Periodic Reporting for period 1 - SCP-Disorder (Disordered and strongly-correlated systems: a new theoretical approach)
Reporting period: 2019-05-01 to 2021-04-30
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.
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.
Objective 1: This objective focused on solving the KS-SCP equations with the spectral renormalisation method. The non-linear KS-SCP equations turned out to be more difficult to converge with respect to our preliminary attempts done before the fellowship. However, we were ultimately able to complete this objective, in particular to converge the very challenging second-order (ZP) case. The results are published in
Kohn-Sham equations with functionals from the strictly-correlated regime: Investigation with a spectral renormalization method, J. Grossi, Z. H. Musslimani, M. Seidl, P. Gori-Giorgi,
J. Phys.: Condens. Matter, 32 475602 (2020) arXiv:2004.10436
Objective 2: This objective aimed at implementing a time evolution algorithm for the KS SCP method. Preliminary work has been done, but it turned out to be rather challenging numerically. The project is being presently continued in a collaboration between the host institution and Florida State University.
Objective 3: The study of disordered systems with the KS SCP method is almost complete. We have several results and we are collecting them into one ore probably two publications. We have carried out both exact calculations in 1D for up to 4 particles (in collaboration with K.J.H Giesbertz) in presence of disorder. We have then tested several different flavors of approximate DFT, namely Hartree and Hartree with the local density approximation (LDA). We have then also implemented the KS SCP method, which was the only one giving good agreement with the exact results (Hartree and LDA were found to completely delocalize the system, a manifestation of the self-interaction error). Publication is in preparation.
Objective 4: This objective was focusing on using the spectral renormalisation method to compute self-consistent results with approximations to KS SCP. As a preliminary step, we have considered a variant of the strong-coupling limit in the framework of the Møller-Plesset adiabatic connection. The spectral renormalisation method proved very successful in finding the exact solution for the special case of the H atom with fractional spins. During this work the fellow co-supervised and instructed the PhD student Tim Daas. The results are published in
T. J. Daas, J. Grossi, S. Vuckovic, Z. H. Musslimani, D. P. Kooi, M. Seidl, K. J. H. Giesbertz, P. Gori-Giorgi, Large coupling-strength expansion of the Møller-Plesset adiabatic connection: From paradigmatic cases to variational expressions for the leading terms, J. Chem. Phys. 153, 214112 (2020) arXiv:2009.04326
Kohn-Sham equations with functionals from the strictly-correlated regime: Investigation with a spectral renormalization method, J. Grossi, Z. H. Musslimani, M. Seidl, P. Gori-Giorgi,
J. Phys.: Condens. Matter, 32 475602 (2020) arXiv:2004.10436
Objective 2: This objective aimed at implementing a time evolution algorithm for the KS SCP method. Preliminary work has been done, but it turned out to be rather challenging numerically. The project is being presently continued in a collaboration between the host institution and Florida State University.
Objective 3: The study of disordered systems with the KS SCP method is almost complete. We have several results and we are collecting them into one ore probably two publications. We have carried out both exact calculations in 1D for up to 4 particles (in collaboration with K.J.H Giesbertz) in presence of disorder. We have then tested several different flavors of approximate DFT, namely Hartree and Hartree with the local density approximation (LDA). We have then also implemented the KS SCP method, which was the only one giving good agreement with the exact results (Hartree and LDA were found to completely delocalize the system, a manifestation of the self-interaction error). Publication is in preparation.
Objective 4: This objective was focusing on using the spectral renormalisation method to compute self-consistent results with approximations to KS SCP. As a preliminary step, we have considered a variant of the strong-coupling limit in the framework of the Møller-Plesset adiabatic connection. The spectral renormalisation method proved very successful in finding the exact solution for the special case of the H atom with fractional spins. During this work the fellow co-supervised and instructed the PhD student Tim Daas. The results are published in
T. J. Daas, J. Grossi, S. Vuckovic, Z. H. Musslimani, D. P. Kooi, M. Seidl, K. J. H. Giesbertz, P. Gori-Giorgi, Large coupling-strength expansion of the Møller-Plesset adiabatic connection: From paradigmatic cases to variational expressions for the leading terms, J. Chem. Phys. 153, 214112 (2020) arXiv:2009.04326
An increasingly important part of chemistry, solid-state physics, and materials science takes place at the computer, allowing for new understanding, a reduction of experimental trial-and-error procedures, and energy and money savings. This requires highly accurate reproduction of phenomena at different scales. The quantum scale (electrons) is by far the most challenging but also the most crucial in many cases (for example, the breaking and making of chemical bonds is a quantum phenomenon). The approach to quantum many particle systems that is being developed in the host institution is very promising, but faces several numerical challenges. In this project we have addressed some of these numerical challenges and we have been able to overcome some of them. We have also performed a the study of systems in presence of disorder within the new method, which opens several future interesting applications.