We study two different classes of strongly interacting many-body quantum systems in this thesis. The first concerns interacting quantum system with strong disorder. The second is a quantum spin liquid problem. Since many-body quantum systems with strong interaction are difficult in analytic approaches, numerical methods will be used to study these systems. In the first part of this thesis, we investigate the many-body localization of the 1D spin-1/2 XXZ model with random magnetic field by Exact Diagonalization. We confirm previous results that there exists a mobility edge between ergodic and localized phases at weak disorder. Furthermore, we study the difference between two many-body localization phases in the XXZ model and confirm that there exists a phase transition between two kinds of many-body localization. Surprisingly, we observe a narrow region of ergodic phase that exists between the two MBL phases. We call the phase Many-Body Ergodic phase since the phase is not analytically connected to the trivial ergodic phase at weak order. In the second part of this thesis, we study the spin -1/2 XXZ spin model on the triangular lattice by mean of Variational Monte Carlo simulation on a class of resonant valence bond (RVB) trial wave-functions. We find that as a function of J⊥/Jz, there exists a transition point where magnetic order vanishes (at J⊥/Jz ~ 0.45). Moreover, two kinds of RVB states competed in the Ising regime when J⊥/Jz < 0.23.
We study two different classes of strongly interacting many-body quantum systems in this thesis. The first concerns interacting quantum system with strong disorder. The second is a quantum spin liquid problem. Since many-body quantum systems with strong interaction are difficult in analytic approaches, numerical methods will be used to study these systems. In the first part of this thesis, we investigate the many-body localization of the 1D spin-1/2 XXZ model with random magnetic field by Exact Diagonalization. We confirm previous results that there exists a mobility edge between ergodic and localized phases at weak disorder. Furthermore, we study the difference between two many-body localization phases in the XXZ model and confirm that there exists a phase transition between two kinds of many-body localization. Surprisingly, we observe a narrow region of ergodic phase that exists between the two MBL phases. We call the phase Many-Body Ergodic phase since the phase is not analytically connected to the trivial ergodic phase at weak order. In the second part of this thesis, we study the spin -1/2 XXZ spin model on the triangular lattice by mean of Variational Monte Carlo simulation on a class of resonant valence bond (RVB) trial wave-functions. We find that as a function of J⊥/Jz, there exists a transition point where magnetic order vanishes (at J⊥/Jz ~ 0.45). Moreover, two kinds of RVB states competed in the Ising regime when J⊥/Jz < 0.23.
In this paper, we report our numerical analysis of energy level spacing statistics for one-dimensional spin-1/2 XXZ model with random on-site longitudinal magnetic fields Bi. We concentrate on the strong disorder limit where the system is expected to be in either a paramagnetic many-body localized (MBL) state or a spin-glass MBL state. By analyzing the energy-level spacing statistics as a function of strength of random magnetic field h/J⊥ (−h≤Bi≤h), interaction strength Jz/J⊥, energy of the many-body state E and the number of spin-↑ particles in the system M=∑i(siz+21), we show that there exists a small region between the paramagnetic and spin-glass MBL phases where an ergodic phase emerges. The ergodic phase emerges at the middle of the many-body energy spectrum when M∼2N and is not adiabatically connected to the ergodic phase that occurs in the weak-disorder, weak-interaction limit.
Understanding the effects of disorder in quantum many-body systems is one of the central challenges of contemporary statistical physics. Its interplay with interactions and quantum fluctuations profoundly modifies the nature of quantum states and their dynamics, giving rise to phenomena such as localization, ergodicity breaking, and the emergence of complex collective behavior. This thesis presents new results on two classes of disordered quantum systems. The first part is devoted to the study of the many-body localization transition, a paradigmatic example of disorder-induced ergodicity breaking. The critical properties of the transition are investigated using a numerical renormalization-group approach, establishing a connection between its scaling behavior and that of the Anderson localization transition on tree-like graphs. The second part focuses on the zero-temperature quantum spin-glass phase of the two-dimensional bond-disordered Heisenberg model. Combining large-scale numerical simulations with a semiclassical expansion, evidence is provided for the existence of the spin-glass phase. The effective model governing the semiclassical fluctuations is further analyzed to characterize the low-energy excitation spectrum and its localization properties. Finally, the weak-disorder regime is investigated and its implications for the ferromagnetic phase are discussed. Overall, the results presented in this thesis contribute to the understanding of the effects of disorder in interacting quantum many-body systems, encompassing both localization phenomena and the collective behavior of disordered quantum magnets.
In this thesis we will study recent examples of exotic, topological, and many body localized quantum phase transitions. In Chapter 2 we study the quantum phase transition between the Z_2 spin liquid and valence bond solid (VBS) orders on a triangular lattice. We find a possible nematic Z_2 spin liquid intermediate phase and predict a continuous 3d XY* transition to the neighboring columnar and resonating-plaquette VBS phases. In Chapter 3 we demonstrate that an extended Kane-Mele Hubbard model on a bilayer honeycomb lattice has two novel quantum phase transitions. The first is a quantum phase transition between the weakly interacting gapless Dirac fermion phase and a strongly interacting fully gapped and symmetric trivial phase, which cannot be described by the standard Gross-Neveu model. The second is a quantum critical point between a quantum spin Hall insulator with spin S^z conservation and the previously mentioned strongly interacting fully gapped phase. We argue that the first quantum phase transition is related to the Z_16 classification of the topological superconductor ^3He-B phase with interactions, while the second quantum phase transition is a topological phase transition described by a bosonic O(4) nonlinear sigma model field theory with a Theta-term. In Chapter 4 we propose that if the highest and lowest energy eigenstates of a Hamiltonian belong to different SPT phases, then this Hamiltonian can't be fully many body localized. In Chapter 5 we study the disordered XYZ spin chain and its marginally many body localized critical lines, which we find to be characterized by an effective central charge c'=ln2 and continuously varying critical exponents.
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 interplay between quenched disorder and interaction effects opens the possibility in a closed quantum many-body system of a phase transition at finite energy density between an ergodic phase, which is governed by the laws of statistical physics, and a localized one, in which the degrees of freedom are frozen and ergodicity breaks down. The possible existence of a quantum phase transition at finite energy density is strongly questioning our understanding of the fundamental laws of nature and has generated an active field of research called many-body localization. This thesis consists of three parts and is dedicated to the understanding and characterization of the phenomenon of many-body localization, approaching it from complementary facets. In particular, borrowing methods and tools from different fields, we analyze timely problems. The first part of the thesis is devoted to detecting the many-body localization transition and to characterize both the ergodic and the localized phase it separates. Here we provide a characterization from two different perspectives: the first one is based on the study of local entanglement properties. In the second one, using tools from quantum-chaos theory, we attempt to answer the question of understanding time-irreversibility, and thus probing the breaking of ergodicity. We analyze experimentally viable observables. Moreover, we propose two different quantities to distinguish an Anderson insulating phase from a many-body localized one, which is one of the issues in experiments. The second part focuses on understanding the existence of a putative subdiffusive multifractal phase. Analyzing the quantum dynamics of the system in this region of the phase diagram, we point out the importance of finite-size effects, questioning the existence of this multifractal phase. We speculate with a possible scenario in which the diffusivity and thus ergodicity could be restored in the thermodynamic limit. Furthermore, we find that the propagation is highly non-Gaussian, which could have an important effect on understanding the critical point of the according transition. We tackle this problem also from a different angle. A possible toy-model to understand many-body localization entails the Anderson model on a random-regular graph. Also in the latter model the possible existence of an intermediate multifractal phase has been conjectured. There, studying the survival return probability of a particle with time, we give a new characterization of multifractal phases and give indication of the possible existence of this phase. Nevertheless, we also outline possible caveats. In the last part of this thesis we study the interplay between symmetry and correlated disorder in a non-interacting fermionic system. We show another possible mechanism for breaking localization. In particular, we focus on studying information and particle transport, emphasizing how the two types of propagation can be different.
In this paper, we theoretically investigate the many-body localization properties of one-dimensional Ising spin-1 chains by using the methods of exact matrix diagonalization. We compare it with the MBL properties of the Ising spin-1/2 chains. The results indicate that the one-dimensional Ising spin-1 chains can also undergo MBL phase transition. There are various forms of disorder, and we compare the effects of different forms of quasi-disorder and random disorder on many-body localization in this paper. First, we calculate the exctied-state fidelity to study the MBL phase transtion. By changing the form of the quasi-disorder, we study the MBL transition of the system with different forms of quasi-disorder and compare them with those of the random disordered system. The results show that both random disorder and quasi-disorder can cause the MBL phase transition in the one-dimensional Ising spin-1 chains. In order to study the effect of spin interactions, we compare Ising spin-1 chains and spin-1/2 chains with the next-nearest-neighbour(N-N) two-body interactions and the next-next-nearest-neighbour (N-N-N)interactions. The results show that the critical point increases with the addition of the interaction. Then we study the dynamical properties of the model by the dynamical behavior of diagonal entropy (DE), local magnetization and the time evolution of fidelity to further prove the occurrence of MBL phase transition in the disordered Ising spin-1 chains with the (N-N) coupling term and distinguish the ergodic phase (thermal phase) and the many-body localized phase. Lastly, we delve into the impact of periodic driving on one-dimensional Ising spin-1 chains. And we compare it with the results obtained from the Ising spin-1/2 chains. It shows that periodic driving can cause Ising spin-1 chains and Ising spin-1/2 chains to occur the MBL transition.