator Mouez Dimassi

- Mathematics, Physics
- 2014

Here V is periodic with respect to the crystal lattice Γ ⊂ R, and it models the electric potential generated by the lattice of atoms in the crystal. The potentialW is a decreasing perturbation and h…

## References

SHOWING 1-10 OF 15 REFERENCES

### Semi-classical asymptotics in solid state physics

- Physics, Mathematics
- 1988

This article studies the Schrödinger equation for an electron in a lattice of ions with an external magnetic field. In a suitable physical scaling the ionic potential becomes rapidly oscillating, and…

### Berry's Phase of Bloch Electrons in Electromagnetic Fields

- Physics
- 1993

Berry recognized in quantum mechanics a topological phase factor arising from the adiabatic transport of a system around a closed loop, which is essentially the Aharonov-Bohm effect in parameter…

### Berry phase, hyperorbits, and the Hofstadter spectrum.

- PhysicsPhysical review letters
- 1995

A semiclassical theory for the dynamics of electrons in a magnetic Bloch band, where the Berry phase plays an important role is developed, and an Onsager-like formula for the quantization of cyclotron orbits is derived.

### Berry phase, hyperorbits, and the Hofstadter spectrum: Semiclassical dynamics in magnetic Bloch bands.

- PhysicsPhysical review. B, Condensed matter
- 1996

Based on a set of semiclassical equations for electrons in magnetic Bloch bands, the pattern of band splitting, the distribution of Hall conductivities, and the positions of energy subbands in the Hofstadter spectrum can be understood in a simple and unified picture.

### Dynamics of Electrons in Solids in External Fields

- Physics
- 1968

A quantum-mechanical representation is defined by means of finite translations in direct and reciprocal space. The eigenfunctions of the finite translations are found and their connection with Bloch…

### Semi-classical constructions in solid state physics

- Physics
- 1991

The problem of the intraband magnetic breakthrough is investndeshgared by scaling methods. Approximate eigenfunctions are constructed in the neightborhood of a saddle point of the Fermi surface. This…

### Semiclassical approximation for equations with periodic coefficients

- Philosophy
- 1987

CONTENTS § 1. Introduction § 2. The effective Hamiltonian § 3. Turning points § 4. Comments and bibliographical indications References

### A mathematical approach to the effective Hamiltonian in perturbed periodic problems

- Mathematics
- 1991

We describe a rigorous mathematical reduction of the spectral study for a class of periodic problems with perturbations which gives a justification of the method of effective Hamiltonians in solid…

### Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase

- Physics
- 1983

It is shown that the "geometrical phase factor" recently found by Berry in his study of the quantum adiabatic theorem is precisely the holonomy in a Hermitian line bundle since the adiabatic theorem…