Topological phases of matter are defined by their nontrivial patterns of ground-state quantum entanglement, which is irremovable so long as the excitation gap and the protecting symmetries, if any, are maintained. Recent studies on noninteracting electrons in crystals have unveiled a peculiar variety of topological phases, which harbor nontrivial entanglement that can be dissolved simply by the the addition of entanglement-free, but charged, degrees of freedom. Such topological phases have a weaker sense of robustness than their conventional counterparts, and are therefore dubbed ``fragile topological phases.'' In this work, we show that fragile topology is a general concept prevailing beyond systems of noninteracting electrons. Fragile topological phases can generally occur when a system has a $\mathrm{U}(1)$ charge conservation symmetry, such that only particles with one sign of the charge are physically allowed (e.g., electrons but not positrons). We demonstrate that fragile topological phases exist in interacting systems of both fermions and of bosons.
Topological phases of matter are defined by their nontrivial patterns of ground-state quantum entanglement, which is irremovable so long as the excitation gap and the protecting symmetries, if any, are maintained. Recent studies on noninteracting electrons in crystals have unveiled a peculiar variety of topological phases, which harbor nontrivial entanglement that can be dissolved simply by the the addition of entanglement-free, but charged, degrees of freedom. Such topological phases have a weaker sense of robustness than their conventional counterparts, and are therefore dubbed ``fragile topological phases.'' In this work, we show that fragile topology is a general concept prevailing beyond systems of noninteracting electrons. Fragile topological phases can generally occur when a system has a U(1) charge conservation symmetry, such that only particles with one sign of the charge are physically allowed (e.g., electrons but not positrons). We demonstrate that fragile topological phases exist in interacting systems of both fermions and of bosons.
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.
Strong interactions can give rise to new fermionic symmetry-protected topological phases which have no analogs in free fermion systems. As an example, we have systematically studied a spinless fermion model with U(1) charge conservation and time-reversal symmetry on a three-leg ladder using density-matrix renormalization group. In the noninteracting limit, there are no topological phases. Turning on interactions, we found two gapped phases. One is trivial and is adiabatically connected to a band insulator, while another one is a nontrivial symmetry-protected topological phase resulting from strong interactions.
Understanding the interplay between symmetries and phases lies at the heart of characterizing states of matter. The conventional Landau's paradigm associates local ordered states with symmetry breaking, but this paradigm fails to include gapped topological phases, which can only be distinguished by the global topological invariant and the present of gapless edge modes at open boundaries. In this thesis, variety of states of matter is enriched from two aspects: a new paradigm of phases that incorporates both symmetry broken phase and topological phase, and exotic topological phases that are realized in different degree of freedoms of the same system. The two aspects are explored by studying corresponding one dimensional fermionic models. Besides gapped phases, exotic quasiparticle interferences of gapless topological systems are also investigated. \n \nThe first chapter serves as a theoretical basis for the next two chapters by giving an introduction to topological phases in one dimension. In particular, a bosonization method is introduced to describe topological degeneracy and edge zero modes, which is favoured by the ability to deal with interactions as well as to uncover the phenomena of spin-charge separation. \n \nIn the second chapter a new paradigm of phases is proposed and investigated that incorporates both symmetry broken phases and symmetry-protected topological phases. Through exact-solvable models bond spin density wave(BSDW) states are proposed as candidates for this new paradigm, which are ZZ2 symmetry breaking due to spontaneous dimerization and yet each is topological nontrivial with edge zero modes of fractional fermions. More general lattice models are constructed, and it is shown that BSDW exists in a considerable regime of parameters for a family of lattice models, from both mean-field analysis and bosonization. \n \nIn the third chapter, we give an effective field description for spin-charge separated edge modes consisting of both Majorana modes and fractional fermions, and demonstrate how to harvest both simultaneously in a spinless fermionic ladder with particle-hole symmetry and single-chain fermion parity. Further, the idea of spin-charge separated edge modes is applied to reinterpret the prominent Haldane phase of spin-1 chain in its equivalent spin-frac12 fermionic model. \n \nIn contrast to the above two theoretically oriented topics that aim to add variety to topological phases, the third topic establishes a bridge between abstract concepts for topological systems to remarkable features that can be detected in experiments. The last chapter reveals how topological charges for topological Fermi points, through exhibiting exotic quasiparticle interference in the presence of impurities, could be visualized in STS measurements. \nThe generalized joint density of states(GJDOS) is applied to address effects from the geometry of band dispersion and topological charges separately. Both analytical results and numerical simulation suggest distinct and robust quasiparticle interference features as indicators for topological charges that can be accessed by FT-STS experiments.
Topological phases are gapped quantum phases of matter classified beyond the paradigm of Landau's symmetry breaking theory. In particular, topologically ordered phases are topological phases with topological order. Unlike the conventional symmetry-breaking order described by local order parameter, topological order is characterized by the excitations. For example, in (2+1)D topologically ordered phases, there are particle-like excitations called anyons. In contrast, in (3+1)D topologically ordered phases, there are both particle-like and loop-like excitations. These excitations exhibit braiding statistics, meaning that carrying one excitation adiabatically around another transform the ground state by a phase (or generally a unitary transformation). The braiding statistics of these excitations serves as a crucial ingredient in characterizing the topological order in these phases. One important aspect is to understand the braiding statistics in topologically ordered phases. While the anyonic braiding statistics in (2+1)D topologically ordered phases is well established, the braiding statistics in (3+1)D topologically ordered phases is far from clear. In (3+1)D, the most well-understood braiding statistics is the particle-loop braiding statistics, where a particle winds around a loop excitation acquires an Aharonov Bohm phase. In the thesis, we introduce a new braiding statistics, which we called the Borromean-Rings braiding statistics. It is a particle-loop-loop braiding statistics where a particle travels around two unlinked loops acquires a braiding phase. Unlike the particle-loop braiding, the particle trajectory in the Borromean-Rings braiding is not linked with any of the two loops. Another important aspect is to probe the topological order in topologically ordered phases. One usual way to probe the topological order is to examine the properties of the phases on various orientable surfaces. For example, for (2+1)D topologically ordered phases, the number of degenerate ground states on Torus counts the number of anyons. Also, ground state transformation under twisting the torus encodes the braiding data. In the thesis, we attempt to probe the topological order by putting the phases on non-orientable surfaces. We find that the ground state degeneracy and the ground state transformation reveals a special kind of anyons, called the parity symmetric anyons, which show a lost of orientation in their braiding statistics. They in turn distinguish different topological order. Even without topological order, there can be non-trivial topological phases in the presence of global symmetry. These topological phases are called the symmetry protected topological phases. For example, (3+1)D fermionic matter with time reversal symmetry and charge conservation symmetry has a non-trivial phase, namely the topological insulators. While the bulk of the topological insulators behaves like trivial insulators, the surface is anomalous with odd number of massless dirac fermions. While most of the understanding of topological insulators is developed upon the free fermion picture, such phase survives under interaction. In the thesis, we derive hydrodynamic field theory for the topological insulators by functional bosonization. Such formulation is robust against interaction. In addition, by suitably incorporating electron interactions, we predict new topological phases, called the fractional topological insulators. Besides, the surface theory of topological insulators also justify the famous duality between free massless dirac fermion and QED_3.
Strong interactions can give rise to new fermionic symmetry protected topological phases which have no analogs in free fermion systems. As an example, we have systematically studied a spinless fermion model with U(1) charge conservation and time reversal symmetry on a three-leg ladder using density-matrix renormalization group. In the non-interacting limit, there are no topological phases. Turning on interactions, we found two gapped phases. One is trivial and is adiabatically connected to a band insulator, while another one is a nontrivial symmetry protected topological phase resulting from strong interactions.
Recently, a new class of quantum phases of matter---symmetry protected topological states, such as topological insulators---has attracted much attention. In presence of interactions, group cohomology provides a classification of these [Chen et al., Phys. Rev. B 87, 155114 (2013)]. These phases have short-ranged entanglement and no topological order in the bulk. However, when long-range entangled topological order is present, it is much less understood how to classify quantum phases of matter in the presence of global symmetries. Here we present a classification of bosonic gapped quantum phases with or without long-range entanglement in the presence or absence of on-site global symmetries. In 2+1 dimensions, the quantum phases in the presence of a global symmetry group SG, and with topological order described by a finite gauge group GG, are classified by the cohomology group ${H}^{3}(SG\ifmmode\times\else\texttimes\fi{}GG,U(1))$. Generally, in d+1 dimensions, such quantum phases are classified by ${H}^{d+1}(SG\ifmmode\times\else\texttimes\fi{}GG,U(1))$. Although we only partially understand to what extent our classification is complete, we present an exactly solvable local bosonic model, in which the topological order is emergent, for each given class in our classification. When the global symmetry is absent, the topological order in our models is described by the general Dijkgraaf-Witten discrete gauge theories. When the topological order is absent, our models become the exactly solvable models for symmetry protected topological phases [Chen et al., Phys. Rev. B 87, 155114 (2013)]. When both the global symmetry and the topological order are present, our models describe symmetry enriched topological phases. Our classification includes, but goes beyond, the previously discussed projective symmetry group classification. Measurable signatures of these symmetry enriched topological phases and generalizations of our classification are discussed.