02-516 Gianluca Panati, Herbert Spohn, Stefan Teufel

Effective dynamics for Bloch electrons: Peierls substitution and beyond (115K, Latex 2e) Dec 12, 02

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Abstract. We reconsider the longstanding problem of an electron moving in a crystal under the influence of weak external electromagnetic fields. More precisely we analyze the dynamics generated by the Schr\"odinger operator $H = \frac{1}{2} ( -\I\nabla_x - A(\varepsilon x) )^2 + V(x) + \phi(\varepsilon x)$, where $V$ is a lattice periodic potential and $A$ and $\phi$ are external potentials which vary slowly on the scale set by the lattice spacing. We study the limit $\varepsilon\to 0$ in several steps: (i) Approximately invariant subspaces associated with isolated Bloch bands are constructed. (ii) We derive an effective quantum Hamiltonian for states inside such a decoupled subspace. The effective Hamiltonian has an asymptotic expansion in $\varepsilon$, starting with the term given through the Peierls substitution. Our construction allows, in principle, to compute also all higher order terms and we give the first order correction to the Peierls substitution explicitly. (iii) The semiclassical limit of the effective Hamiltonian yields the first order corrections to the ``semiclassical model'' of solid states physics, including a new term which has been missed in earlier heuristic studies.

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