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 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.
Frustrated magnetism provides a rich platform for realizing exotic quantum phases in condensed matter systems. Among these, quantum spin liquids (QSLs) are a particularly intriguing class of states that exhibit no magnetic ordering and preserve all symmetries down to zero temperature. Just as conventional magnets can be thought of as spin crystals,QSLs represent their liquid analogs. This thesis explores two broad types of QSLs in two-dimensional magnets, with a focus on their dynamic properties and the nature of transitions from conventional magnetic states into QSL phases. QSLs are topological phases that host fractionalized excitations known as spinons. In these states, mobile spinons are coupled to an emergent gauge field, forming a highly entangled phase of matter. The first part of this thesis focuses on Dirac spin liquids (DSLs), a class of QSLs where spinons exhibit a gapless Dirac dispersion. To study the dynamicsof these spin liquids beyond leading-order mean-field theory, we develop a novel large-N fermionic limit of gapless spinons. In this limit, all gauge excitations are suppressed, and the leading-order corrections instead arise from order parameter fluctuations, which may condense and drive transitions into symmetry-broken phases from the parent DSL. Incorporating random phase approximation (RPA) corrections to the spin susceptibility, we find that sharp, spin-wave–like modes are recovered in the excitation spectrum. This methodology successfully captures characteristic features of the dynamics near phase transitions and in weakly symmetry-broken phases proximate to a DSL. We apply this approach to microscopic models, including the J1-J2 XXZ model on the triangular lattice and the J1-J3 XXZ model on the honeycomb lattice. These models closely represent a wide range of frustrated materials, such as triangular and honeycomb cobaltates and triangular ytterbium compounds. Our theoretical predictions show excellent agreement with experimentally measured inelastic neutron scattering spectra from these materials. Gapless spinons can also sometimes become gapped through chiral symmetry breaking, leading to the formation of a chiral spin liquid (CSL). The CSL is an archetypal bosonic realization of a fractional quantum Hall (FQH) state at half-filling, and has been proposed as a time-reversal–breaking ground state in a variety of frustrated spin models. The second part of this thesis addresses the intriguing possibility of realizing a CSL through the "melting" of a chiral spin crystal—a conventional magnet with non-coplanar ordering. Such a transition offers a physically appealing analogue to the melting of a Wigner crystal into a fermionic FQH liquid. Focusing on the kagome lattice, where chiral interactions are knownto stabilize a CSL ground state, we study the effects of additional long-range Heisenberg interactions and demonstrate a phase transition into a non-coplanar octahedral magnetic order. To describe the universal class of transition between a CSL and a non-coplanar anti-ferromagnetic order, we then develop a novel Chern-Simons–matter field theory framework, constrained by symmetry principles and symmetry fractionalization patterns. This thesis thus provides new insights into the dynamical signatures and phase transitions of quantum spin liquids, offering a bridge between theoretical models and experimental observations in frustrated magnetic systems.
Variational wave functions have enabled exceptional scientific breakthroughs related to the understanding of novel phases of matter. Examples include the Bardeen-Cooper-Schrieffer theory of superconductivity, the description of the fractional quantum Hall effect through the Laughlin state, and Feynman's variational understanding of large-scale quantum effects in liquid helium. More recently, Gutzwiller-projected wave functions, typically constructed from fermionic degrees of freedom, have been employed to examine quantum spin models in the presence of competing interactions, where exotic phases with no spontaneous symmetry breaking and fractional excitations may exist. In this work, we investigate the aforementioned fermionic wave functions supplemented with neural networks, specifically with the so-called restricted Boltzmann machine (RBM), to boost their accuracy and obtain reliable approximations to the ground state of generic spin models. In particular, we apply our neural augmented fermionic construction to the description of both magnetically ordered and disordered phases of increasing complexity, including cases where the ground state displays a nontrivial sign structure. Even though the RBM state is by far more effective for N\'eel states endowed with a particularly simple sign structure, it provides a significant improvement over the original fermionic state in highly frustrated regimes where a complex sign structure is anticipated, thus marking the path to an understanding of strongly correlated spin models on the lattice via neural Gutzwiller-projected variational wave functions.
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.
Understanding the nature of the excitation spectrum in quantum spin liquids is of fundamental importance, in particular for the experimental detection of candidate materials.However, current theoretical and numerical techniques have limited capabilities, especially in obtaining the dynamical structure factor, which gives a crucial characterization of the ultimate nature of the quantum state and may be directly assessed by inelastic neutron scattering.In this work, we investigate the low-energy properties of the S ¼ 1=2 Heisenberg model on the triangular lattice, including both nearest-neighbor J 1 and next-nearestneighbor J 2 superexchanges, by a dynamical variational Monte Carlo approach that allows accurate results on spin models.For J 2 ¼ 0, our calculations are compatible with the existence of a well-defined magnon in the whole Brillouin zone, with gapless excitations at K points (i.e., at the corners of the Brillouin zone).The strong renormalization of the magnon branch (also including rotonlike minima around the M points, i.e., midpoints of the border zone) is described by our Gutzwiller-projected state, where Abrikosov fermions are subject to a nontrivial magnetic π flux threading half of the triangular plaquettes.When increasing the frustrating ratio J 2 =J 1 , we detect a progressive softening of the magnon branch at M, which eventually becomes gapless within the spin-liquid phase.This feature is captured by the band structure of the unprojected wave function (with two Dirac points for each spin component).In addition, we observe an intense signal at low energies around the K points, which cannot be understood within the unprojected picture and emerges only when the Gutzwiller projection is considered, suggesting the relevance of gauge fields for the low-energy physics of spin liquids.
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.
We investigate the ground-state properties of two strongly-correlated systems: the two-dimensional t−J model that can be used to study the high-temperature superconducting phase in the doped antiferromagnet, and the frustrated N\'eel antiferromagnet described by the J1−J2 Heisenberg model on the square lattice, which is widely considered as the prototype model for spin frustration. In this thesis, we apply to these two systems state-of-the-art quantum Monte Carlo techniques, including the variational and the Green's function Monte Carlo with the fixed-node approximation. Few Lanczos steps are used to systematically improve the accuracy of the trial wave functions. By introducing a suitable regularization scheme for the variational calculations with few Lanczos steps, stable and controllable simulations can be performed up to very large cluster sizes with very good accuracy. In the two-dimensional t−J model at J/t=0.4, we show that the accuracy of the Gutzwiller-projected variational state (containing dx2−y2 pairing) can be improved much by few Lanczos steps; in addition, the fixed-node Monte Carlo with these systematically improvable trial wave functions gives results that are comparable with the best accurate DMRG ones. Our main outcome is that the ground state is homogeneous and no evidence of stripes is detected around the doping δ=1/8. Indeed, our best approximation to the ground state does not show any tendency towards charge inhomogeneity. Furthermore, our results show that a uniform state containing superconductivity and antiferromagnetism is stabilized at low hole doping, i.e., δ≲0.1. In the J1−J2 Heisenberg model on the square lattice, we use the projected mean-field state that is built from Abrikosov fermions having a Z2 gauge structure and four Dirac points in the spinon spectrum. No spin or dimer order is found in the strongly frustrated regime and our calculations imply that a spin liquid phase may faithfully represent the exact ground state around J2/J1=0.5. The few Lanczos step technique is used to systematically improve the accuracy of the variational states both for the ground state and for few relevant low-energy excitations. This procedure allows us to estimate, in a valuable and systemic way, the spin gaps within thermodynamical limit and to show a solid evidence of an unconventional gapless excitation spectrum in the strongly frustrated regime, i.e., J2/J1≃0.5. In particular, we found gapless triplet excitations at momenta (π,0) and (0,π), which are compatible with the presence of four Dirac points at momenta (±2π,±2π) in the spinon spectrum.