Reduced Density Matrices in Quantum Physics and Role of Fermionic Exchange Symmetry

In my talk I will attempt to provide an overview on the application of the density matrix renormalization group (DMRG) algorithm in quantum chemistry. I will compare to traditional approaches with respect to limitations and capabilities. I will highlight the ma- trix product operator structure of our second-generation DMRG program and discuss its features. Emphasis will be put on how orbital-entanglement measures can be exploited to automatically select proper active orbital spaces, a major problem of all multi-configurational approaches.

In this talk, I will introduce DMRG both from the historical (statistical) and modern (matrix product state) perspective, highlighting why it has become the method of choice for one-dimensional quantum systems in and out of equilibrium. I will talk as well as why it has made interesting forays into the world of two-dimensional quantum systems. I will also discuss its limitations, as well as the resulting directions of future development.

Information about the interaction of a many-electron quantum system with its environment is encoded within the one-electron density matrix (1-RDM). While the 1-RDM from an ensemble many-electron quantum system must obey the Pauli exclusion principle, the 1-RDM from a pure quantum system must obey additional constraints known as the generalized Pauli conditions. By examining the 1-RDM’s violation of these generalized Pauli conditions, we obtain a sufficient condition at the level of a single electron for a many-electron quantum system’s openness. As the system interacts with the environment, the more stringent generalized Pauli conditions relax to the Pauli principle, the nature and extent of this relaxation serving to quantify the degree openness of a many-electron quantum system. In an application to photosynthetic light harvesting we show that the interaction of the system with the environment (quantum noise) relaxes significant constraints imposed on the exciton dynamics by the generalized Pauli conditions. This relaxation provides a geometric (kinematic) interpretation for the role of noise in enhancing exciton transport in quantum systems.

Reduced Density Matrix Functional Theory is a method that relies on the 1-1 correspondence between the ground state wavefunction of many electron systems and the first order reduced density matrix(1RDM) and uses the second one as its fundamental valuable. The ground state of a system is determined within this approach by minimizing the energy functional with respect to the 1RDM while satisfying that the 1RDM corresponds to a fermionic ensemble (Coleman’s conditions). As the explicit expression of the energy functional with respect to the 1RDM is not known, different approximate functionals are employed. If we had the exact functional performing the energy minimization using the ensemble representability constraints would be enough to find a 1RDM that corresponds to a pure state (if our ground state is not degenerate so we really have a pure state). However, performing the energy minimization with approximate functionals, as we found for 3 electron systems test cases, results in occupation numbers that do not satisfy the generalized Pauli constraints (GPC). One then could in principle employ the GPC as additional constraints during the energy minimization to ensure that the ground state 1RDM that finds can result from a pure state. However due to the big number of these constraints this is not feasible in practice apart from a few cases of really small systems.

An idea to be explored is constructing energy functionals that satisfy at least some of the GPC. This could serve as a change of paradigm for functional derivation because until now 1RDM functionals were mostly tested on whether they reproduced or not ground state energies correctly. Another idea that we would like to discuss is whether we could apply the GPC in an approximate way to only the electrons that have occupations smaller than one. The electrons with occupation one do not play any role to whether the 1RDM corresponds to a pure state or not. Although it is doubtful whether 1RDMs with occupations that are exactly one can correspond to a ground state of a real fermionic system, in many cases this is a sensible approximation that would significantly reduce the amount of GPC to be considered in a RDMFT minimization.

It is now known that fermionic natural occupation numbers (NON) do not only obey Pauli’s exclusion principle but are even stronger restricted by the so-called generalized Pauli constraints (GPC). So far, the nature of these constraints has been explored in some systems: a model of few spinless fermions confined to a one-dimensional harmonic potential, the lithium isoelectronic series and ground and excited states of some three-, four- and five-electron atomic and molecular systems. Whenever given NON lie on the boundary of the allowed region the corresponding N-fermion quantum state has a significantly simpler structure. By employing this structure a variational optimization method for few fermion ground states is elaborated. We quantitatively confirm its high accuracy for systems with the vector of NON in a small distance to the boundary of the polytope. In particular, we derive an upper bound on the error of the correlation energy given by the ratio of the distance to the boundary of the polytope and the distance of the vector of NON to the Hartree-Fock point. Moreover, these geometric insights shed some light on the concept of active spaces, correlation energy, frozen electrons and virtual orbitals.

The Pauli exclusion principle has a strong impact on the properties and the behavior of most fermionic quantum systems. Remarkably, even stronger restrictions on fermionic natural occupation numbers follow from the fermionic exchange symmetry. We develop an operationally meaningful measure which allows one to quantify the potential physical relevance of those generalized Pauli constraints beyond the well-established relevance of Pauli’s exclusion principle. It is based on a geometric hierarchy induced by Pauli exclusion principle constraints. The significance of that measure is illustrated for a few-fermion model which also confirms such nontrivial relevance of the generalized Pauli constraints.

The Pauli exclusion principle is a constraint on the natural occupation numbers of fermionic states. It has been suspected since at least the 1970’s, and only proved very recently, that there is a multitude of further constraints on these numbers, generalizing the Pauli principle. Here, we provide the first analytic analysis of the physical relevance of these constraints. We compute the natural occupation numbers for the ground states of a family of interacting fermions in a harmonic potential. Intriguingly, we find that the occupation numbers are almost, but not exactly, pinned to the boundary of the allowed region (quasi-pinned). The result suggests that the physics behind the phenomenon is richer than previously appreciated. In particular, it shows that for some models, the generalized Pauli constraints play a role for the ground state, even though they do not limit the ground-state energy. Our findings suggest a generalization of the Hartree-Fock approximation.

In the talk I am planning to explain two different solutions of N-representability problem and then give the algorithm to calculate GPCs. After that, as examples I will show my calculations for 3 fermions systems of rank 6 and 7 for which the algorithm works smoothly. For higher rank systems we need an additional tool which I will try to explain also.

I will give an introduction to the univariate quantum marginal problem using an elementary mathematical point of view. In particular, I will explain how extremality of the local spectrum carries structural information about the global wave function. The talk will also give some quantum information background, touching e.g. on the computational complexity of general quantum marginal problems and relations to entanglement.

In the talk I will present recent progress in proving closeness of the microscopic and effective description for systems of many fermions. Solving the time dependent Schrödinger equation for a system of many interacting Fermions is in many cases impossible, both analytically and numerically. Instead of the microscopic description one uses an effective description which describes the behaviour of the fermions in good approximation and can be treated with a computer. In many cases one uses the fermionic Hartree or Hartree-Fock equation as effective description.