Two-electron wavefunctions are matrix product states with bond dimension Three

We prove the statement in the title, for a suitable (wavefunction-dependent) choice of the underlying orbitals, and show that Three is optimal. Thus for two-electron systems, the QC-DMRG method with bond dimension Three combined with fermionic mode optimization exactly recovers the FCI energy

A Tensor Train Continuous Time Solver for Quantum Impurity Models

The simulation of strongly correlated quantum impurity models is a significant challenge in modern condensed matter physics that has multiple important applications. Thus far, the most successful methods for approaching this challenge involve Monte Carlo techniques that accurately and reliably sample perturbative expansions to any order. However, the cost of obtaining high precision through these methods is high. Recently, tensor train decomposition techniques have been developed as an alternative to Monte Carlo integration. In this study, we apply these techniques to the single-impurity Anderson model at equilibrium by calculating the systematic expansion in power of the hybridization of the impurity with the bath. We demonstrate the performance of the method in a paradigmatic application, examining the first-order phase transition on the infinite dimensional Bethe lattice, which can be mapped to an impurity model through dynamical mean field theory. Our results indicate that using tensor train decomposition schemes allows the calculation of finite-temperature Green's functions and thermodynamic observables with unprecedented accuracy. The methodology holds promise for future applications to frustrated multi-orbital systems, using a combination of partially summed series with other techniques pioneered in diagrammatic and continuous-time quantum Monte Carlo

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

Bayesian Modelling Approaches for Quantum States -- The Ultimate Gaussian Process States Handbook

Capturing the correlation emerging between constituents of many-body systems accurately is one of the key challenges for the appropriate description of various systems whose properties are underpinned by quantum mechanical fundamentals. This thesis discusses novel tools and techniques for the (classical) modelling of quantum many-body wavefunctions with the ultimate goal to introduce a universal framework for finding accurate representations from which system properties can be extracted efficiently. It is outlined how synergies with standard machine learning approaches can be exploited to enable an automated inference of the most relevant intrinsic characteristics through rigorous Bayesian regression techniques. Based on the probabilistic framework forming the foundation of the introduced ansatz, coined the Gaussian Process State, different compression techniques are explored to extract numerically feasible representations of relevant target states within stochastic schemes. By following intuitively motivated design principles, the resulting model carries a high degree of interpretability and offers an easily applicable tool for the numerical study of quantum systems, including ones which are notoriously difficult to simulate due to a strong intrinsic correlation. The practical applicability of the Gaussian Process States framework is demonstrated within several benchmark applications, in particular, ground state approximations for prototypical quantum lattice models, Fermi-Hubbard models and $J_1-J_2$ models, as well as simple ab-initio quantum chemical systems.Comment: PhD Thesis, King's College London, 202 page

One-dimensional Tensor Network Recovery

We study the recovery of the underlying graphs or permutations for tensors in tensor ring or tensor train format. Our proposed algorithms compare the matricization ranks after down-sampling, whose complexity is $O(d\log d)$ for $d$-th order tensors. We prove that our algorithms can almost surely recover the correct graph or permutation when tensor entries can be observed without noise. We further establish the robustness of our algorithms against observational noise. The theoretical results are validated by numerical experiments

Dynamical mean-field theory studies on real materials

Numerical studies on strongly correlated fermionic systems are very complicated and still provide essential problems. The main reason is the exponential growth of the un- derlying Hilbert state space with the system size and the fermionic sign problem for Monte Carlo studies. Among the most widely employed numerical techniques for study- ing two-dimensional quantum many-body systems are cluster extensions of the dynamical mean-field theory (DMFT), e.g. dynamical cluster approximation (DCA). They map an infinitely large multi-dimensional lattice problem to a one-dimensional impurity problem. In 2015 it was shown that the density matrix renormalisation group (DMRG) used as an impurity solver for DMFT (DMFT+DMRG) on the imaginary-frequency axis allows to solve multi-site and multi-band problems extremely fast compared to other solvers. Within this thesis, we further develop this DMRG+DMFT approach to apply the method on real material settings. The step from artificial, completely degenerate multi-band mod- els with simple dispersion relations on a Bethe lattice, studied in 2015, to systems with realistic band structures and lifted degeneracies involves more challenges than originally suspected. In this thesis, we will first recapitulate relevant methods for our approach like matrix prod- uct states, the density matrix renormalisation group and several time evolution methods. In this context we will present several improvements ranging from optimised time evo- lutions to entanglement based optimisations of tensor networks. Second, we will present a very detailed description of the dynamical mean field theory. We will focus on both methodological aspects and implementation details. This chapter is intended to allow other researcher to implement their own DMFT code using DMRG as an impurity solver. Third, we will discuss three different models to show the extent of problems DMRG+ DMFT is able to solve. We will focus on multi-site DCA calculations in the case of the two-dimensional Hubbard model and show that DMRG allows to tackle systems with intermediate interaction strengths at low temperatures, which are unsolvable with other solvers. In the second case, the real material Sr2VO4, we will show the first two-site DCA results for a realistic three-band model. In contrast to assumptions, partly reintroducing the momentum dependence of the self-energy does not improve agreement between exper- imental observations and theoretical results. Finally, we will move on to another realistic three-band model, which describes Sr2RuO4, to show how to deal with the influence of spin-orbit coupling on DMFT. We will present the first low-temperature results for this material and will confirm previous results of simplified model calculations.Numerische Untersuchungen stark korrelierter fermionischer Systeme sind schwierig und beinhalten noch heute essentielle Probleme. Die Hauptgründe dafür sind das exponen- tielle Wachstum des Hilbertraumes der Quantenzustände mit der Systemgröße und das fermionische Vorzeichenproblem bei Monte-Carlo-Rechnungen. Eine der am häufigsten verwendeten Methoden zur Untersuchung zweidimensionaler Gittersysteme sind Cluster- Erweiterungen der dynamische Molekularfeld Theory (DMFT), wie zum Beispiel die dy- namische Cluster Approximation (DCA). Diese Methoden bilden mehrdimensionale Git- tersysteme auf eindimensionale Störstellen-Probleme ab. 2015 wurde gezeigt, dass DMFT auf der imaginären Frequenzachse kombiniert mit der Dichtematrix-Renormierungsgruppe (DMFT+DMRG) Mehrband- und Multisite-Systeme schneller lösen kann, als wenn an- dere Störstellen-Löser verwendet werden. In dieser Arbeit entwickeln wir diesen Ansatz weiter und wenden ihn auf Modelle realer Materialen an. Am Anfang dieser Arbeit besprechen wir relevante Methoden für DMRG+ DMFT, wie zum Beispiel Matrix-Produkt-Zustände, die Dichtematrix-Renormierungs- gruppe und mehrere Zeitentwicklungs-Methoden. In diesem Zusammenhang werden wir auch mehrere Verbesserungen besprechen, die von methodischen Anpassungen von Zeit- entwicklungen bis hin zur Neuordnung des Tensornetzwerkes basierend auf Verschrän- kungs-Eigenschaften reichen. Danach werden wir uns detailliert mit den methodologischen und programmiertechnischen Aspekten von DMFT beschäftigen. Dieses Kapitel dient als Grundlage für andere Forscher, die eigene DMRG+DMFT-Codes programmieren wollen. Abschließend werden wir drei verschiedene Modelle besprechen, um das Ausmaß der Sys- teme zu zeigen, die mit diesem Ansatz gelöst werden können. Wir werden uns im Kon- text des Hubbard-Modells detailliert mit Multisite-DCA beschäftigen und zeigen, dass DMRG+DMFT Ergebnisse für Systeme mit mittleren Wechselwirkungsstärken bei niedri- gen Temperaturen erzeugen kann. Das ist mit anderen Störstellen-Lösern bisher nicht möglich. Im zweiten Fall beschäftigen wir uns mit Strontiumvanadat Sr2VO4 und werden die ersten Zweisite-DCA-Ergebnisse für ein realistisches Dreiband-Modell präsentieren. Im Gegensatz zu bisherigen Erwartungen führt die teilweise Wiedereinführung der Im- pulsabhängigkeit der Selbstenergie nicht zu einer besseren Übereinstimmung von Theorie und Experiment. Das dritte Modell beschreibt Strontiumruthenat Sr2RuO4. In diesem Fall besprechen wir den Einfluss der Spin-Bahn-Kopplung auf DMFT und wie die damit verbundenen Probleme optimal gelöst werden können. Abschließend zeigen wir die ersten Ergebnisse für dieses Modell bei niedrigen Temperaturen