In this thesis, we investigate frustrated systems with focus on the triangular lattice anisotropic Heisenberg antiferromagnet. Firstly, we examined the material CeMgAl11O19 as a potential quantum spin liquid utilizing the density matrix renormalization group method (DMRG). Two different low-temperature effective models have been proposed. The first was motivated by similar systems and contains long-range exchange interac- tion. The second, introduced much later, assumes only nearest-neighbor interactions and exhibits strong anisotropy, mixing ferromagnetic and antiferromagnetic couplings. We showed that the first model does not reproduce the experimental results even qual- itatively, but the second is consistent with the measured data. Our analysis demon- strated that CeMgAl11O19 is farther away from the critical point than previously claimed in the literature. We then investigated the second model in more detail and examined its complicated ground state phase diagram in magnetic field. The complicated struc- ture of the diagram was analyzed by unsupervised machine learning techniques as well as by employing order parameters constructed from structure factors. Finally, we used the recently introduced variational Monte Carlo based on neural network quantum states (NNQS) to investigate the ground state of...
In this thesis, we investigate frustrated systems with focus on the triangular lattice anisotropic Heisenberg antiferromagnet. Firstly, we examined the material CeMgAl11O19 as a potential quantum spin liquid utilizing the density matrix renormalization group method (DMRG). Two different low-temperature effective models have been proposed. The first was motivated by similar systems and contains long-range exchange interac- tion. The second, introduced much later, assumes only nearest-neighbor interactions and exhibits strong anisotropy, mixing ferromagnetic and antiferromagnetic couplings. We showed that the first model does not reproduce the experimental results even qual- itatively, but the second is consistent with the measured data. Our analysis demon- strated that CeMgAl11O19 is farther away from the critical point than previously claimed in the literature. We then investigated the second model in more detail and examined its complicated ground state phase diagram in magnetic field. The complicated struc- ture of the diagram was analyzed by unsupervised machine learning techniques as well as by employing order parameters constructed from structure factors. Finally, we used the recently introduced variational Monte Carlo based on neural network quantum states (NNQS) to investigate the ground state of...
In spin systems, geometrical frustration describes the impossibility of minimizing simultaneously all the interactions in a Hamiltonian, often giving rise to macroscopic ground-state degeneracies and emergent low-temperature physics. In this thesis, combining tensor network (TN) methods to Monte Carlo (MC) methods and ground-state energy lower bound approaches, we study two-dimensional frustrated classical Ising models. In particular, we focus on the determination of the residual entropy in the presence of farther-neighbor interactions in kagome lattice Ising antiferromagnets (KIAFM). In general, using MC to determine the residual entropy is a significant challenge requiring ad-hoc updates, a precise evaluation of the energy at all temperatures to allow for thermodynamic integration, and a good control of the finite-size scaling behavior. As an alternative, we turn to TNs; however, we argue that, in the presence of frustration and macroscopic ground-state degeneracy, standard algorithms fail to converge at low temperatures on the usual TN formulation of partition functions. Inspired by methods for constructing ground-state energy lower bounds, we propose a systematic way to find the ground-state local rule using linear programming. Characterizing the rules as tiles that can be tessellated to form ground states of the model gives rise to a natural contractible TN formulation of the partition function. This method provides a direct access to the ground-state properties of frustrated models and, in particular, allows an extremely precise determination of their residual entropy. We then study two models inspired by artificial spin systems on the kagome lattice with out-of-plane (OOP) anisotropy. The first model is motivated by experiments on an array of chirally coupled nanomagnets. We argue that the farther-neighbor to nearest-neighbor couplings ratios in this system are much smaller than in the dipolar case, J2/J1 being of the order of 2%. A comparison of the experimental correlations with the results of extensive TN and MC simulations shows that (1) the experimental second- and third-neighbor correlations are inverted as compared to those of a pure nearest-neighbor model at equilibrium (even with a magnetic field), and (2) second-neighbor couplings as small as 1% of the nearest-neighbor couplings will affect the spin-spin correlations even at fairly high temperatures. Motivated by dipolar coupled artificial spin systems, we turn to the progressive lifting of the ground-state degeneracy of the KIAFM. We provide a detailed study of the ground-state phases of this model with up to third neighbor interactions, for arbitrary J2, J3 such that J1 >> J2, J3, obtaining exact results for the ground-state energies. When all couplings are antiferromagnetic, we exhibit three macroscopically degenerate ground-state phases and establish their residual entropy using our TN approach. Furthermore, in the phase corresponding to the dipolar KIAFM truncated to third neighbors, we use the ground-state tiles to establish the existence of a mapping to the ground-state manifold of the triangular Ising antiferromagnet.
This thesis is devoted to the numerical investigation of quantum spin models which describe the low-energy physics of frustrated magnets. At extremely low temperatures, these systems can host the so-called spin liquid phase, an unconventional state of matter characterized by a high degree of quantum entanglement and the absence of magnetic order. The experimental identification of the spin liquid phase in actual materials relies on the detection of its distinctive excitations (named spinons), which possess fractional quantum numbers and can be probed by inelastic neutron scattering experiments. From the theoretical point of view, variational methods have been largely employed to tackle ground state properties of frustrated spin models. In particular, variational Monte Carlo techniques based on Gutzwiller-projected fermionic wave functions have been shown to provide accurate results for several frustrated systems. In this thesis, we pursue an extension of this variational scheme to target dynamical spectra, which are directly measured by inelastic neutron scattering experiments. Specifically, we compute the dynamical structure factor by constructing approximate excited states, which are obtained by applying two-spinon operators to the ground state wave function. Our results prove that this variational method can accurately describe the spectral features of different spin systems. Focusing on prototypical frustrated models on the square and triangular lattices, we observe how the dynamical structure factor reflects the phase transition between a magnetically ordered phase with spin wave excitations, to a spin liquid state with fractional degrees of freedom. In addition to spectral properties, we also explore new directions to improve the accuracy of Gutzwiller-projected states by the application of a neural network correlator, in the form of a restricted Boltzmann machine. While this hybrid variational scheme provides a considerable improvement of the variational energy in the case of unfrustrated spin models, less satisfactory results are obtained for frustrated systems, which call for further refinements of the neural network variational Ansätze.
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.
The study of quantum frustrated systems remains one of the most challenging subjects of quantum magnetism, as they can hold quantum spin liquids, whose characterization is quite elusive. The presence of gapped quantum spin liquids possessing long-range entanglement while being locally indistinguishable often demands highly sophisticated numerical approaches for their description. Here we propose an easy computational method based on exact diagonalization with engineered boundary conditions in very small plaquettes. We apply the method to study the quantum phase diagram of diverse antiferromagnetic frustrated Heisenberg models in the triangular lattice. Our results are in qualitative agreement with previous results obtained by means of sophisticated methods like density matrix renormalization group (2D-DMRG) or variational quantum Monte Carlo.
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.
This thesis is dedicated to the analytical and numerical study of quantum entanglement and nonlocality in strongly correlated quantum spin tube models in connection with various thermodynamic properties and magnetization processes. More specifically, the study builds on the analysis of the behavior of these physical quantities within a particular spinning network that includes competitive interactions and quantum fluctuations known as a three-leg spins ladder or triangular spin tube, due to the periodic boundary conditions imposed on the three spins that form the Heisenberg steps of this network. In systems with this structure described above, magnetization plateaus are formed due to the competition imposed by the geometric frustration that the network imposes on the spins and the applied external field. Such competition is the mechanism that defines the orientation of the magnetic domains that the network will present, which within specific regions of the network parameters space create regions where there is a discontinuous response of local magnetization to external field variations, such regions are known as magnetization plateaus. To evaluate the frustrated spin- ½ Ising-Heisenberg triangular tube consisting of a spin tube formed by triangular unit cells with 3 spins-½ interconnected by Heisenberg couplings between neighboring first spins and the pairs of neighboring first spins belonging to the different cells have their interaction intermediated through classical Ising couplings, an exact analytical procedure, known as a transfer-matrix technique, was used to obtain the analytical solutions for various thermodynamic properties of the model, including magnetization, specific heat and magnetization profiles. These quantities were evaluated as a function of temperature, external magnetic field and the relevant couplings. However, it was necessary to apply a different methodology to obtain thermodynamic measurements for the other model proposed in this thesis: the frustrated spin- ½ Heisenberg triangular tube, which consists of a spin tube with similar structure to the previous one, except that the pairs of the first neighboring spins belonging to the different cells interact through Heisenberg-type quantum connections. In this case, a numerical procedure known as the quantum density matrix renormalization group method, known as DMRG, is used to obtain the ground state diagrams, magnetization processes, and bipartite quantum entanglement of the Heisenberg triangular tubes as a function of the external magnetic field and the relevant couplings. In both models, the magnetization curves exhibit discontinuous magnetization jumps and intermediate plateaus due to discontinuous phase transitions induced by the external field, These transitions are also reflected in the exotic behavior of some thermodynamic quantities of the models, such as the presence of a striking low temperature peak for the specific heat curve and a highlighted magnetocaloric effect.