Classical and quantum spin systems are widely used models in both experimental and theoretical condensed matter physics. In many materials, the electronic interactions can be difficult to model exactly. However, in some insulators (Mott insulators), the magnetic (spin) interactions can be captured well with spin-only models. Several models are studied in this thesis. First, I report the solution of a long-standing issue in a classical frustrated spin model (i.e., where the quantum effects are neglected and the complexity is due to competing interactions): the nature of the thermal phase transition to a stripe state in the two dimensional (2D) J1-J2 model. Here J1 and J2 are nearest- and next-nearest-neighbor couplings. Monte Carlo simulations with single-spin updates are used for the calculations, and an extended-ensemble method, a generalization of the Wang-Landau algorithm, is also developed and tested. I focus on the study of "weak universality" behavior (continuously varying critical exponents, with one of the exponents staying fixed), for which I show a correspondence with a known class of conformal field theories with charge c = 1. Next, moving to quantum spins, to shed light on magnetic systems studied experimentally and to investigate new types of quantum states of interest in developing theories of quantum magnetism, I study the S = 1/2 Heisenberg model on the 2D square lattice with added six-spin interactions (the so-called J-Q3 model) as well as a set of 3D quantum antiferromagnets on dimerized lattices. Here I use the stochastic series expansion quantum Monte Carlo method. In the study of the J-Q3 model, I report on a similar weak-universality behavior as in the classical J1-J2 model, but with a mapping to a different known class of c = 1 conformal field theories. The critical behavior of the system again shows continuously changing exponents, in a way which corresponds to a gradual weakening of Z4 symmetry-breaking to an emergent U(1) symmetry. In the study of dimerized antiferromagnets, I report on a universal behavior of the Néel temperature TN, which can be related to ground state parameters independently of the microscopic interaction details in several different models.
Classical and quantum spin systems are widely used models in both experimental and theoretical condensed matter physics. In many materials, the electronic interactions can be difficult to model exactly. However, in some insulators (Mott insulators), the magnetic (spin) interactions can be captured well with spin-only models. Several models are studied in this thesis. First, I report the solution of a long-standing issue in a classical frustrated spin model (i.e., where the quantum effects are neglected and the complexity is due to competing interactions): the nature of the thermal phase transition to a stripe state in the two dimensional (2D) J1-J2 model. Here J1 and J2 are nearest- and next-nearest-neighbor couplings. Monte Carlo simulations with single-spin updates are used for the calculations, and an extended-ensemble method, a generalization of the Wang-Landau algorithm, is also developed and tested. I focus on the study of "weak universality" behavior (continuously varying critical exponents, with one of the exponents staying fixed), for which I show a correspondence with a known class of conformal field theories with charge c = 1. Next, moving to quantum spins, to shed light on magnetic systems studied experimentally and to investigate new types of quantum states of interest in developing theories of quantum magnetism, I study the S = 1/2 Heisenberg model on the 2D square lattice with added six-spin interactions (the so-called J-Q3 model) as well as a set of 3D quantum antiferromagnets on dimerized lattices. Here I use the stochastic series expansion quantum Monte Carlo method. In the study of the J-Q3 model, I report on a similar weak-universality behavior as in the classical J1-J2 model, but with a mapping to a different known class of c = 1 conformal field theories. The critical behavior of the system again shows continuously changing exponents, in a way which corresponds to a gradual weakening of Z4 symmetry-breaking to an emergent U(1) symmetry. In the study of dimerized antiferromagnets, I report on a universal behavior of the Néel temperature TN, which can be related to ground state parameters independently of the microscopic interaction details in several different models.
In this work the ordering processes in classical frustrated Ising models and the ground-state behavior of frustrated quantum spin models are investigated. In the first part the phase transition from a high-temperature paramagnetic phase to a antiferromagnetic ground state is analyzed for the square-lattice Ising model with competing antiferromagnetic interactions between nearest- and next-nearest-neighbor spins. In the case of spatially isotropic interactions the phase transition to a collinear state is of special interest. Using Monte-Carlo simulations on finite lattices as well as field-theoretical methods for the continuous model it is shown that depending on the frustration three different sectors can be distinguished that show (i) Ising-universal behavior (ii) Ashkin-Teller-like non-universal behavior (varying critical exponents) and (iii) non-critical first-order behavior. For spatially anisotropic interactions the emergence of an intermediate incommensurate phase is verified by means of Monte-Carlo simulations which has been predicted by earlier mean-field calculations. The nature of this phase is studied analyzing spin-spin correlation functions and its ordering wave-vector reveals a strong temperature and size dependence. In the second part the influence of quantum fluctuations on classical antiferromagnetic ground states are investigated in anisotropic quantum spin-1/2 Heisenberg models with competing interactions. Using quantum Monte-Carlo simulations, exact diagonalization and perturbation theory the ground-state phase diagrams are calculated for a square lattice with nearest- and next-nearest-neighbor interactions and a honeycomb lattice with up to third-nearest-neighbor interactions whereby the fluctuations are chosen to be non-frustrating. For both cases the stability regions of classical magnetic states are identified and an intermediate phase is detected that shows no sign of long-range magnetic order. Further calculations also rule out valence-bond-crystal or topological order for this state, thus a quantum disordered ground state is stabilized in these models.
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.
Strongly correlated low-dimensional quantum spin models provide a well-established frame- \nwork to study magnetic properties of insulators, and are of great theoretical interest and experimental relevance in condensed-matter physics. In this thesis, I use quantum Monte Carlo methods to numerically study quantum critical behavior in low-dimensional quantum spin models and wavefunctions. \n \nFirst, I study spinons &ndash emergent spin-1/2 bosonic excitations &ndash at certain one- and two-dimensional quantum phase transitions (QPTs) in spin models, by characterizing their size and confinement length quantitatively. In particular, I focus on the QPT from an antiferromagnetic (AFM) phase into a valence-bond solid (VBS) phase, which is an example of a violation of the standard Landau-Ginzburg-Wilson paradigm for phase transitions. This transition in two dimensions (2D) is instead likely described by a novel theory called "deconfined quantum criticality" (DQC). According to the theory, spinons should be deconfined. The degree of deconfinement is quantified in my calculations. \n \nSecond, I present a comprehensive study of so-called short-bond resonating-valence-bond (RVB) spin liquids in 2D, which have been suggested as a good starting point for understanding the spin physics of high-temperature cuprates. I find that these RVB states can also be classified as quantum-critical VBS states, which indicates that RVB is less disordered than expected. This work suggests a possible mapping from the quantum RVB states to classical dimer models via a classical continuum field theory--the height model. This map explicitly bridges well-established classical results to future quantum studies. \n \nThird, I consider 1D amplitude product (AP) states, which are generalized versions of RVB states, with different wavefunction weightings of bonds according to their lengths. AP states constitute a good ansatz for certain Hamiltonians and are of broad interest in quantum magnetism. I study phase transitions from AFM-VBS phases in AP states by tuning their amplitudes, and obtain continuously varying critical exponents. In addition, I classify the 1D AP states through entanglement entropy calculations of the central charge in (1+1)D conformal field theory. This new classification could serve as guide for AP states as trial wavefunctions to search for ground states of corresponding quantum spin models.
Frustrated spin systems exhibit a wide range of phenomena such as novel phases, residual entropy and glassy dynamics. In contrast to the usual assumption of universality, microscopic details such as the underlying lattice structure often play a crucial role in these systems. In this dissertation the frustrated J1-J2 Ising model on the honeycomb lattice is considered. Earlier work shows that the coupling ratio R = J2 /|J1 | = −1/4 separates regions of two different ground states. Above this threshold the ground state is (anti-)ferromagnetically ordered for (anti-)ferromagnetic J1 , whereas below it infinitely many states of minimal energy exist. A recent Monte Carlo (MC) study by Žukovič finds a continuous transition with Ising critical exponents for R ≥ −0.2, leaving open the question of criticality for −1/4 < R < −0.2, for which the study was obfuscated by the presence of strong metastability. The regime with the highly degenerate ground states (R < −1/4), on the other hand, is characterized by feature-rich specific heat curves, and slow low-temperature dynamics reminiscent of spin glasses, although there is no disorder in this system. Prior to this work, no order parameter has been identified, and there is no clear understanding of its potentially critical behavior. Here, the question of criticality for −1/4 < R < −0.2 is addressed, and Ising universality is found for lower values of R, namely, at least for R ≥ −0.23. This result is achieved by carrying out finite-size scaling analyses of response functions like the magnetic susceptibility, and of the complex partition function zeros. In both cases, the underlying numerical data was generated using population annealing (PA) MC, a framework devised for spin glasses but generally applicable to systems with metastability. Furthermore, the region of degenerate ground states is studied using different boundary conditions, which are found to strongly affect the thermodynamic observations. A broken symmetry at zero temperature is identified, which can be quantified using a complex nematic order parameter. The MC data are consistent with a continuous symmetry-breaking transition. However, due to the limited system sizes that could be studied, and the presence of strong finite-size effects, one cannot rule out a higher-order transition or a cross-over without any singularities. In the process of studying this system, also the understanding of the PA framework was further developed. Specifically, the role of the resampling step in the algorithm was investigated. While it is crucial for efficient equilibration, no benefit is found in noisier resampling schemes such as the multinomial resampling method. Instead, two resampling schemes, which have the lowest sampling variance, are identified as preferential.:1 Introduction I Background and Methodology 2 Background 2.1 Fundamentals of Statistical Mechanics 2.1.1 Equilibrium ensembles 2.1.2 Phase transitions and critical behavior 2.1.3 Common spin models 2.2 Frustrated Spin Models 2.2.1 Frustrated J1-J2 Ising model on the square lattice 2.2.2 Gonihedric model in two dimensions 2.3 Monte Carlo Simulations 2.3.1 Fundamentals 2.3.2 Autocorrelation and error bars in MCMC time series 2.3.3 Possible implementations 2.3.4 Generalized ensemble methods 3 Methodology 3.1 General Algorithmic Design 3.2 Population Annealing 3.2.1 Algorithm 3.2.2 Assessing population annealing performance 3.2.3 Adaptive temperature steps 3.2.4 Weighted averages 3.2.5 Population annealing and first-order transitions 3.3 Estimating Critical Exponents from Finite Systems 3.3.1 Standard finite size scaling 3.3.2 Complex partition function zeros II Results 4 Algorithmic Improvements and Extensions to the Population Annealing Framework 4.1 Resampling Step in Population Annealing 4.1.1 Conceptual overview 4.1.2 Possible resampling methods 4.1.3 Benchmarking different resampling methods 4.1.4 Simulation results 4.1.5 PA simulations with finite populations in the limit of perfect equilibration 4.1.6 PA simulations in the limit of infinite populations and perfect equilibration 4.1.7 Conclusion 4.2 Using Population Annealing on Many Computing Nodes 4.2.1 Literature overview and different rebalancing schemes 4.2.2 Simulations of simulations 4.2.3 Conclusion and outlook 4.3 Adaptive Number of Spin Updates in Population Annealing 4.4 Microcanonical Population Annealing 5 The Frustrated J1-J2 Ising Model on the Honeycomb Lattice 5.1 Overview 5.2 Regime of Ising Universality Class 5.2.1 Finite-size scaling analysis 5.2.2 Energy barriers 5.2.3 Exploring the complex partition function zeros 5.3 Finite Ground State Entropy per Spin for J2/|J1| = −1/4 5.4 Highly Degenerate Striped Phase for J2/|J1| < −1/4 5.4.1 Boundary conditions 5.4.2 Energy minimization and ground states 5.4.3 Topological defects 5.4.4 Observables 5.4.5 Simulation details 5.4.6 Simulation results for R = −0.5 5.4.7 Simulation results for R = −1 5.4.8 Discussion 5.4.9 Conclusion and outlook 6 Parallel tempering and its infinite-swap limit 6.1 Formalism 6.2 Details on Implementation 6.3 Numerical Results from INS-PT Simulations 6.3.1 One-dimensional Ising model 6.3.2 Two-dimensional Ising model 6.3.3 q-state Potts model 6.3.4 Lennard-Jones clusters 6.4 Discussion 7 Conclusions and outlook Bibliography Appendices A Algorithmic Improvements and Extensions to the Population Annealing framework: Further details A.1 Generating correlated energy time series B J1-J2 honeycomb Ising model: Further plots and details B.1 GPU domain decomposition B.2 Finite-size scaling using Metropolis population annealing B.3 The decoupled antiferromagnet limit B.4 Zero temperature value of the nematic order parameter B.5 Percolation properties for J2 = −0.5 B.6 Binder parameter of the one-dimensional Ising model C Application for Ethical Approval
This thesis investigates exotic phases of matter in magnetic materials that present anisotropic properties in two dimensions. Using the microscopic spin Hamiltonians, the study addresses two problems: (i) the appearance of a novel class of magnetic excitations in the spectra of anisotropic magnets at zero temperature, and (ii) the quantum-to-classical correspondence of spins for the description of magnetic solids in the disordered phase above the critical temperature. Different analytical and numerical means are used to this purpose.In the first part, motivated by experimental observations of multipolar excitations in afamily of strongly-anisotropic van der Waals antiferromagnets, we have developed ageneral theory for these excitations as a distinct type of quasi-particles. We called themlongitudinal magnons. They appear in easy-axis magnetic materials, and are characterized by the quantum number Sz = ±2S. They can exist as coherent long-lived excitations for D ≥ |J|, where D and J are the typical single-ion and exchange coupling constants, respectively. Starting with the strong-coupling limit, this work presented different approaches to compute the dispersion of longitudinal magnons. First, a mapping to an effective spin-1/2 model was performed and illustrated for several lattice spin models: the honeycomb-lattice zigzag antiferromagnet and the square-lattice ferro- and antiferromagnet. The excitation energy was derived for different spin values S = 1, 3/2, 2and 5/2. Second, a thorough study explored the S = 1 anisotropic square-lattice, wherethe analytical results of the effective model are compared to the multi-boson spin-wavetheory, the linked-cluster calculations in antiferromagnetic coupling J > 0, and the exactsolution of the two-magnon problem in ferromagnetic coupling J < 0. In the limit of largeJ/D, the longitudinal magnons start to overlap with the multi-magnon continuum, andtheir finite lifetime can be computed using the multi-boson theory, which is specificallyshown for two-magnon decay processes in S = 1. Moreover, critical values of J/D forthe stability of longitudinal magnons are obtained along high-symmetry lines of the Brillouin zone. Third, we investigated the appearance and condensation of the longitudinal magnons in triangular S = 1 antiferromagnets under high magnetic field. As suggested by recent experiments on a candidate material, a spin nematic phase may appear upon condensation of the multipolar excitations, opening the way for realizations of new phases of matter in classically-ordered magnets.In the second part, the correspondence between quantum and classical spins is investigated for spin Hamiltonians at finite temperature. Given a model of quantum spins, it is useful to determine the classical model that best describes its thermodynamics, which is of interest for semi-classical analyses, e.g. by classical Monte Carlo methods. Based on high-temperature series expansions, we established the scaling laws of the different microscopic constants in a quantum spin model, involving quadratic and biquadratic exchange interactions, single-ion anisotropy and the Zeeman energy. We showed that quantum corrections to the thermodynamic quantities are power series in S(S + 1). Subsequently, the findings were applied to numerically determine the phase transition temperature in several ferro- and antiferromagnetic compounds, using classical Monte Carlo simulations. To this aim, the lattice spin models were constructed with microscopic coupling constants from the literature, and the transition temperatures were compared to the experimental values.This illuminates complex collective magnetic behaviors in different models and materials.
Magnetic frustrations are the fundamental cause of many interesting phenomena in materials that have been intensely studied since the middle of the 20th century. Magnetic systems with competing interactions, ranging from insulating antiferromagnets to arrays of superconducting qubits, exhibit nontrivial quantum e ects that arise from the need to satisfy a jagged free energy landscape. The search for and characterization of quantum spin liquids, nontrivial topological phases, and quantum phase transitions is conducted through investigations of such frustrated models. Within the large umbrella of frustrated magnetism I have studied some speci c models with the purpose of answering the following questions: (1) What are the interesting phases and less-interesting phases, and how do they fall in the phase diagram relative to one another? (2) What properties do the phases have, and how can we characterize what happens when one phase transitions to another? Armed with the answers to these questions, researchers can further explore these interesting phenomena and apply them towards new materials and devices. In this thesis, I rst develop a computational methodology for studying both classical and quantum spin models using various forms of Monte Carlo simulation. I use classical Monte Carlo to study highly-frustrated models of Ising spins both at nite temperatures and in the T = 0 limit, to map the phase diagram and determine the universality class of phase transitions. Using these methods, I show that in a dissipative chain of qubits, bath-induced qubit-qubit interactions lead to collective decoherence above a critical dissipation strength. This causes the qubits to lose all stored information and become fully correlated with one another, an e ect that one would need to mitigate in order to perform quantum computation. In another study, I show that in antiferromagnetic insulators with anisotropic interactions, frustration can cause an intermediate paramagnetic phase with oblong \droplets" of correlated spins. This phenomenon is quite general and the presence of these droplets can help explain experimental signatures that occur at unexpected temperatures in such materials. In addition to the classical simulations, I develop a variational Monte Carlo (VMC) technique to search for ground states in Heisenberg-type models, speci cally for spins S = 1. The case of S = 1 is more complicated than the case for S = 1=2 due to an additional local degree of freedom, which has interesting implications for the possible states that can be realized. Using semiclassical energy minimization as well as VMC, I show that spin S = 1 diamond-lattice antiferromagnets experience higher-order spinspin interactions that destroy previously theorized liquid-like spiral spin orderings. However, the presence of these higher-order interactions can also potentially allow other phases such as quantum spin liquids and topological paramagnets. Such results can provide an avenue for explaining anomalous phenomena like low-temperature paramagnetism in future experimental materials.