Capturing the effects of the interactions between the particles of a quantum system in an accurate and efficient way is of crucial importance in many areas of Physics and Chemistry. Particle-particle interactions not only determine many of the fundamental physical properties of the system under study, but also play a crucial role regarding practical applications in fields ranging from materials sciences to theoretical and computational chemistry, quantum information, spintronics, ultracold chemistry, just to name a few.
Computationally efficient methods that treat the quantum many-body interactions in real space (i.e. without resorting to model hamiltonians) are mainly based on effective single-particle equations (Gross-Pitaevskii for bosons, Hartree-Fock and Kohn-Sham Density Functional Theory for fermions, etc), and rely on an ansatz for transforming the many-body interactions into an effective single-particle potential. Usually, these effective single-particle methods fail when the effects of the interactions become relatively more important than the kinetic energy. In the recent years, the host scientist and coworkers have introduced a new formalism to treat strongly-interacting many-particle quantum systems within the framework of Kohn-Sham (KS) Density Functional Theory (DFT). This “strictly-correlated-particles” (SCP) approach has shown the promise to treat both bosons and fermions, by simply changing the kinetic energy functional,4 and has been shown to be able to capture strong correlation effects (in systems with long-range interactions) that are usually regarded as beyond mean field.This new KS SCP method requires the self-consistent solution of a set of single-particle Schroedinger equations whose non-linear term (the effective potential, which depends on the particle density) has a totally different form with respect to other traditional (e.g. Gross-Pitaevskii, Hartree-Fock or KS DFT with semi-local functionals) single-particle methods, with an extreme non-locality. This special nonlinearity poses several numerical challenges, for example convergence problems in the strong-correlation regime with many particles even in the one-dimensional case.
In this project we combine the fellow's expertise in numerical methods for nonlinear problems, in real-time evolution algorithms, and in qauntum systems in the presence of disorder, with the host expertise in DFT for strongly-interacting systems with long-range interactions, to reach four main objectives:
Objective 1 – We extend an algorithm proposed by the fellow and used for the Gross-Pitaevskii problem, called spectral renormalization method, to the Kohn-Sham Strictly-Correlated-Particles (KS SCP) self-consistent problem.
Objective 2 –We will use a time evolution algorithm to perform real time evolution with the one-dimensional KS SCP equations, in the framework of time-dependent DFT.
Objective 3 – As a first interesting application of Objectives 1 and 2, we study Anderson localization in one-dimensional systems with long-range interactions at different correlation regimes. We will study several interesting features of the problem, for example the influence of the kind of particle-particle interaction (e.g. dipolar with different strenghts), and of the correlation regime, by using both the novel SCP functional and standard DFT functionals like the local density approximation (LDA).
Objective 4 –We will test approximations inspired to the SCP mathematical form, which can be easily adapted to different interactions and dimensionalities, comparing it to the SCP functional results.We will also investigate the effects of the leading correction to SCP.
