The essence of quantum magnetism lies in correlated many-body states. They determine the physical properties beyond paramagnetism. Even constructions as simple as singlets or triplets cannot be understood without the mathematical concept of superpositions. This is even more evident if time evolution of quantum states and phenomena such as quantum coherenceQuantum coherence are considered. The major approach to assess correlated many-body states in quantum magnetism of molecules is exact diagonalizationDiagonalization of the many-body Hamiltonian describing the physical system. Symmetries are of great help when tackling the numerical task of matrix diagonalization since they allow to decompose a problem of large Hilbert spaceHilbert space dimension into a number of smaller problems. This approach will be presented first. It rests on the identification of the available symmetries—e.g., U(1), SU(2), as well as point groupsPoint group—and their application. However, the applicability of exact diagonalizationDiagonalization is limited by the exponential growth of the Hilbert spaceHilbert space of the spin system with the number of spins: $$\dim \left( {\mathcal{H}} \right) = \mathop \prod \nolimits_{i} \left( {2 s_{i} + 1} \right)$$ . Nowadays, complex Hermitian matricesHermitian matrices of a linear dimension of about 100,000 can be completely diagonalized on supercomputers which limits the number of spins depending on their size $$s_{i}$$ to just a few. I will therefore present a very accurate approximation to exact diagonalization that rests on Lanczos procedures and the concept of typicality. The most popular version, the finite-temperature Lanczos methodFinite-Temperature Lanczos Method (FTLM), allows to evaluate low-energy spectroscopic data, thermal averages as well as time evolutions for systems with Hilbert space dimensions of up to about 1011. Hands-on experience and examples will be provided.
The essence of quantum magnetism lies in correlated many-body states. They determine the physical properties beyond paramagnetism. Even constructions as simple as singlets or triplets cannot be understood without the mathematical concept of superpositions. This is even more evident if time evolution of quantum states and phenomena such as quantum coherenceQuantum coherence are considered. The major approach to assess correlated many-body states in quantum magnetism of molecules is exact diagonalizationDiagonalization of the many-body Hamiltonian describing the physical system. Symmetries are of great help when tackling the numerical task of matrix diagonalization since they allow to decompose a problem of large Hilbert spaceHilbert space dimension into a number of smaller problems. This approach will be presented first. It rests on the identification of the available symmetries—e.g., U(1), SU(2), as well as point groupsPoint group—and their application. However, the applicability of exact diagonalizationDiagonalization is limited by the exponential growth of the Hilbert spaceHilbert space of the spin system with the number of spins: \dim \left( {\mathcal{H}} \right) = \mathop \prod \nolimits_{i} \left( {2 s_{i} + 1} \right) . Nowadays, complex Hermitian matricesHermitian matrices of a linear dimension of about 100,000 can be completely diagonalized on supercomputers which limits the number of spins depending on their size si to just a few. I will therefore present a very accurate approximation to exact diagonalization that rests on Lanczos procedures and the concept of typicality. The most popular version, the finite-temperature Lanczos methodFinite-Temperature Lanczos Method (FTLM), allows to evaluate low-energy spectroscopic data, thermal averages as well as time evolutions for systems with Hilbert space dimensions of up to about 1011. Hands-on experience and examples will be provided.
One of the most intriguing areas of physics is the study of strongly correlated many-body systems. In these kinds of systems, a large number of particles interacts via interactions that are strong enough to play a major role in determining the properties of the system, which is particularly interesting when multiple interaction terms compete and cannot be satisfied simultaneously. Examples of phases they realize range from simple magnetism to puzzling phenomena such as superconductivity or topological order. Of special interest in condensed matter are fermionic models which have an extra layer of complexity added by the possibly intricate sign structure of the wave function. \nHowever, exactly solvable models that exhibit the sought-after phenomena are scarce and Hamiltonians that are supposed to model experiments often elude analytical solutions precisely due to the strong interactions. Numerical methods are thus essential to gain insight into these systems and provide an important link between theoretical models and real world experiments. There are many different methods that in principle allow solving a problem numerically exact, most prominently exact diagonalization, tensor network based methods such as DMRG and (quantum) Monte Carlo. Exact diagonalization is certainly the most straightforward technique which allows measuring any observable of interest but its application to interacting models is severely limited by the exponential growth of the Hilbert space with the system size which only allows studying rather small systems. Tensor network methods excel in the study of gapped, one-dimensional systems and while they have shown some impressive results for two-dimensional systems, they need a lot of fine tuning and careful analysis to be considered reliable in this very important domain. Quantum Monte Carlo, on the other hand, is in principle neither limited by the size of the Hilbert space nor by the strength of the interactions and has thus been a tremendously important tool for studying condensed matter systems which is why it is the focus of this thesis. \nIt is necessary to continuously advance existing and develop new techniques to keep up with theoretical as well as with experimental progress. In this thesis, new methods to identify and characterize conventional and novel phases of matter within quantum Monte Carlo are developed and tested on archetypical models of condensed matter. \nIn the first part of this thesis, a method to study entanglement properties of strongly interacting fermionic models in quantum Monte Carlo is presented. Measurements of this kind are needed because some characteristics of a system remain hidden from conventional approaches based on the calculation of correlation functions. Prime examples are systems that realize so-called topological order entailing long-range entanglement that can be positively diagnosed using entanglement techniques. Realizing such measurements in a concrete algorithm is not straightforward for fermions. A large section of this part is thus devoted to developing a solution to the problem of implementing entanglement measurements and benchmarking the results against known data. \nThe second part is concerned with a truly novel and recent approach to the many-body problem, namely the symbiosis of machine learning and quantum Monte Carlo techniques. Out of the many different possibilities to combine the two, this part is concerne with exploring ways in which machine learning can help to effortlessly explore phase diagrams of hitherto unknown Hamiltonians. \nDespite the power and versatility of quantum Monte Carlo techniques, it does suffer from one significant weakness called the fermion sign problem. At its core related to the peculiar exchange statistics of fermions, it is in principle merely a technical problem of the Monte Carlo procedure but one that actually prohibits the use of Monte Carlo methods for a large number of interesting problems. The sign problem has been known for a long time but remains unsolved to this day. Thus, it is time to approach the problem from a different perspective with the hope of learning something new that can either help to better understand the problem itself or the properties of the models affected by it. Both of the two major research areas, entanglement measures and machine learning, are capable of providing such new perspectives and the sign problem will be revisited in more detail in the context of each of them.
Magnetic molecules promise progress in a wide variety of fields ranging from high density memory storage to quantum computing as well as magnetic cooling. In order to gain better insight this work aims at the study of theoretical models and evaluation of thermodynamic observables by implementing and testing a new numerical approximation method - the Finite-Temperature Lanczos-Method (FTLM). Magnetic centres of molecules are modelled as spins of fixed quantum number, position and interactions. The model is specified in a spin Hamiltonian consisting of three parts: Heisenberg exchange, single ion anisotropy and Zeeman interaction. Although finite, the dimension of the Hilbert space grows exponentially with the number of spin sites, so full diagonalisation of the Hamiltonian is rendered infeasible: Even for a small number of spins the Hilbert space dimension exceeds several millions, which is beyond exact diagonalisation techniques. In order to calculate observables yet, it is necessary to implement approximation methods. Following previous work, where the Hilbert space could be partitioned into invariant subspaces with respect to magnetic quantum numbers, now systems with anisotropy are studied. These anisotropic terms generally do not commute with total spin operators, therefore partitioning no longer helps reducing the problem size. There are several techniques for approximate solutions to eigenvalue equations, among these adaptations of the original Lanczos method. As a Krylov subspace method, its main advantage is that the most complex operations required are matrix-vector multiplications. When matrix elements are calculated on-the-fly, the most economical version requires storage of only two vectors. The second ingredient for FTLM is a typicality based Monte Carlo approach: Traces are estimated as an average over a tiny set of initial vectors compared to the full Hilbert space. First tests revealed a major flaw due to numerical instability of traces that normally are zero. By partitioning with respect to magnetic quantum numbers, especially exploiting their inversion symmetry, this problem was unknowingly dodged in previous calculations. It was solved while preserving the advantage of minimal computational effort: An arbitrary set of initial vectors would regain this time-reversal symmetry only, if a substantial number of initial vectors where used. Instead, a doubling of the initial vectors by inverting them with respect to magnetic quantum numbers yields the desired effect. Time-consuming evaluations of matrix elements were also avoided by calculating the magnetisation as a difference quotient. Finally FTLM is implemented and tested for several magnetic molecules. Analogues of the hour glass molecules synthesised by Glaser et al. were first model systems, showing good agreement of exact diagonalisation results and approximations. The method can be applied to any Hamiltonian, given that matrix-vector products in the underlying Hilbert space can be calculated. Here limiting factors are memory usage and computing time. As a proof of principle observables in a Hilbert space consisting of d=100,000,000 states were calculated using the parameter set suggested by Mazurenko et al. for the Mn12-acetate molecule. Single crystal as well as powder averaged results are compared to experimental data. So this previously impossible calculation is now executed within hours by use of FTLM.
Abstract Tensor network methods as presented in our open source matrix product states software have opened up the possibility to study many-body quantum physics in one and quasi-one-dimensional systems in an easily accessible package similar to density functional theory codes but for strongly correlated dynamics. Here, we address methods which allow one to capture the full entanglement without truncation of the Hilbert space. Such methods are suitable for validation of and comparisons to tensor network algorithms, but especially useful in the case of new kinds of quantum states with high entanglement violating the truncation in tensor networks. Quantum cellular automata are one example for such a system, characterized by tunable complexity, entanglement, and a large spread over the Hilbert space. Beyond the evolution of pure states as a closed system, we adapt the techniques for open quantum systems simulated via the Lindblad master equation. We present three algorithms for solving closed-system many-body time evolution without truncation of the Hilbert space. Exact diagonalization methods have the advantage that they not only keep the full entanglement but also require no approximations to the propagator. Seeking the limits of a maximal number of qubits on a single core, we use Trotter decompositions or Krylov approximation to the exponential of the Hamiltonian. All three methods are also implemented for open systems represented via the Lindblad master equation built from local channels. We show their convergence parameters and focus on efficient schemes for their implementations including Abelian symmetries, e.g. symmetry used for number conservation in the Bose–Hubbard model or discrete symmetries in the quantum Ising model. We present the thermalization timescale in the long-range quantum Ising model as a key example of how exact diagonalization contributes to novel physics.
This paper explores the numerical study of quantum many-body systems with an emphasis on exact diagonalization techniques. The complexity of strongly correlated systems, often governed by large Hilbert spaces, presents significant computational challenges, making exact solutions difficult. In this work, we examine methods to simplify these systems by leveraging techniques such as the Schrieffer-Wolff transformation, which decouples high-energy and low-energy states, and the use of symmetry operators to block-diagonalize Hamiltonians and so on. These approaches are demonstrated with examples such as the hydrogen atom and a lambda system. The second part of the paper focuses on specific case studies, including a one-dimensional spin model and Bose-Hubbard model. The latter is crucial for understanding the behavior of interacting bosons in lattice systems and phenomena such as the superfluid-Mott insulator transition. We present a detailed investigation of the phase diagram for the one-dimensional Bose-Hubbard model using both exact diagonalization and mean field theory, providing insights into its quantum phase transitions. This study underscores the potential of exact diagonalization in quantum simulations and highlights its relevance for experimental setups involving trapped ions and superconducting qubits.
The dynamical mean-field theory (DMFT) maps a correlated lattice problem onto an impurity problem of a single correlated site coupled to an uncorrelated bath. Most implementations solve the DMFT equations using quantum Monte-Carlo sampling on the imaginary time and frequency (Matsubara) axis. We will here review alternative methods using exact diagonalization, i.e., representing the many-body ground state of the impurity as a sum over Slater determinants and calculating Green’s functions using iterative Lanczos procedures. The advantage being that these methods have no sign problem, can handle involved multi-orbital Hamiltonians (low crystal symmetry, spin-orbit coupling) and – when working completely on the real axis – do not need a mathematically ill-posed analytical continuation. The disadvantage of traditional implementations of exact diagonalization has been the exponential scaling of the calculation problem as a function of number of bath discretization points. In the last part we will review how recent advances in exact diagonalization can evade the exponential barrier thereby increasing the number of bath discretization points to reach the thermodynamic limit.
This thesis is focused on the application and development of numerical methods for studying quantum many- body spin systems. Part I sets the stage for research projects. In chapter 1, I go over major milestones in the field of frustrated magnetism, how it fits into a larger category of condensed matter physics, and recent developments that provide context for the research projects presented in part II. Chapters 2 3 provide the reader with necessary concepts in physics of phase transitions and numerical methods that are often employed when studying ground states of quantum many body systems. Part II focuses on the research projects carried out during my PhD research. In chapter 4 I present a machine-learning inspired method for solving quantum many body problems. This method addresses issues of representation of variational wave functions and their optimization. In chapter 5, I present a numerical study on the embedding of an exactly solvable point in a phase diagram of an extended Heisenberg model on a kagome lattice. Chapter 6 studies a quantum Heisenberg model on the stuffed honeycomb lattice. Phase transitions in quantum many body systems manifest themselves as a reorganization of low energy spectra of the system. They come in a variety of forms ranging from symmetry breaking transitions, where a local order parameter is formed and to topological phase transitions where the change of the ground state can be attributed to non-local changes in the entanglement patterns. In chapter 2, I review classic models studied in the context of frustrated magnetism and briefly go over the formulation of Landau symmetry breaking theory and common order parameters. The breakdown of Landau symmetry-breaking theory for topological phase transition is discussed using the example of Resonance Valence Bond (RVB) state, which serves as a qualitative exemplification of spin liquid phases. I discuss means of detecting phase transitions with a focus on methods that are applicable in a numerical setting. Chapter 3 presents elements of the numerical machinery that the rest of the thesis employs and builds upon. I discuss traditional trade-offs and provide rough metrics used for analysis of strengths and drawbacks of different approaches. Then I focus on Exact Diagonalization (ED) and Variational Monte Carlo (VMC) methods that are used in later chapters. In chapter 4, I describe the application of machine learning techniques to the variational approach to the quantum many body problem. I introduce Computational Graph States (CGS), a new class of variational wave functions based on the dataflow programming paradigm. The use of computational graphs makes composing new wave functions easy due to its modular structure. Automatic differentiation allows us to obtain efficient derivatives with respect to variational parameters automatically. We also develop a novel optimization scheme, Supervised Wave-function Optimization. It combines ideas of supervised learning and imaginary time evolution. Our method resolves a number of technical challenges and allows seamless optimization of complicated variational models. During the exploration process we constructed a new class of system size invariant model wave functions that can be trained on one system size and generalized to other system sizes with high accuracy. In chapter 5, we study the phase diagram of the kagome antiferromagnet in the J2, Jz plane, which corresponds to second nearest neighbor interactions and spin anisotropy. We report an exactly solvable point with macroscopic degeneracy at J2 = 0, Jz = −0.5 and show that many phases emanate from around this point. In addition, we find a previously unknown transition within the spin-liquid region suggesting a possibility of two distinct spin liquids. In chapter 6, we consider a Heisenberg model on the stuffed honeycomb lattice which interpolated between the triangular and honeycomb lattices. Using exact diagonalization we compute the phase diagram of this model. We find that for this model, the triangular lattice J1-J2 spin liquid region is extended to a larger domain that has lower symmetries than that of the triangular lattice. We perform a variational study of projected BCS wave functions which we directly compare to exact ground states on 36 site cluster as well as consider variational energies from larger system sizes. We argue that construction of symmetric states is essential to obtain good variational energies and give evidence that Dirac spin-liquid state is likely to characterize the spin liquid phase.