The understanding of the effects of disorder in condensed matter systems has been of great importance at each stage of the development of solid state physics and engineering. This importance has not waned in recent years when the theoretical and experimental study of topological insulators has become one of the most active areas of research in the field. The role of quantum entanglement in these topological phases of matter has also become an exciting frontier of research. In this mostly numerical senior thesis, we explore the methods of random matrix theory along with the established information theoretic framework for probing the topological phase transitions in disordered insulators. Specifically, we seek to understand the spectral properties of the entanglement (modular) Hamiltonian associated with the disordered quantum spin Hall insulator and the p-wave superconducting chain. To this end, we first review the recent developments in the application of random matrix theory methods to the entanglement spectrum and demonstrate the necessary numerical techniques. Guided by these developments, we introduce a new measure, the entanglement participation ratio, to reduce the computational costs of using methods of random matrix theory and provide a theoretical avenue for gaining further physical insight into the entanglement spectrum of disordered topological matter. We find that this new measure is successful in identifying topological phase transitions and in providing a complementary viewpoint to the random matrix theory approach.
The understanding of the effects of disorder in condensed matter systems has been of great importance at each stage of the development of solid state physics and engineering. This importance has not waned in recent years when the theoretical and experimental study of topological insulators has become one of the most active areas of research in the field. The role of quantum entanglement in these topological phases of matter has also become an exciting frontier of research. In this mostly numerical senior thesis, we explore the methods of random matrix theory along with the established information theoretic framework for probing the topological phase transitions in disordered insulators. Specifically, we seek to understand the spectral properties of the entanglement (modular) Hamiltonian associated with the disordered quantum spin Hall insulator and the p-wave superconducting chain. To this end, we first review the recent developments in the application of random matrix theory methods to the entanglement spectrum and demonstrate the necessary numerical techniques. Guided by these developments, we introduce a new measure, the entanglement participation ratio, to reduce the computational costs of using methods of random matrix theory and provide a theoretical avenue for gaining further physical insight into the entanglement spectrum of disordered topological matter. We find that this new measure is successful in identifying topological phase transitions and in providing a complementary viewpoint to the random matrix theory approach.
The concept of non-trivial electronic topology in condensed matter physics is crucial for understanding the electronic spectrum of solids and for tailoring various thermodynamic and transport properties, useful for real-world applications. Research on topological insulators and superconductors primarily utilizes the electronic band theory and symmetry classes of gapped quantum systems, with the existence of spectral or mobility gaps ensured even in the presence of impurities and defects. However, structural disorder, naturally manifested in materials such as glasses and amorphous solids, can significantly impact the critical properties of phase transitions in topological quantum matters. This dissertation systematically studies topological phase transitions in non-crystalline free-fermion systems via tight-binding quantum Hamiltonians on 2D random geometries. Leveraging various theoretical and experimentally accessible measures, we demonstrate the resilience of topological phases in non-crystalline systems against structural randomness, described by classical percolation theory, and characterize their unique critical behavior through finite-size scaling. We also extend the notion of amorphous topological states to non-integer dimensional random systems. We reveal that these 2D-embedded random fractals can admit well-defined thermodynamic topological phases with unconventional critical properties. We further study quantum dynamics in these random topological systems. By employing a family of discrete-time topological quantum walks on random percolation lattices, we address wavefunction propagation, superdiffusive quantum speedup, and localization transition by identifying the critical probabilities and timescales. Lastly, we introduce topological lattices with adjustable spatial correlation to investigate Landau levels and quantum oscillations in a low magnetic field regime.
The recent discovery of topological insulators has led to a tremendous interest in the exploration of topological phases of matter which do not fit into Landau's symmetry breaking paradigm. Numerous exotic topological materials are theoretically predicted. Some of them have been experimentally reported, but many remain not. In this thesis, we explore topological phases of matter from three aspects: their classification, realization and application. We first review some basic classification theories, which provide us a "big picture" and lay the foundation for the rest of the thesis. We then move on to propose a systematic method based on quaternion algebra to construct toy tight-binding Hamiltonians for all the exotic phases in a recently developed periodic table for topological insulators and superconductors. We also introduce two peculiar families of topological phases that are beyond the table---the Hopf and four-dimensional topological insulators without time reversal symmetry. Prototypical Hamiltonians are constructed and their topological properties, such as robust edge states, are numerically studied. Motivated by rapid experimental progress in engineering spin-orbit coupling and artificial gauge field for cold atoms, we continue the thesis by proposing a feasible experimental scheme to realize a three-dimensional chiral topological insulator with cold fermionic atoms in an optical lattice. To unambiguously probe topological phases, we also bring forth systematic and generic methods to measure the characteristic topological invariants, for both free and strongly interacting systems. Moreover, we demonstrate that a kaleidoscope of knot and link structures is encoded in the spin texture of Hopf insulators and show how to observe different knots and links in cold atoms via time-of-flight images. The last part of the thesis is about the application of topological materials. After a demonstration of how to create, braid and detect Majorana fermions with cold atoms, we put forward a proposal to construct a self-test quantum random number generator by using Majorana fermions. Majorana random number generators are able to generate certifiable true random numbers with unconditional security. They offer a new perspective to the utilization of topological materials and may have vital applications in cryptography and related areas.
At the interface of quantum information and condensed matter physics, the study of entanglement in quantum many-body systems requires a new toolset which combines concepts from each. This thesis introduces a set of computational methods to study phases and phase transitions in lattice models of quantum systems, using the Renyi entropies as a means of quantifying entanglement. The scaling of entanglement entropy can give valuable insight into the phase of a condensed matter system. It can be used to detect exotic types of phases, to pinpoint transitions between phases, and can give us universal information about a system. \nThe first approach in this thesis is a technique to measure entanglement in finite size \nlattice systems using zero-temperature quantum Monte Carlo simulations. The algorithm \nis developed, implemented, and used to explore anomalous entanglement scaling terms in \nthe spin-1/2 Heisenberg antiferromagnet. \nIn the second part of this thesis, a new and complementary numerical technique is introduced to study entanglement not just in finite size systems, but as we approach the thermodynamic limit. This “numerical linked-cluster expansion” is used to study two different systems at their quantum critical points — continuous phase transitions occurring at zero temperature, at which these systems exhibit universal properties. Remarkably, these universal properties can be reflected in the scaling of entanglement. \nEntanglement offers a new perspective on condensed matter systems, one which takes us closer to genuinely understanding what goes on in these materials at the quantum mechanical level. This thesis demonstrates the first steps in developing an extensive list of computational tools that can be used to study entanglement over a wide range of interacting quantum many-body systems. With the ever increasing computational power available, it may be only a matter of time before these tools are used to create a comprehensive framework for the characterization of condensed matter phases and phase transitions.
Abstract Entanglement spectroscopy has stimulated an extensive range of studies. The entanglement spectrum is the spectrum of the reduced density matrix when the system is partitioned into two. For many quantum systems, when computed from the bulk ground-state wavefunction, it gives access to the physics of edge excitations. It is thus a valuable tool to diagnose topological ordering. These lecture notes provide an overview of entanglement spectroscopy, mostly in the context of the fractional quantum Hall effect. The basic concepts are introduced through the example of quantum spin chains. The connection with entanglement entropy and the matrix product state representation is discussed. The entanglement spectrum can be computed for non-interacting topological phases, revealing edge excitation from the ground state. Application to fractional quantum Hall phases shows how much information is encoded within the ground state and how different partitions probe different types of excitations. Application to fractional Chern insulators is discussed.
Topological states of matter in disordered systems without translation symmetry have attracted great interest in recent years. These states with topological characters are not only robust against certain disorders, but also can be counterintuitively induced by disorders from a topologically trivial phase in the clean limit. In this review, we summarize the current theoretical and experimental progress on disorder-induced topological phases in both condensed-matter and artificial systems. We first introduce the topological Anderson insulators (TAIs) induced by random disorders and their topological characterizations and experimental realizations. We then discuss various extensions of TAIs with unique localization phenomena in quasiperiodic and non-Hermitian systems. We also review the theoretical and experimental studies on the disorder-induced topology in dynamical and many-body systems, including topological Anderson-Thouless pumps, disordered correlated topological insulators and average-symmetry protected topological orders acting as interacting TAI phases. Finally, we conclude the review by highlighting potential directions for future explorations.
Of the available classes of insulators which have been shown to contain topologically nontrivial properties, one of the most important is class AII, which contains systems that possess time-reversal symmetry T with ${T}^{2}=\ensuremath{-}1.$ This class has been the subject of significant attention as it encompasses nontrivial Z2 topological insulators such as the quantum spin Hall (QSH) state and the three-dimensional strong topological insulator. One of the defining properties of this system is the robustness of the state under the addition of disorder that preserves T. In this Rapid Communication, we explore the phase diagram of the disordered QSH state as a function of disorder strength and chemical potential by examining the entanglement spectrum for disordered class-AII symplectic systems. As for the case of the T-breaking Chern insulator, we show that there is a correspondence between the level-spacing statistics of the Hamiltonian and that of the level-spacing statistics of the entanglement spectrum. We observe a feature in the statistics of the entanglement spectrum that aids the identification of delocalized states and consequently critical energies across which phase transitions occur.