This thesis, comprising three publications, explores the efficacy of novel generalization of the fidelity susceptibility and their numerical application to the study of frustrated quantum phase transitions in two and one dimensions. Specifically, they will be used in exact diagonalization studies of the various limiting cases of the anisotropic next-nearest neighbour triangular lattice Heisenberg model (ANNTLHM). These generalized susceptibilities are related to the order parameter susceptibilities and spin stiffness and are believed to exhibit similar behaviour although with greater sensitivity. This makes them ideal for numerical studies on small systems. Additionally, the utility of the excited-state fidelity and twist boundary conditions will be explored. All studies are done through numerical exact diagonalization. In the limit of interchain couplings going to zero the ANNTLHM reduces to the well studied $J_1-J_2$ chain with a known, difficult to identify, BKT-type transition. In the first publication of this work the generalized fidelity susceptibilities introduced therein are shown to be able to identify this transition as well as characterize the already understood phases it straddles. The second publication of this work then seeks to apply these generalized fidelity susceptibilities, as well as the excited-state fidelity, to the study of the general phase diagram of the ANNTLHM. It is shown that the regular and excited-state fidelities are useful quantities for the mapping of novel phase diagrams and that the generalized fidelity susceptibilities can provide valuable information as to the nature of the phases within the mapped phase regions. The final paper sees the application of twisted boundary conditions to the anisotropic triangular model (next-nearest neighbour interactions are zero). It is demonstrated that these boundary conditions greatly enhance the ability to numerically explore incommensurate physics in small systems.
This thesis, comprising three publications, explores the efficacy of novel generalization of the fidelity susceptibility and their numerical application to the study of frustrated quantum phase transitions in two and one dimensions. Specifically, they will be used in exact diagonalization studies of the various limiting cases of the anisotropic next-nearest neighbour triangular lattice Heisenberg model (ANNTLHM). These generalized susceptibilities are related to the order parameter susceptibilities and spin stiffness and are believed to exhibit similar behaviour although with greater sensitivity. This makes them ideal for numerical studies on small systems. Additionally, the utility of the excited-state fidelity and twist boundary conditions will be explored. All studies are done through numerical exact diagonalization. In the limit of interchain couplings going to zero the ANNTLHM reduces to the well studied J1−J2 chain with a known, difficult to identify, BKT-type transition. In the first publication of this work the generalized fidelity susceptibilities introduced therein are shown to be able to identify this transition as well as characterize the already understood phases it straddles. The second publication of this work then seeks to apply these generalized fidelity susceptibilities, as well as the excited-state fidelity, to the study of the general phase diagram of the ANNTLHM. It is shown that the regular and excited-state fidelities are useful quantities for the mapping of novel phase diagrams and that the generalized fidelity susceptibilities can provide valuable information as to the nature of the phases within the mapped phase regions. The final paper sees the application of twisted boundary conditions to the anisotropic triangular model (next-nearest neighbour interactions are zero). It is demonstrated that these boundary conditions greatly enhance the ability to numerically explore incommensurate physics in small systems.
Ground- and excited-state quantum fidelities in combination with generalized quantum fidelity susceptibilites, obtained from exact diagonalizations, are used to explore the phase diagram of the anisotropic next-nearest-neighbour triangular Heisenberg model. Specifically, the J'-J2 plane of this model, which connects the J1-J2 chain and the anisotropic triangular lattice Heisenberg model, is explored using these quantities. Through the use of a quantum fidelity associated with the first excited-state, in addition to the conventional ground-state fidelity, the BKT-type transition and Majumdar-Ghosh point of the J1-J2 chain (J'=0) are found to extend into the J'-J2 plane and connect with points on the J2=0 axis thereby forming bounded regions in the phase diagram. These bounded regions are then explored through the generalized quantum fidelity susceptibilities χρ, χ₁₂₀°, χD and χCAF which are associated with the spin stiffness, 120° spiral order parameter, dimer order parameter and collinear antiferromagnetic order parameter respectively. These quantities are believed to be extremely sensitive to the underlying phase and are thus well suited for finite-size studies. Analysis of the fidelity susceptibilities suggests that the J', J2≪J phase of the anisotropic triangular model is either a collinear antiferromagnet or possibly a gapless disordered phase that is directly connected to the Luttinger phase of the J1-J2 chain. Furthermore, the outer region is dominated by incommensurate spiral physics as well as dimer order.
This work introduces a new technique for the determination of quantum phase transitions known as the Conjugate Field Fidelity Susceptibility (CFFS) approach. The CFFS method draws highly on the Quantum Information concept of quantum fidelity susceptibility. An initial introduction to the computational tricks needed to feasibly calculate quantum fidelities will be given. Following this will be a discussion of the nature, derivation and applications of the quantum fidelity and fidelity susceptibilities. Then the CFFS technique will be described before applying it to identify a BKT(Berezinsky-Kosterlitz-Thouless) transition in the Next-Nearest Neighbour Heisenberg Spin Chain. Finally, the method will be applied to confirm the more robust phase diagram of the Next-Nearest Neighbour Heisenberg model on an anisotropic triangular lattice.
We study slightly generalized quantum fidelity susceptibilities where the differential change in the fidelity is measured with respect to a different term than the one used for driving the system toward a quantum phase transition. As a model system we use the spin-1/2J1-J2 antiferromagnetic Heisenberg chain. For this model, we study three fidelity susceptibilities, ${\ensuremath{\chi}}_{\ensuremath{\rho}}$, ${\ensuremath{\chi}}_{D}$, and ${\ensuremath{\chi}}_{\mathrm{AF}}$, which are related to the spin stiffness, the dimer order, and antiferromagnetic order, respectively. All these ground-state fidelity susceptibilities are sensitive to the phase diagram of the J1-J2 model. We show that they all can accurately identify a quantum critical point in this model occurring at ${J}_{2}^{c}\ensuremath{\sim}0.241{J}_{1}$ between a gapless Heisenberg phase for J2<J2c and a dimerized phase for J2>J2c. This phase transition, in the Berezinskii-Kosterlitz-Thouless universality class, is controlled by a marginal operator and is therefore particularly difficult to observe.
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.
We present three different methods of finding the critical spin value Sc of the quantum phase transition between the spin liquid and the Néel order phase on the mean field triangular Heisenberg antiferromagnet at T=0K. These methods are: using the sublattice magnetization as an order parameter, looking at the scaling of the energy difference of topologically degenerate states in different phases, and the quantum fidelity approach. All of these methods are able to pick up the signal of the phase transition, and their estimated numerical values of Sc agree for systems where the relative strength of next nearest neighbour interaction compared to the nearest neighbour interaction is small. For the relative interaction strength j=0 and j=0.1, all three methods predict Scapprox0.21 and Scapprox0.25 respectively, while j=0.2 gives deviating results. This deviation might be caused by the influence of other magnetically ordered states that we have not accounted for in our calculations, and so we restrict the validity of our results to j<0.125.
In spin lattices, frustration happens when there exists a competition between different terms of the Hamiltonian, which cannot be simultaneously minimized in all local physical bases. Frustration can destabilize the bipartite Neel order, which is the conventional groundstate of antiferromagnetic Heisenberg-type models in two dimensions. Hence, the focus of two-dimensional quantum magnetism studies has been directed toward elucidating the zero-temperature properties of Heisenberg-type models on the Archimedean lattices that exhibit geometrical frustration. This type of frustration is of great interest since it can lead to exotic forms of the quantum matter, such as topological spin-liquid (SL) and many-sublattice long-range-ordered phases. The spin-1/2 triangular Heisenberg model (THM) is a prototypical model with geometrical frustration, which has the highest coordination number of all Archimedean lattices. However, there exist two major, distinct difficulties for numerical methods to efficiently determine the groundstate properties of the THM: the immense dimension of the Hilbert space and the geometrical frustration phenomenon. In this thesis, to deal with such difficulties, we employ the state-of-the- art non-Abelian matrix product states (MPS) and density matrix renormalization group (DMRG) methods. We exploit the symmetries of the Hamiltonian to reduce the working dimension of the Hilbert space and, to lessen the effects of the geometrical frustration, we depend on the local entanglement (area-law) nature of the MPS as it scales with the size of a lattice subspace with reduced spatial dimensions. In such a way, we thoroughly examine the phase diagram of the THM with nearest and next-nearest neighbor (NN and NNN) exchange couplings on finite- and infinite-length cylinders of widths up to 12 sites. In addition, we develop new numerical tools to analyze topological and symmetry-broken phases in the context of the SU(2)-symmetric translation-invariant MPS algorithm. We establish that on infinite cylinders, the anyonic sectors of a topological order can be classified through the fractionalization of global symmetries and degeneracy patterns of the entanglement spectrum (ES). On the other hand, we detect symmetry-breaking phase transitions by a combination of the correlation lengths and second and fourth cumulants of the magnetic order parameters, even though symmetry implies that the order parameter itself is strictly zero. Furthermore, the appearance of Nambu-Goldstone modes in the excitation spectrum can be detected using “tower of states” (TOS) levels in the momentum-resolved ES. By applying the new tools to the THM, we discover a variety of exotic groundstates such as a Z2-gauge toric-code-type topological order (with vison and spinon excitations), a NNN Majumdar-Gosh, and a long-range 120◦-ordered state.