We investigate the BCS critical temperature $$T_c$$ T c in the high-density limit and derive an asymptotic formula, which strongly depends on the behavior of the interaction potential V on the Fermi-surface. Our results include a rigorous confirmation for the behavior of $$T_c$$ T c at high densities proposed by Langmann et al. (Phys Rev Lett 122:157001, 2019) and identify precise conditions under which superconducting domes arise in BCS theory.

We first review the problem of a rigorous justification of Kubo's formula for transport coefficients in gapped extended Hamiltonian quantum systems at zero temperature. In particular, the theoretical understanding of the quantum Hall effect rests on the validity of Kubo's formula for such systems, a connection that we review briefly as well. We then highlight an approach to linear response theory based on non-equilibrium almost-stationary states (NEASS) and on a corresponding adiabatic theorem for such systems that was recently proposed and worked out by one of us in [51] for interacting fermionic systems on finite lattices. In the second part of our paper we show how to lift the results of [51] to infinite systems by taking a thermodynamic limit.

We show that recent results on adiabatic theory for interacting gapped many-body systems on finite lattices remain valid in the thermodynamic limit. More precisely, we prove a generalized super-adiabatic theorem for the automorphism group describing the infinite volume dynamics on the quasi-local algebra of observables. The key assumption is the existence of a sequence of gapped finite volume Hamiltonians, which generates the same infinite volume dynamics in the thermodynamic limit. Our adiabatic theorem also holds for certain perturbations of gapped ground states that close the spectral gap (so it is also an adiabatic theorem for resonances and, in this sense, “generalized”), and it provides an adiabatic approximation to all orders in the adiabatic parameter (a property often called “super-adiabatic”). In addition to the existing results for finite lattices, we also perform a resummation of the adiabatic expansion and allow for observables that are not strictly local. Finally, as an application, we prove the validity of linear and higher order response theory for our class of perturbations for infinite systems. While we consider the result and its proof as new and interesting in itself, we also lay the foundation for the proof of an adiabatic theorem for systems with a gap only in the bulk, which will be presented in a follow-up article.

We prove a generalised super-adiabatic theorem for extended fermionic systems assuming a spectral gap only in the bulk. More precisely, we assume that the infinite system has a unique ground state and that the corresponding GNS-Hamiltonian has a spectral gap above its eigenvalue zero. Moreover, we show that a similar adiabatic theorem also holds in the bulk of finite systems up to errors that vanish faster than any inverse power of the system size, although the corresponding finite volume Hamiltonians need not have a spectral gap.