Reviews in Mathematical Physics

We revisit the celebrated Peierls–Onsager substitution for weak magnetic fields with no spatial decay conditions. We assume that the non-magnetic $\mathrm{\Gamma }$-periodic Hamiltonian has an isolated spectral band whose Riesz projection has a range which admits a basis generated by $N$ exponentially localized composite Wannier functions. Then we show that the effective magnetic band Hamiltonian is unitarily equivalent to a Hofstadter-like magnetic matrix living in ${[{\ell }^2(\mathrm{\Gamma })]}^N$. In addition, if the magnetic field perturbation is slowly variable in space, then the perturbed spectral island is close (in the Hausdorff distance) to the spectrum of a Weyl quantized minimally coupled symbol. This symbol only depends on $\xi$ and is ${\mathrm{\Gamma }}^{\ast }$-periodic; if $N=1$, the symbol equals the Bloch eigenvalue itself. In particular, this rigorously formulates a result from 1951 by J. M. Luttinger.

We prove an adiabatic theorem for general densities of observables that are sums of local terms in finite systems of interacting fermions, without periodicity assumptions on the Hamiltonian and with error estimates that are uniform in the size of the system. Our result provides an adiabatic expansion to all orders, in particular, also for the initial data that lie in eigenspaces of degenerate eigenvalues. Our proof is based on ideas from [], where Bachmann et al. proved an adiabatic theorem for interacting spin systems.

As one important application of this adiabatic theorem, we provide the first rigorous derivation of the adiabatic response formula for the current density induced by an adiabatic change of the Hamiltonian of a system of interacting fermions in a ground state, with error estimates uniform in the system size. We also discuss the application to quantum Hall systems.

For the massless Nelson model, we provide detailed information about the dependence of the normalized ground states ${\check{\psi }}_{P,\sigma }$ of the fiber single-electron Hamiltonians $H_{P,\sigma }$ on the total momentum $P$ and the infrared cut-off $\sigma$. This information is obtained with the help of the iterative analytic perturbation theory. In particular, we derive bounds of the form