Complex Quantum Systems

The following sections are included:

• CONTENTS

• FOREWORD

• PREFACE

These are extended notes based on a series of lectures on the Stability of Matter given by one of us (RB) as part of the workshop on “Complex Quantum Systems” held at the National University of Singapore, February 17-March 27, 2010.

These notes give a survey on mathematical results concerning approximations to the ground state and the ground state energy of large atoms and molecules in terms of density functional theory of density matrix functional theory. In particular, stability of matter is proved, as well as, the asymptotic validity of the Hartree-Fock approximation in the limit of large nuclear charges. The latter requires the use of correlation in-equalities which are derived here and related to some notions from the theory of N-representability, such as the G, P, and Q conditions and also Erdahl's T₁ and T₂ conditions. The latter have recently drawn a lot of attention in quantum chemistry.

This article addresses some aspects of the dynamics of a Fermi gas in a random medium, at weak disorders. We present some joint results with I. Sasaki on the Boltzmann limit for the thermal momentum distribution function, and on the persistence of quasifreeness, for the case of a free Fermi gas in a random medium. Subsequently, we present some recent joint results with I. Rodnianski on the derivation of the Boltzmann limit for a Fermi gas in a random medium with nonlinear self-interactions modeled in dynamical Hartree-Fock theory.

A Hamiltonian defined as a polynomial in creation and annihilation operators is considered. After a minimization of its expectation value over pure Gaussian states, the Hamiltonian is Wick-ordered in creation and annihilation operators adapted to the minimizing state. It is shown that this procedure eliminates from the Hamiltonian terms of degrees 1 and 2 that do not preserve the particle number, and leaves only the terms that can be interpreted as quasiparticles excitations. We propose to call this fact Beliaev's Theorem, since to our knowledge it was mentioned for the first time in a paper by Beliaev from 1959.

The reduced-density-matrix method is a promising next-generation electronic structure calculation method; it is equivalent to solving the Schrödinger equation for the ground state. The number of variables is the same as a four electron system and constant regardless of the number of electrons in the system. Many researchers have been hoping for a simpler method of doing quantum mechanical calculations and this one may be it. In this chapter, we give an overview of the method covering the theory behind it and its methodology. We also give a brief history of the subject and report new computational results. Typically, the results obtained by the reduced-density-matrix method are comparable to those of the CCSD(T), which is a rather sophisticated traditional approach in quantum chemistry.

These lecture notes show how Quantum Information Theory can help us to gain deeper insight into the physics of fermionic quantum many-body systems. In the first part, we start from fermionic correlations as they appear in recent experiments with cold gases and develop a pairing theory for fermionic particles applicable to the current experimental setups. In the second part we introduce and discuss a new class of variational states, the fermionic Projected Entangled Pair States (fPEPS) that allow for an efficient approximation of ground and thermal states of local fermionic Hamiltonians.

In this review we consider two different models of a hydrogenic atom in a quantized electromagnetic field that treat the electron relativistically. The first one is a no-pair model in the free picture, the second one is given by the semi-relativistic Pauli-Fierz Hamiltonian. For both models we discuss the semi-boundedness of the Hamiltonian, the strict positivity of the ionization energy, and the exponential localization in position space of spectral subspaces corresponding to energies below the ionization threshold. Moreover, we prove the existence of degenerate ground state eigenvalues at the bottom of the spectrum of the Hamiltonian in both models. All these results hold true, for arbitrary values of the finestructure constant, e², and the ultra-violet cut-off, and for a general class of electrostatic potentials including the Coulomb potential with nuclear charges less than (sometimes including) the critical charges without radiation field, namely e-²2/π for the semi-relativistic Pauli-Fierz operator and e-²2/(2/π+π/2) for the no-pair operator. Apart from a detailed discussion of diamagnetic inequalities in QED (which are applied to study the semi-boundedness) all results stem from earlier articles written by the authors. While a few proofs are merely sketched, we streamline earlier proofs or present alternative arguments at many places.