Diagonalization- and Numerical Renormalization-Group-Based Methods for Interacting Quantum Systems

In these lecture notes, we present a pedagogical review of a number of related {\it numerically exact} approaches to quantum many-body problems. In particular, we focus on methods based on the exact diagonalization of the Hamiltonian matrix and on methods extending exact diagonalization using renormalization group ideas, i.e., Wilson's Numerical Renormalization Group (NRG) and White's Density Matrix Renormalization Group (DMRG). These methods are standard tools for the investigation of a variety of interacting quantum systems, especially low-dimensional quantum lattice models. We also survey extensions to the methods to calculate properties such as dynamical quantities and behavior at finite temperature, and discuss generalizations of the DMRG method to a wider variety of systems, such as classical models and quantum chemical problems. Finally, we briefly review some recent developments for obtaining a more general formulation of the DMRG in the context of matrix product states as well as recent progress in calculating the time evolution of quantum systems using the DMRG and the relationship of the foundations of the method with quantum information theory.Comment: 51 pages; lecture notes on numerically exact methods. Pedagogical review appearing in the proceedings of the "IX. Training Course in the Physics of Correlated Electron Systems and High-Tc Superconductors", Vietri sul Mare (Salerno, Italy, October 2004

Full configuration interaction quantum Monte Carlo for coupled electron--boson systems and infinite spaces

We extend the scope of full configuration interaction quantum Monte Carlo (FCIQMC) to be applied to coupled fermion-boson hamiltonians, alleviating the a priori truncation in boson occupation which is necessary for many other wave function based approaches to be tractable. Detailing the required algorithmic changes for efficient excitation generation, we apply FCIQMC in two contrasting settings. The first is a sign-problem-free Hubbard--Holstein model of local electron-phonon interactions, where we show that with care to control for population bias via importance sampling and/or reweighting, the method can achieve unbiased energies extrapolated to the thermodynamic limit, without suffering additional computational overheads from relaxing boson occupation constraints. Secondly, we apply the method as a `solver' within a quantum embedding scheme which maps electronic systems to local electron-boson auxiliary models, with the bosons representing coupling to long-range plasmonic-like fluctuations. We are able to sample these general electron-boson hamiltonians with ease despite a formal sign problem, including a faithful reconstruction of converged reduced density matrices of the system

CMB-S4 Science Book, First Edition

This book lays out the scientific goals to be addressed by the next-generation ground-based cosmic microwave background experiment, CMB-S4, envisioned to consist of dedicated telescopes at the South Pole, the high Chilean Atacama plateau and possibly a northern hemisphere site, all equipped with new superconducting cameras. CMB-S4 will dramatically advance cosmological studies by crossing critical thresholds in the search for the B-mode polarization signature of primordial gravitational waves, in the determination of the number and masses of the neutrinos, in the search for evidence of new light relics, in constraining the nature of dark energy, and in testing general relativity on large scales

Evidence for symplectic symmetry in ab initio no-core shell model results

Advances in the construction of realistic internucleon interactions together with the advent of massively parallel computers have resulted in a successful utilization of the ab initio approaches to the investigation of properties of light nuclei. The no-core shell model is a prominent ab initio method that yields a good description of the low-lying states in few-nucleon systems as well as in more complex p-shell nuclei. Nevertheless, its applicability is limited by the rapid growth of the many-body basis with larger model spaces and increasing number of nucleons. To extend the scope of the ab initio no-core shell model to heavier nuclei and larger model spaces, we analyze the possibility of augmenting the spherical harmonic oscillator basis with symplectic Sp(3,R) symmetry-adapted configurations of the symplectic shell model which describe naturally the monopole-quadrupole vibrational and rotational modes, and also partially incorporate α-cluster correlations. In our study we project low-lying states of 12C and 16O determined by the no-core shell model with the JISP16 realistic interaction onto Sp(3,R)-symmetric model space that is free of spurious center-of-mass excitations. The eigenstates under investigation are found to project at the 85-90% level onto a few of the most deformed symplectic basis states that span only a small fraction (≈0.001%) of the full model space. The results are nearly independent of whether the bare or renormalized effective interactions are used in the analysis. The outcome of this study points to the relevance of the symplectic extension of the ab initio no-core shell model. Further, it serves to reaffirm the Elliott SU(3) model upon which the symplectic scheme is built. While extensions of this work are clearly going to be required if the theory is to become a model of choice for nuclear structure calculations, these early results seem to suggest that there may be simplicity within the complexity of nuclear structure that has heretofore not been fully appreciated. As follow-on work to what is reported in this thesis, we expect to develop a stand alone shell-model code that builds upon the underlying symmetries of the symplectic model

An analysis for some methods and algorithms of quantum chemistry

In der theoretischen Berechnung der Eigenschaften von Atomen, Molekülen und Festkörpern spielt die Lösung der elektronischen Schrödingergleichung, einer Operatoreigenwertgleichung für den Hamiltonoperator H des jeweiligen Systems, eine zentrale Rolle. Besondere Bedeutung kommt hierbei dem kleinsten Eigenwert von H zu, der die Grundzustandsenergie des Systems angibt. Um den unterschiedlichen Anforderungen in der Fülle von Anwendungsgebieten der elektronischen Schrödingergleichung gerecht zu werden, wurden in den letzten Jahrzehnten verschiedenste Näherungsverfahren entwickelt, um die Lösung dieses extrem hochdimensionalen Minimierungsproblems zu approximieren. Das Ziel der vorliegenden Arbeit ist es, eine (mathematische) Analysis für Aspekte einiger der verwendeten Methoden der Quantenchemie zu liefern. Zu diesem Zweck gliedert sich die Arbeit in vier Teile: Der erste Teil gibt eine Einführung in den mathematischen, hauptsächlich der Funktionalanalysis zuzuschreibenden Hintergrund, der bei der Behandlung der elektronischen Schrödingergleichung als Operatoreigenwertgleichung notwendig ist, und stellt viele der in den späteren Kapiteln benötigten Handwerkszeuge zur Verfügung. Der zweite Teil beschäftigt sich mit einem Gradientenalgorithmus mit Orthogonalitätsnebenbedingungen, der zur der Lösung der in der Beschreibung größerer Systeme wichtigen Hartree-Fock- und Kohn-Sham-Gleichungen, aber auch zur algorithmischen Behandlung der CI-Methode und außerhalb der Elektronenstrukturberechnung in der Berechnung invarianter Unterräume verwendet wird. Wir formulieren den Algorithmus allgemeiner als Verfahren zur Behandlung von Minimierungsproblemen auf der sogenannten Grassmann-Mannigfaltigkeit [1] und beweisen mit Hilfe dieses Formalismus unter anderem lineare Konvergenz des Algorithmus und die quadratische Konvergenz der zugeh√∂rigen Energien. Im dritten Teil der Arbeit wird die in der Praxis für hochgenaue Rechnungen bedeutsame Coupled-Cluster-Methode, traditionell ein Ansatz zur Approximation der Galerkinlösung der Schrödingergleichung innerhalb einer gegebenen Diskretisierung [2], als Verfahren im unendlichdimensionalen, undiskretisierten Raum formuliert. Zu diesem Zweck wird zunächst die Stetigkeit des Clusteroperators T als Operator vom Sobolevraum H1 in sich bewiesen: hieraus lässt sich dann die (unendlichdimensionale Verallgemeinerung der bekannten) Nullstellengleichung für die Coupled-Cluster-Funktion formulieren. Wir zeigen die lokale starke Monotonie der CC-Funktion, mit deren Hilfe wir Existenz- und Eindeutigkeitsaussagen und einen zielorientierten Fehlerschätzer nach [3] beweisen. Schließlich diskutieren wir die algorithmische Behandlung der oben genannten Nullstellengleichung. Teil 4 beschäftigt sich mit der DIIS-Methode, einem im Rahmen der Quantenchemie standardmäßig verwendeten Verfahren zur Konvergenzbeschleuningung iterativer Algorithmen. Wir identifizieren DIIS mit einer Variante des projezierten Broyden-Verfahrens [4] und zeigen, dass sich das Verfahren, angewandt auf lineare Probleme, als Variante des GMRES-Verfahrens auffassen lässt. Für den allgemeinen Fall beweisen wir schließlich zwei lokale Konvergenzaussagen und diskutieren die Umstände, unter denen DIIS superlineares Konvergenzverhalten zeigen kann. [1] T. A. Arias, A. Edelman, S. T. Smith, SIAM J. Matrix Anal. and Appl. 20, 2, 1999. [2] R. Schneider, Num. Math. 113, 3, 2009. [3] R. Becker, R. Rannacher, Acta Numerica 2000 (A. Iserlet, ed.), Cambridge University Press, 2001. [4] D. M. Gay, R. B. Schnabel, Nonlinear Programming 3, Academic Press, 1978.In the field of ab-initio calculation of the properties of atoms, molecules and solids, the solution of the electronic Schrödinger equation, an operator eigenvalue equation for the Hamiltonian of the system, plays a major role. Of utmost significance is the lowest eigenvalue of H, representing the ground state energy of the system. To meet the requirements of the multitude of possible applications of the elctronic Schrödinger equation, the last decades have seen the development of a variety of different methods designed to approximate the solution of this extremely high-dimensional minimization problem. The aim of the present work is to deliver a (mathematical) analysis for some aspects of some of these methods used in the context of quantum chemistry. The work consists of four parts: The first part gives an introduction to the mathematical background, mainly belonging to the field of functional analysis, that is needed for the rigogous treatment of the electronic Schrödinger equation as an operator eigenvalue equation, and provides many of the technical means needed in the later chapters. The second part is concerned with a gradient algorithm with orthogonality constraints, which is used for the solution Hartree-Fock and Kohn-Sham equations playing an important role in the description of larger systems and which also serves for the algorithmic treatment of the CI method and - outside of the field of electronic structure calculation - for the calculation of invariant subspaces. The algorithm is formulated as an abstract method for the treatment of minimization problems on the so-called Grassmann manifold [1]; with the help of this formalism, linear convergence of the algorithm and quadratic convergence of the corresponding eigenvalues is proven. The third part of the work is concerned with the Coupled Cluster method, being of high practical significance in calculations where high accuracy is demanded. We lift the method, usually formulated as an ansatz for the approximation of the Galerkin solution in a finite dimensional, discretised subspace [2] to the continuous, undiscretised space, resulting in what we will call the continuous Coupled Cluster method. To define the continuous method, we first prove the continuity of the cluster operator T as an operator mapping the Sobolev space H1 to itself; with the help of this result, the infinite dimensional globalization of the canonical) Coupled Cluster equations can be formulated. Afterwards, we prove local strong monotonicity of the CC function, from which we derive existence and (local) uniqueness statements and a goal-oriented a-posteriori error estimator in the fashion of [3]. Finally, we discuss the algorithmic treatment of the CC root equation. The last part of this work features an analysis for the acceleration technique DIIS that is commonly used in quantum chemistry codes. We identify DIIS with a variant of a projected Broyden's method [4] and show that when applied to linear systems, the method can be interpreted as a variant of the well-known GMRES method. For the global nonlinear case, we finally prove two local convergence results and discuss the circumstances under which DIIS can show superlinear convergence. [1] T. A. Arias, A. Edelman, S. T. Smith, SIAM J. Matrix Anal. and Appl. 20, 2, 1999. [2] R. Schneider, Num. Math. 113, 3, 2009. [3] R. Becker, R. Rannacher, Acta Numerica 2000 (A. Iserlet, ed.), Cambridge University Press, 2001. [4] D. M. Gay, R. B. Schnabel, Nonlinear Programming 3, Academic Press, 1978

Accurate variational electronic structure calculations with the density matrix renormalization group

During the past fifteen years, the density matrix renormalization group (DMRG) has become increasingly important for ab initio quantum chemistry. Its underlying wavefunction ansatz, the matrix product state (MPS), is a low­-rank decomposition of the full configuration interaction tensor. The virtual dimension of the MPS, the rank of the decomposition, controls the size of the corner of the many­-body Hilbert space that can be reached with the ansatz. This parameter can be systematically increased until numerical convergence is reached. Whereas the MPS ansatz can only capture exponentially decaying correlation functions in the thermodynamic limit, and will therefore only yield an efficient description for noncritical one-dimensional systems, it can still be used as a variational ansatz for finite­-size systems. Rather large virtual dimensions are then required. The two most important aspects to reduce the corresponding computational cost are a proper choice and ordering of the active space orbitals, and the exploitation of the symmetry group of the Hamiltonian. By taking care of both aspects, DMRG becomes an efficient replacement for exact diagonalization in quantum chemistry. For hydrogen chains, accurate longitudinal static hyperpolarizabilities were obtained in the thermodynamic limit. In addition, the low-lying states of the carbon dimer were accurately resolved. DMRG and Hartree-­Fock theory have an analogous structure. The former can be interpreted as a self­-consistent mean­-field theory in the DMRG lattice sites, and the latter in the particles. It is possible to build upon this analogy to introduce post-­DMRG methods. Based on an approximate MPS, these methods provide improved ansätze for the ground state, as well as for excitations. Exponentiation of the single­-particle excitations for a Slater determinant leads to the Thouless theorem for Hartree-­Fock theory, an explicit nonredundant parameterization of the entire manifold of Slater determinants. For an MPS with open boundary conditions, exponentiation of the single-site excitations leads to the Thouless theorem for DMRG, an explicit nonredundant parameterization of the entire manifold of MPS wavefunctions. This gives rise to the configuration interaction expansion for DMRG. The Hubbard-­Stratonovich transformation lies at the basis of auxiliary field quantum Monte Carlo for Slater determinants. An analogous transformation for spin-­lattice Hamiltonians allows to formulate a promising variant for matrix product states

Accurate variational electronic structure calculations with the density matrix renormalization group

During the past 15 years, the density matrix renormalization group (DMRG) has become increasingly important for ab initio quantum chemistry. The underlying matrix product state (MPS) ansatz is a low-rank decomposition of the full configuration interaction tensor. The virtual dimension of the MPS controls the size of the corner of the many-body Hilbert space that can be reached. Whereas the MPS ansatz will only yield an efficient description for noncritical one-dimensional systems, it can still be used as a variational ansatz for other finite-size systems. Rather large virtual dimensions are then required. The two most important aspects to reduce the corresponding computational cost are a proper choice and ordering of the active space orbitals, and the exploitation of the symmetry group of the Hamiltonian. By taking care of both aspects, DMRG becomes an efficient replacement for exact diagonalization in quantum chemistry. DMRG and Hartree-Fock theory have an analogous structure. The former can be interpreted as a self-consistent mean-field theory in the DMRG lattice sites, and the latter in the particles. It is possible to build upon this analogy to introduce post-DMRG methods. Based on an approximate MPS, these methods provide improved ans\"atze for the ground state, as well as for excitations. Exponentiation of the single-particle (single-site) excitations for a Slater determinant (an MPS with open boundary conditions) leads to the Thouless theorem for Hartree-Fock theory (DMRG), an explicit nonredundant parameterization of the entire manifold of Slater determinants (MPS wavefunctions). This gives rise to the configuration interaction expansion for DMRG. The Hubbard-Stratonovich transformation lies at the basis of auxiliary field quantum Monte Carlo for Slater determinants. An analogous transformation for spin-lattice Hamiltonians allows to formulate a promising variant for MPSs.Comment: PhD thesis (225 pages). PhD thesis, Ghent University (2014), ISBN 978946197194