Superfluidity in Fermi systems within the framework of Density Functional Theory

TL;DR

This review develops and surveys Density Functional Theory for fermionic superfluids, applying the Hohenberg–Kohn framework and its Kohn–Sham extension to include pairing via the anomalous density $\chi$ and the pairing field $\Delta$. It introduces the Superfluid Local Density Approximation (SLDA) and its time-dependent extension (TDSLDA), enabling self-consistent, three-dimensional real-time simulations of nonequilibrium dynamics in nuclei, neutron-star crusts, and ultracold atomic gases. The work discusses memory effects, the adiabatic approximation, and ultraviolet renormalization necessary for local pairing, and surveys applications to nuclear reactions, induced fission, vortex dynamics, neutron-star vortices, and ferron states in spin-imbalanced Fermi gases. Overall, the framework provides a unified, versatile tool for predicting pairing dynamics, collective modes, and dissipation in diverse strongly interacting fermionic systems, with significant implications for nuclear physics, astrophysics, and quantum gases.

Abstract

This review is based on lectures given by the author at the Enrico Fermi Summer School in Varenna. It presents the basics of Density Functional Theory (DFT) for Fermi superfluids, with particular emphasis on nuclear systems. Special attention is given to the foundations of both DFT and time-dependent DFT (TDDFT). The review explores the advantages and challenges involved in the practical application of TDDFT to superfluid systems, as well as the typical approximations employed. Various applications of the TDDFT framework to the description of phenomena related to nonequilibrium superfluidity in atomic nuclei, neutron stars, and ultracold atoms are discussed.

Figures (6)

• Figure 1: Schematic illustration of two fundamental excitation modes in a superfluid system.
• Figure 2: Snapshot from TDSLDA simulation of head-on collision of $^{96}Zr + ^{96}Zr$ at the center-of-mass energy $182$MeV PhysRevC.105.064602. The upper subfigure shows density distributions for protons (upper half) and neutrons (lower half). The lower subfigure shows $|\Delta|$ for protons (upper half) and neutron (lower half). The solitonic structure appearing between colliding nuclei is visible -- $|\Delta|$ is decreased at the contact of two nuclei. The phase difference between nuclear pairing fields is equal to $\Delta\phi=\pi$. Legends on the left correspond to protons, whereas on the right - to neutrons. Magnitudes of $|\Delta|$ are in MeV. Densities are in $fm^{-3}$.
• Figure 3: Schematic picture showing vortex line interacting with nuclear impurity. The line becomes bent due to the interaction and the nucleus becomes deformed.
• Figure 4: In systems with spin imbalance, the Fermi sphere of the minority spin component (in this case, spin-down fermions with Fermi momentum $k_{F\downarrow}$) is displaced by a momentum vector $\bm{q}$ such that it comes into contact with the Fermi surface of the majority (spin-up) component. This momentum mismatch leads to a spatially modulated pairing field. Depending on the structure of this modulation, two distinct phases may emerge: Fulde-Ferrell (FF) phase - $|\Delta|\exp(i\bm{q}\cdot\bm{r})$, Larkin-Ovchinnikov (LO) phase - $|\Delta|\cos(\bm{q}\cdot\bm{r})$.
• Figure 5: Snapshot from TDSLDA simulations PhysRevA.100.033613, illustrating the formation of a stable structure, referred to as a ferron, which emerges following the application of a localized, spin-selective potential to an initially spin-unpolarized Fermi gas. This perturbation induces a characteristic nodal structure in the pairing field (upper left panel), closely resembling the spatial modulation found in the Larkin-Ovchinnikov phase. Notably, the sign of the pairing field reverses at the center relative to the surrounding bulk (upper right panel), while the spin polarization becomes concentrated along the nodal line (lower left panel), contributing to the stability of the ferron configuration.