We give a self-contained and enriched review about topology properties in the rapidly growing field of topological states of matter (TSM). This review is mainly focus on the beautiful interplay of topology mathematics and condensed matter physics that issuing TSM. Fiber bundle theory is a powerful concept to describe the non-trivial topology properties underlying the physical system. So we briefly present some motivation of fiber bundle theory and following that several effective topological methods have been introduced to judge whether a fiber bundle is trivial or not. Next, we give some topological invariants that characterizes the non-trivial TSM in the non-interacting systems in all dimensions, which is called topological band theory. Following that, we review and generalize the topological response using topological field theory called Chern-Simons effective theory. Finally, the classification of free-fermion systems have been studied by loop space and K-theory.
We give a self-contained and enriched review about topology properties in the rapidly growing field of topological states of matter (TSM). This review is mainly focus on the beautiful interplay of topology mathematics and condensed matter physics that issuing TSM. Fiber bundle theory is a powerful concept to describe the non-trivial topology properties underlying the physical system. So we briefly present some motivation of fiber bundle theory and following that several effective topological methods have been introduced to judge whether a fiber bundle is trivial or not. Next, we give some topological invariants that characterizes the non-trivial TSM in the non-interacting systems in all dimensions, which is called topological band theory. Following that, we review and generalize the topological response using topological field theory called Chern-Simons effective theory. Finally, the classification of free-fermion systems have been studied by loop space and K-theory.
States of matter which are described by concepts in topology are of great interest in the community of condensed matter physics. Both from a pure theoretical perspective, where many non-intuitive and mathematically rich results can be found from seemingly simple models, and also since recent experimental developments allows for explicit construction of devices where the underlying topology has observable consequences. This thesis provides a conceptual introduction to the framework describing such topological states of matter (TSM). It starts by explaining ordinary band theory which is followed by a discussion of quantum mechanical symmetries. These two concepts provides a method of classifying all non-interacting fermionic TSM. Following are explicit applications to a few models in one spatial dimension. In the accompanying paper, we study one-dimensional superconductors with Josephson-junctions allowing phase shifts of π. We make a detailed analysis of the junction bound states and their properties in some different settings and compare the behaviour of trivial and topological superconducting junctions. In addition, we provide a phenomenological topological field theory in the low energy limit.
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.
T HE 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 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 activity 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 summaries 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 [48, 402, 153] and review papers [211, 212, 357, 22] specifically dedicated to this subject.
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.