Homogeneous electron gas in arbitrary dimensions

The homogeneous electron gas is one of the most studied model systems in condensed matter physics. It is also at the basis of the large majority of approximations to the functionals of density functional theory. As such, its exchange-correlation energy has been extensively studied, and is well-known for systems of 1, 2, and 3 dimensions. Here, we extend this model and compute the exchange and correlation energy, as a function of the Wigner-Seitz radius $r_s$, for arbitrary dimension $D$. We find a very different behavior for reduced dimensional spaces ($D=1$ and 2), our three dimensional space, and for higher dimensions. In fact, for $D > 3$, the leading term of the correlation energy does not depend on the logarithm of $r_s$ (as for $D=3$), but instead scales polynomialy: $ -c_D /r_s^{gamma_D}$, with the exponent $gamma_D=(D-3)/(D-1)$. In the large-$D$ limit, the value of $c_D$ is found to depend linearly with the dimension. In this limit, we also find that the concepts of exchange and correlation merge, sharing a common $1/r_s$ dependence.

Reduced Density Matrix Functional Theory for Bosons

Based on a generalization of Hohenberg-Kohns theorem, we propose a ground state theory for bosonic quantum systems. Since it involves the one-particle reduced density matrix $gamma$ as a natural variable but still recovers quantum correlations in an exact way it is particularly well-suited for the accurate description of Bose-Einstein condensates. As a proof of principle we study the building block of optical lattices. The solution of the underlying $v$-representability problem is found and its peculiar form identifies the constrained search formalism as the ideal starting point for constructing accurate functional approximations: The exact functionals for this $N$-boson Hubbard dimer and general Bogoliubov-approximated systems are determined. The respective gradient forces are found to diverge in the regime of Bose-Einstein condensation, $ abla_{gamma} mathcal{F} propto 1/sqrt{1-N_{mathrm{BEC}}/N}$, providing a natural explanation for the absence of complete BEC in nature.

Implications of pinned occupation numbers for natural orbital expansions. I: Generalizing the concept of active spaces

The concept of active spaces simplifies the description of interacting quantum many-body systems by restricting to a neighbourhood of active orbitals around the Fermi level. The respective wavefunction ansatzes which involve all possible electron configurations of active orbitals can be characterized by the saturation of a certain number of Pauli constraints $0 leq n_i leq 1$, identifying the occupied core orbitals ($n_i=1$) and the inactive virtual orbitals ($n_j=0$). In Part I, we generalize this crucial concept of active spaces by referring to the generalized Pauli constraints. To be more specific, we explain and illustrate that the saturation of any such constraint on fermionic occupation numbers characterizes a distinctive set of active electron configurations. A converse form of this selection rule establishes the basis for corresponding multiconfigurational wavefunction ansatzes. In Part II, we provide rigorous derivations of those findings. Moroever, we extend our results to non-fermionic multipartite quantum systems, revealing that extremal single-body information has always strong implications for the multipartite quantum state. In that sense, our work also confirms that pinned quantum systems define new physical entities and the presence of pinnings reflect the existence of (possibly hidden) ground state symmetries.

Machine Learning the Physical Non-Local Exchange-Correlation Functional of Density-Functional Theory

We train a neural network as the universal exchange-correlation functional of density-functional theory that simultaneously reproduces both the exact exchange-correlation energy and potential. This functional is extremely non-local, but retains the computational scaling of traditional local or semi-local approximations. It therefore holds the promise of solving some of the delocalization problems that plague density-functional theory, while maintaining the computational efficiency that characterizes the Kohn-Sham equations. Furthermore, by using automatic differentiation, a capability present in modern machine-learning frameworks, we impose the exact mathematical relation between the exchange-correlation energy and the potential, leading to a fully consistent method. We demonstrate the feasibility of our approach by looking at one-dimensional systems with two strongly-correlated electrons, where density-functional methods are known to fail, and investigate the behavior and performance of our functional by varying the degree of non-locality.

Reduced Density Matrix Functional Theory for Superconductors

We present an textit{ab initio} theory for superconductors, based on a unique mapping between the statistical density operator at equilibrium, on the one hand, and the corresponding one-body reduced density matrix $gamma$ and the anomalous density $chi$, on the other. This new formalism for superconductivity yields the existence of a universal functional $mathfrak{F}_beta[gamma,chi]$ for the superconductor ground state, whose unique properties we derive. We then prove the existence of a Kohn-Sham system at finite temperature and derive the corresponding Bogoliubov-de Gennes-like single particle equations. By adapting the decoupling approximation from density functional theory for superconductors we bring these equations into a computationally feasible form. Finally, we use the existence of the Kohn-Sham system to extend the Sham-Schluter connection and derive a first exchange-correlation functional for our theory. This reduced density matrix functional theory for superconductors has the potential of overcoming some of the shortcomings and fundamental limitations of density functional theory of superconductivity.

On the time evolution of fermionic occupation numbers

We derive an equation for the time evolution of the natural occupation numbers for fermionic systems with more than two electrons. The evolution of such numbers is connected with the symmetry-adapted generalized Pauli exclusion principle, as well as with the evolution of the natural orbitals and a set of many-body relative phases. We then relate the evolution of these phases to a geometrical and a dynamical term, attached to each one of the Slater determinants appearing in the configuration-interaction expansion of the wave function.

Static correlated functionals for reduced density matrix functional theory

Based on recent progress on fermionic exchange symmetry we propose a way to develop new functionals for reduced density matrix functional theory. For some settings with an odd number of electrons, by assuming saturation of the inequalities stemming from the generalized Pauli principle, the many-body wave-function can be written explicitly in terms of the natural occupation numbers and natural orbitals. This leads to an expression for the two-particle density matrix and therefore for the correlation energy functional. This functional was then tested for a three-electron Hubbard model where it showed excellent performance both in the weak and strong correlation regimes.

A representability problem in density functional theory for superconductors

Aiming at a unified treatment of correlation and inhomogeneity effects in superconductors, Oliveira, Gross and Kohn proposed in 1988 a density functional theory for the superconducting state. This theory relies on the existence of a Kohn-Sham scheme, i.e., an auxiliary noninteracting system with the same electron and anomalous densities of the original superconducting system. However, the question of noninteracting $v$-representability has never been properly addressed and the existence of the Kohn-Sham system has always been assumed without proof. Here, we show that indeed such a noninteracting system does not exist in at zero temperature. In spite of this result, we also show that the theory is still able to yield good results, although in the limit of weakly correlated systems only.

Towards a formal definition of static and dynamic electronic correlations

Some of the most spectacular failures of density-functional and Hartree-Fock theories are related to an incorrect description of the so-called static electron correlation. Motivated by recent progress on the N-representability problem of the one-body density matrix for pure states, we propose a way to quantify the static contribution to the electronic correlation. By studying several molecular systems we show that our proposal correlates well with our intuition of static and dynamic electron correlation. Our results bring out the paramount importance of the occupancy of the highest occupied natural spin-orbital in such quantification.

Relating correlation measures: the importance of the energy gap

The concept of correlation is central to all approaches that attempt the description of many-body effects in electronic systems. Multipartite correlation is a quantum information theoretical property that is attributed to quantum states independent of the underlying physics. In quantum chemistry, however, the correlation energy (the energy not seized by the Hartree-Fock ansatz) plays a more prominent role. We show that these two different viewpoints on electron correlation are closely related. The key ingredient turns out to be the energy gap within the symmetry-adapted subspace. We then use a few-site Hubbard model and the stretched H$_2$ to illustrate this connection and to show how the corresponding measures of correlation compare.