Topological quantum states are important research fields in condensed matter physics, which can be divided into two categories: (1) symmetry-protected topological states (SPTs) with short-range entanglement and (2) symmetry-enriched topological states (SETs) with long-range entanglement. Recently, research on non-Hermitian topological quantum states (SPTs and SETs) has become a hot topic. In this review paper, we discuss the theory for non-Hermitian topological insulators—a particular type of SPTs and that for non-Hermitian Z$_2$ topological orders—a particular type of SETs based on the progress of our research group.These theories would help understand the physics of non-Hermitian topological quantum states.
Topological quantum states are important research fields in condensed matter physics, which can be divided into two categories: (1) symmetry-protected topological states (SPTs) with short-range entanglement and (2) symmetry-enriched topological states (SETs) with long-range entanglement. Recently, research on non-Hermitian topological quantum states (SPTs and SETs) has become a hot topic. In this review paper, we discuss the theory for non-Hermitian topological insulators—a particular type of SPTs and that for non-Hermitian Z2 topological orders—a particular type of SETs based on the progress of our research group.These theories would help understand the physics of non-Hermitian topological quantum states.
Recently, it was realized that quantum states of matter can be classified as long-range entangled states (i.e., with nontrivial topological order) and short-range entangled states (i.e., with trivial topological order). We can use group cohomology class Hd(SG,R/Z) to systematically describe the SRE states with a symmetry SG [referred as symmetry-protected trivial (SPT) or symmetry-protected topological (SPT) states] in d-dimensional space-time. In this paper, we study the physical properties of those SPT states, such as the fractionalization of the quantum numbers of the global symmetry on some designed point defects and the appearance of fractionalized SPT states on some designed defect lines/membranes. Those physical properties are SPT invariants of the SPT states which allow us to experimentally or numerically detect those SPT states, i.e., to measure the elements in Hd(G,R/Z) that label different SPT states. For example, 2+1-dimensional bosonic SPT states with Zn symmetry are classified by a Zn integer $m\ensuremath{\in}{\mathcal{H}}^{3}({Z}_{n},\mathbb{R}/\mathbb{Z})={\mathbb{Z}}_{n}$. We find that n identical monodromy defects, in a Zn SPT state labeled by m, carry a total Zn charge 2m (which is not a multiple of n in general).
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.
In the presence of symmetry-protection, topological invariants of topological phases of matter in free fermion systems, e.g., topological band insulators, can be directly computed via the properties of band structure. Nevertheless, it is usually difficult to extract topological invariants in strongly-correlated topological phases of matter in which band structure is not well-defined. One typical example is the fractional quantum Hall effect whose low-energy physics is governed by Chern-Simons topological gauge theory and Hall conductivity plateaus involve extremely fruitful physics of strong correlation. In this article, we focus on intrinsic topological order (iTO), symmetry-protected topological phases (SPT), and symmetry-enriched topological phases (SET) in boson and spin systems. Through gauge field-theoretical approach, we review some research progress on these topological phases of matter from the aspects of projective construction, low-energy effective theory and topological response theory.
Symmetry-protected topological phases (SPTs) characterized by short-range entanglement include many states essential to the understanding of topological condensed matter physics, and the extension to gapless SPTs provides essential understanding of their consequences. In this work, we identify a fundamental connection between gapless SPTs and recently introduced multiplicative topological phases, demonstrating that multiplicative topological phases are an intuitive and general approach to realizing concrete models for gapless SPTs. In particular, they are naturally well suited to realizing higher-dimensional, stable, and intrinsically gapless SPTs through the combination of canonical topological insulator and semimetal models with critical gapless models in symmetry-protected tensor product constructions, opening avenues to far broader and deeper investigation of topology via short-range entanglement.
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 phenomena associated with topological defects have had an enormous impact in condensed matter physics for more than 50 years. Beginning with an understanding of topological defects in ordered phases, the field is now sharply focused on defects in topological phases of matter. In this review, we cover aspects of defects in conventional ordered media, bound states on defects in strong topological insulators (TIs) and topological superconductors (TSCs), and bound states on defects in crystalline topological phases protected by spatial symmetries. As a unifying theme, we present the structure of many types of index theorems that relate the existence of topological bound states to the bulk topology of the host phase of matter and the topological charge of the relevant defects.