The classification of distinct phases of matter is a central theme in condensed matter physics. A phase of matter can be viewed as an equivalence class of physical states that share a certain set of properties. In the case of topological quantum matter, these properties are associated with topological invariants that take specific values for each equivalence class. Perturbing a gapped state that corresponds to a given topological phase (i.e., smoothly deforming the Hamiltonian of the system) does not affect the topological invariant, as long as the bulk energy gap remains finite. The value of this invariant changes only if the system undergoes a quantum phase transition, which is signaled by the collapse of the gap. Full topological protection can only emerge in interacting systems and requires the presence of long-range entanglement, which is at the origin of intrinsic topological order. However, topological properties protected by symmetries emerge even in noninteracting systems and, perhaps surprisingly, can be described within the framework of good old band theory of solids. Systems belonging to this category are generically called topological insulators, after the class of materials that is largely responsible for the recent surge in research on topological matter. In addition, topological band theory considerations can also be applied to superconductors (and superfluids), as described at the mean-field level using Bogoliubov–de Gennes theory. This chapter summarizes the key ideas behind the topological classification of noninteracting systems and discusses the basic properties of various classes of topological insulators and superconductors. For in-depth discussions and related developments the reader is referred to a number of books [ 54 , 175 , 469 ] and review papers [ 23 , 239 , 240 , 421 ] specifically dedicated to this subject.
The classification of distinct phases of matter is a central theme in condensed matter physics. A phase of matter can be viewed as an equivalence class of physical states that share a certain set of properties. In the case of topological quantum matter, these properties are associated with topological invariants that take specific values for each equivalence class. Perturbing a gapped state that corresponds to a given topological phase (i.e., smoothly deforming the Hamiltonian of the system) does not affect the topological invariant, as long as the bulk energy gap remains finite. The value of this invariant changes only if the system undergoes a quantum phase transition, which is signaled by the collapse of the gap. Full topological protection can only emerge in interacting systems and requires the presence of long-range entanglement, which is at the origin of intrinsic topological order. However, topological properties protected by symmetries emerge even in noninteracting systems and, perhaps surprisingly, can be described within the framework of good old band theory of solids. Systems belonging to this category are generically called topological insulators, after the class of materials that is largely responsible for the recent surge in research on topological matter. In addition, topological band theory considerations can also be applied to superconductors (and superfluids), as described at the mean-field level using Bogoliubov–de Gennes theory. This chapter summarizes the key ideas behind the topological classification of noninteracting systems and discusses the basic properties of various classes of topological insulators and superconductors. For in-depth discussions and related developments the reader is referred to a number of books [ 54 , 175 , 469 ] and review papers [ 23 , 239 , 240 , 421 ] specifically dedicated to this subject.
Topological insulators and superconductors are subsets of (almost) noninteracting symmetry protected topological (SPT) phases. More generally, SPT phases represent equivalence classes of (equilibrium) gapped quantum states with short-range entanglement that are protected by certain symmetries. Two quantum states are “equivalent” if they can be adiabatically deformed into one another without closing the bulk gap. Thus, quantum states belonging to different SPT phases cannot be smoothly deformed into each other without a phase transition, as long as the underlying symmetry is preserved. However, upon removing the symmetry all SPT states can be adiabatically deformed to a trivial product state (e.g., an atomic insulator) without undergoing a phase transition. In this chapter, we generalize our discussion of (almost) noninteracting topological states by removing some of the explicit and implicit constraints underlying the classification and properties of topological insulators and superconductors discussed in Chapter 3 . In Section 4.1 , we enlarge the type of symmetries that protect the topological phases by supplementing internal symmetries (e.g., time-reversal and particle-hole symmetries) with crystalline symmetries (e.g., mirror and rotation symmetries) and discuss topological crystalline insulators and superconductors , a subsets of SPT phases protected by internal and crystalline symmetries. In Section 4.2 , we remove the requirement of having a finite (bulk) gap and discuss Weyl semimetals – crystalline systems characterized by topologically protected bulk band degeneracy points separated in momentum space, which support low energy excitations that are massless chiral fermions, i.e., Weyl fermions. Finally, in Section 4.3 , we remove the condition of thermodynamic equilibrium and consider topological quantum states in periodically driven systems, focusing on the so-called Floquet topological insulators , which exhibit a richer topological structure than their static counterparts. We note that aspects regarding the effects of particle-particle interactions, including the emergence of topological order in systems with long-range entanglement, will be addressed in Chapter 5 . We also point out that the areas summarized in this chapter have grown tremendously in recent years and continue to grow; even a brief mentioning of all recent developments would be impossible within the constraints of a book chapter. Instead, we point out the key elements that determine the nontrivial topology of the quantum systems mentioned above, discuss specific manifestations of their topological properties (e.g., the presence of protected gapless states), and provide representative examples.
What theoretical approaches are appropriate for describing topological quantum matter? The answer depends on the type of topological phase we want to study, e.g., noninteracting or interacting, and on the type of information we are interested in, e.g., generic properties or system-specific properties that are relevant to a particular experiment or type of experiment. Noninteracting topological phases can be described within topological band theory using the basic tools of the band theory of solids. However, the complexity of the model, i.e., how much detail has to be included, depends on whether we are only interested in the topological properties of the system or, in addition, we need to consider certain non-topological features. For example, predicting whether or not a specific material is a topological insulator requires detailed knowledge of its band structure, which can be acquired using density functional theory (DFT) methods. Similar treatments may be required to account for certain experimental features involving both topological and nontopological contributions. On the other hand, to understand the classification of topological insulators, it is sufficient to consider “bare bone” effective models, such as continuum Dirac models or simple tight-binding models. Topologically, these descriptions with extremely different levels of complexity are equivalent whenever one can smoothly connect the corresponding Hamiltonians (up to the addition/subtraction of trivial bands) without closing the bulk gap. However, the effective models provide only a crude description of experimentally relevant properties such as energy gaps, band dispersion, and real space properties of boundary states. In some sense, the situation is analogous to the characterization of a given mathematical object within topology, which only deals with properties that are robust against deformations, and geometry, which accounts for properties described by distances, angles, and shapes.
One problem of paramount importance in physics over various fields such as elementary particle physics, condensed matter physics, and statistical mechanics has been to make classification of phases of matter, including nontrivial structures of the vacuum. The triumphant history of theory on the phase classification and the phase transition can be traced back to Landau, which is based on the spontaneous breaking of symmetry. It is known, however, that other kinds of phases exist that cannot be characterized by conventional arguments of symmetry breaking, namely, topological phases. Topological phases are gapped phases at zero temperature without symmetry breaking. Examples of the topological phases include a quantum phase in a fractional quantum Hall system, where the ground state exhibits degeneracy, and an anyon, a quasi-particle with nontrivial statistical nature, neither boson nor fermion, emerges as a low-energy excitation. In contemporary physics of the 21st century, it has been revealed that, among various topological states, there exists a nontrivial ground state that does not have ground state degeneracy and yet can be distinguished from a trivial state. This nontrivial ground state, i.e., a topological insulator, has been recognized first in free electron systems [1–3] and later extended to more general quantum many-body systems with interactions including bosonic (spin) systems. Now, such a generalized situation is called a symmetry protected topological (SPT) phase [4–7]. From the point of a modern view of the SPT phase, nontrivial states, such as the Haldane state proposed in antiferromagnetic quantum spin systems in the 1980’s [8] and the ground state in the Affleck-Kennedy-Lieb-Tasaki model [9], belong to the SPT phase. A remarkable feature of the SPT phase is that, if the system has a surface, gapless surface states appear and they are stable against disturbances which do not break the symmetry. One way to understand these surface states is the quantum anomaly on the surface. That is, as long as the surface is concerned, the quantum anomaly arises, but the whole system has no quantum anomaly, and the symmetry is preserved due to cancellation from the bulk contribution. This canceling mechanism of the quantum anomaly has been long known as the anomaly inflow in elementary particle physics [10]. In fact, the classification of the SPT phases can be identified as the classification of the quantum anomaly through the bulk-boundary correspondence. This type of the quantum anomaly closely related to the SPT phase can provide us with a useful theoretical device to constrain phases that can or cannot be realized under the ’t Hooft anomaly matching condition [11–13]. It is quite difficult to scrutinize nonperturbatively interacting systems, but the ’t Hooft anomaly matching condition can be applied to those systems, and possible phases can be constrained. Nowadays, our understanding of the SPT phases has been so developed that applications of the anomaly matching condition can cover systems with finite groups and/or extending objects [14–16]. In this way, the concept of the topological phases has been essential by now in a wide range of physics, particularly in elementary particle physics, condensed matter physics, and statistical mechanics, as mentioned in the beginning. This special section is a collection of papers related to the topological aspects of physics including the topological phases. We would like to express our sincere gratitude to all the authors who made excellent contributions.
Despite great success we have achieved in the past two decades on systematically understanding gapped topological phases, a firm foundation for understanding gapless topological phases is still lacking. In this dissertation I develop a general framework to describe, classify and construct gapped and gapless phases with or without symmetry. I then apply this framework to construct new gapless phases, and classify some classes of them completely. The first chapter is a non-technical summary of the subjects, ideas and results of the rest of the dissertation. In the first chapter I introduce the notion of gapless topological phases, and briefly discuss their general structures. We also discuss how topological holography can be useful for classifying and constructing gapless phases with generalized symmetries. In chapter two I develop a framework for understanding general quantum phases and phase transitions with generalized symmetries. After reviewing the established theory of gapped phases, we introduce two key notions: quantum order and symmetry-enriched quantum order. These concepts are the correct generalized of topological order and symmetry-enriched topological orders. I show that these new notions subsume conventional phase transitions, gapless spin liquids, and gapless symmetry-protected topological phases. For a class of symmetry-enriched quantum orders with finite many topological defects, we present a general classification using higher tensor category theory. The rest of the dissertation provides more detailed studies of specific gapless topological phases, including classification and lattice model construction of gapless symmetry-protected topological phases and the connection to the theory of topological holography.
Topology is woven into the subtle fabric of quantum theory. Topological concepts and ideas developed starting in the 1980s by the high energy physics and quantum gravity communities, first viewed by many as nothing but aesthetically appealing abstract possibilities, are central components of modern condensed matter physics and play a key role in the field of quantum information and quantum computation. The discovery of integer and fractional quantum Hall effects and, more recently, of topological insulators and superconductors has brought topological quantum matter in the laboratory, with the perspective of realizing functional platforms for topological quantum computation. Rooted in a solid mathematical formalism and spanning its relevance from fundamental questions in quantum gravity and quantum information to practical aspects regarding new materials and platforms for quantum computation, quantum topology is positioned at a fertile confluence of mathematics, physics, and computer and information science. This book focuses on the aspects of quantum topology that are most relevant from the perspective of condensed matter physics and its relation to quantum computation. We start our journey by sketching the relation between topology and quantum physics and summarizing a few basic mathematical concepts that will be used throughout this work. Beyond dry formal definitions, our main goal is to grasp the delicate root that makes the topological quantum world utterly strange and beautiful and follow it into the realm of condensed matter many-body systems.