Response of superfluid fermions at finite temperature

TL;DR

The paper develops a consistent finite-temperature microscopic framework for the response of superfluid fermionic systems, formulated in the Bogoliubov quasiparticle basis with a $4\times4$ matrix structure. It derives the general equation of motion for the thermally averaged two-quasiparticle propagator and decomposes the interaction into static and dynamical kernels, mapping the latter to a finite-temperature quasiparticle-vibration coupling (qPVC) with a new $qPVC$ vertex. The formalism extends to finite temperature via Matsubara techniques, incorporating thermally unblocked amplitudes ${\cal U},{\cal V}$ and detailed balance in the strength functions, and recovers FT-QRPA as a baseline while enabling fully correlated dynamical kernels. This approach bridges the zero-temperature superfluid regime and the normal phase above the critical temperature $T_c$, with broad applicability to nuclear structure and condensed-matter contexts, and sets the stage for numerical implementations of temperature-dependent collective excitations.

Abstract

A consistent finite-temperature microscopic theory for the response of strongly coupled superfluid fermionic systems is formulated. We start from the general many-body Hamiltonian with the vacuum (bare) two-fermion interaction and derive the equation of motion (EOM) for the thermally averaged two-time two-fermion correlation function, which determines the spectrum of the system under study. The superfluidity is introduced via the Bogoliubov transformation of the fermionic field operators, and the entire formalism is carried out in the basis of Bogoliubov's quasiparticles, keeping the complete 4x4 block matrix structure of the two-fermion EOM. Fully correlated static and dynamical interaction kernels of the resulting EOM are discussed. A special focus is then placed on the latter kernel, which is advanced to a factorized form, enabling a minimal truncation of the many-body problem while keeping important effects of emergent collectivity and mapping to the quasiparticle-vibration coupling (qPVC). As in the zero-temperature and non-superfluid cases, the qPVC can be associated with a new order parameter, qPVC vertex. In the thermal superfluid theory, the latter vertex, as well as the components of the dynamical kernel, acquire an extended form including thermally unblocked transition amplitudes.