[Paper Review] Applications of Magnetic PsiDO Techniques to Space-adiabatic Perturbation Theory

Giuseppe De Nittis, Max Lein|HAL (Le Centre pour la Communication Scientifique Directe)|Jun 15, 2010

Topological Materials and Phenomena19 references18 citations

TL;DR

This paper extends space-adiabatic perturbation theory (SAPT) to systems with weak, slowly varying electromagnetic fields by incorporating magnetic pseudodifferential operator (magnetic ΨDO) techniques. It proves that even under constant magnetic fields, the effective dynamics of a single particle in a periodic potential are governed by a modified Peierls substitution Hamiltonian, with corrections up to any order in the semiclassical parameter ε, preserving the almost-invariant subspace structure of Bloch bands and enabling accurate semiclassical approximation of quantum evolution up to O(ε²).

ABSTRACT

In this review, we show how advances in the theory of magnetic pseudodifferential operators (magnetic $Ψ$DO) can be put to good use in space-adiabatic perturbation theory (SAPT). As a particular example, we extend results of [PST03] to a more general class of magnetic fields: we consider a single particle moving in a periodic potential which is subjectd to a weak and slowly-varying electromagnetic field. In addition to the semiclassical parameter $\eps \ll 1$ which quantifies the separation of spatial scales, we explore the influence of additional parameters that allow us to selectively switch off the magnetic field. We find that even in the case of magnetic fields with components in $C_b^{\infty}(\R^d)$, e. g. for constant magnetic fields, the results of Panati, Spohn and Teufel hold, i.e. to each isolated family of Bloch bands, there exists an associated almost invariant subspace of $L^2(\R^d)$ and an effective hamiltonian which generates the dynamics within this almost invariant subspace. In case of an isolated non-degenerate Bloch band, the full quantum dynamics can be approximated by the hamiltonian flow associated to the semiclassical equations of motion found in [PST03].

Motivation & Objective

• To extend space-adiabatic perturbation theory (SAPT) to systems with weak, slowly varying electromagnetic fields, particularly constant magnetic fields.
• To address the breakdown of Bloch band structure under constant magnetic fields, where standard SAPT fails due to lack of periodicity.
• To establish the existence of an almost-invariant subspace and an effective Hamiltonian for isolated, non-degenerate Bloch bands under magnetic fields.
• To derive a systematic expansion of the effective dynamics in powers of the semiclassical parameter ε, including corrections from the magnetic field.
• To prove that the full quantum evolution can be approximated by the effective Hamiltonian flow up to O(ε²), even in the presence of a constant magnetic field.

Proposed method

• Introduces a double-scale parameterization with ε ≪ 1 (spatial adiabatic scale) and λ (magnetic field strength), allowing selective switching of the magnetic field.
• Applies magnetic Weyl calculus and magnetic pseudodifferential operators (magnetic ΨDO) to handle magnetic fields with components in C∞b(Rd), including constant fields.
• Uses the Bloch-Floquet-Zak (BFZ) transform to decompose the Hamiltonian into fibered operators over the Brillouin zone M∗.
• Derives an effective Hamiltonian heff via the Peierls substitution, defined as a magnetic pseudodifferential operator, which generates dynamics within the almost-invariant subspace.
• Establishes an Egorov-type theorem showing that the quantum evolution of macroscopic observables is approximated by the classical flow of heff up to O(ε²).
• Combines the Egorov theorem with a unitary transformation and asymptotic expansion to relate the full quantum evolution to the effective dynamics, proving the main approximation result.

Experimental results

Research questions

• RQ1 Can space-adiabatic perturbation theory be extended to systems with constant magnetic fields, where Bloch band structure is destroyed?
• RQ2 Does the Peierls substitution remain valid as an effective Hamiltonian in the presence of a weak, constant magnetic field?
• RQ3 Can the full quantum dynamics be approximated by the flow of an effective Hamiltonian up to O(ε²) error, even when the magnetic field is constant?
• RQ4 How does the magnetic field modify the symplectic structure of the effective dynamics, and what role does the pseudomagnetic field Ω play?
• RQ5 Is the effective dynamics still governed by a non-commutative Poisson bracket structure when the magnetic field is included?

Key findings

• Even for constant magnetic fields with components in C∞b(Rd), an almost-invariant subspace exists for each isolated, non-degenerate Bloch band, preserving the core structure of SAPT.
• The effective Hamiltonian is given by the Peierls substitution: heff = E∗(−i∇x − λA(εx)) + φ(εx), defined as a magnetic pseudodifferential operator.
• The effective dynamics are governed by a modified symplectic form (5.8) that includes contributions from the magnetic field B and a quantum pseudomagnetic field Ω, leading to non-commuting position coordinates: {reffl, reffj}λB,εΩ = −εΩlj.
• The quantum evolution of macroscopic observables is approximated by the effective Hamiltonian flow up to O(ε²), as proven by an Egorov-type theorem (Theorem 5.1).
• The full quantum evolution, when restricted to the almost-invariant subspace, is approximated by the macroscopic flow Φmacro_t up to O(ε²), as stated in Theorem 5.2.
• The method is robust to general magnetic fields (not requiring smooth vector potentials), relying only on the magnetic field components B ∈ C∞b(Rd), which allows treatment of constant and slowly varying fields.