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Topological invariants and adiabatic principle in correlated systems

工藤, 耕司 筑波大学 DOI:10.15068/0002000940

2021.08.03

概要

One of the most important challenges in condensed matter physics is to develop unified and simple picture for understanding phases of matter. The so-called symmetry breaking is one of the most fundamental concepts to classify material phases, which has been quite successful to describe a wide variety of phase transitions. While the Landau’s symmetry breaking theory have been becoming important cornerstone for modern physics, recent in- tensive studies have revealed that there are other types of materials in which phases are not identified by their symmetry breaking patterns. This new class is called topological phases, which are characterized by topological invariants instead of the conventional order param- eters. The topological invariants may emerge as physical observables, but most of them cannot be measured directly from the bulk. Instead, these appear at edges as low energy boundary states, which are characteristic properties of topological phases. For theoretical discoveries of topological phase transitions and topological phases of matter, D. J. Thou- less, F. D. M. Haldane, and J. M. Kosterlitz were awarded the 2016 Nobel Prize in physics. Recent years have seen a massive growth of interests in topological phases. Although the topological phases were historically discovered through efforts to understand the underlying physics behind a quantum phenomenon, its concept have been nowadays extended to various systems and have provided a new versatile platform in modern condensed matter physics. Because of its universality and diversity, theoretical and experimental research of topological phases is extremely essential for building a broad foundation of material science.

Historically, the discovery of the integer QH (IQH) effect in 1980 [1] opened the door to the era of topological phases in condensed matter physics. The IQH systems exhibit quantum phenomena associated with phase transitions, which are identified by not conventional order parameters but the topological index known as the Chern number [2–6]. It is closely related to the emergence of gapless edge modes around boundaries, which is called the bulk-edge cor- respondence [7, 8]. The Haldane phases of integer Heisenberg spin chains [9, 10] are also the typical example of topological phases without symmetry breaking. The electron-electron in- teractions in the QH phases give birth to even more topologically nontrivial material phases. The fractional QH (FQH) effect [11, 12] is known today as one of topologically ordered phases [13, 14], which is an incompressible quantum liquid with fractionalized excitations. The quasiparticles describing the excitations carry the fractional charges and the fractional statistics [12, 15–19]. These fractionalizations are closely related to the topological degener- acy [20–24]. The non-Abelian FQH effect [25–28], some quantum spin liquids [29–34], etc., are also known as the topologically ordered phase with long-range entanglement [35–37]. Some of topologically ordered states exhibits excitations with non-Abelian anyons, which can be used for the so-called topological quantum computation. [33, 38, 39].

In recent years, the topological insulators/superconductors [40–43] have been studied intensively. The topological insulators are interpreted as a kind of QH phases with time reversal symmetry, where the spin-orbit interactions play an important role. Recent intensive studies have revealed that internal symmetry brings further diversity into topological phases if the idea of topology is augmented by symmetry. Indeed, many kinds of topological phases of noninteracting fermions are classified systematically [44–47]. These systematic classifications plays an important role for exploring hitherto unknown various topological phases. The electron-electron interactions in these topological materials engender new physics that does not arise in the noninteracting problem. Recently, a wide variety of correlated topological phases have been clarified, some of which include the fractional Chern insulators [48–50], the fractional topological insulators [51, 52], etc. Furthermore, the concepts of the Mott transition and the Kondo effect, which are the typical example of strongly correlate materials, have been applied to physics of topological phases, which have given novel phases of matter such as the topological Mott insulators [53–57] and the topological Kondo insulators [58– 61]. Also, the electron correlations have an even more significant effects on the topological classification [62–75]. Because of this diversity of correlation effects in topological phases, theoretical study on correlated topological materials has become increasingly important for developing the modern theory of the topological phases.

The purpose of this thesis is to establish fundamental concepts and to develop novel materials of topological phases in correlated systems. This thesis consists of two parts, Part I “topological invariants” and Part II “adiabatic principle”. Part I gives arguments on topological invariants for characterizing correlated topological systems. In Chapter 1, we first give a generic setup for the Chern number and the Berry phase and discuss their gauge structures. After the general arguments, the ZQ Berry phases that are quantized due to symmetry of the system are introduced. We also review the TKNN formula and the Niu-Thouless-Wu formula to describe the relation of physical observables and topological invariants. In Chapter 2, it is numerically demonstrated that the integration in evaluating the many-body Chern number for correlated systems can be actually skipped if the system size is sufficiently large. Analyzing the Hofstadter model with or without electron correlations, we show the exponential accuracy of the single-plaquette approximation with respect to the system size. We also discuss the usefulness of the one-plaquette Chern number in systems in which topological phase transitions occur.

Part II gives arguments on the adiabatic principle for characterizing correlated topological systems, which consists of three sections. The adiabatic deformation is a fundamental concept in the theory of topological phases (see in Fig. 1). In Chapter 3, we first revisit a simple problem to demonstrate that the (non-Abelian) Berry phases characterize the geometrical structures of wave functions involved in the adiabatic development. The concept of the adiabatic deformation gives a sophisticated view for characterizing phases of matter. From this perspective, in Chapter 4, we propose a new topological state “higher-order topological Mott insulator” that exhibits a generalized bulk-edge correspondence. It is numerically demonstrated that this correlated topological states are realized in the Hubbard model on a kagome lattice. Their topological nature are simply understood from adiabatically deformed systems, whose information is encoded in the quantized Berry phase. In Chapter 4, we move onto the problem of the adiabatic deformations for the QH effects. According to the adiabatic heuristic arguments, the FQH and IQH states are adiabatically connected with each other by flux-attachment. In this chapter, we numerically demonstrate that the energy gap of the QH states on a torus are indeed smooth and finite even though the topological degeneracy changes wildly. We also analytically derive the relation between the many-body Chern number and the wild change of the topological degeneracy. This is a generalization of the Streda formula of the Hofstadter butterfly in the single-particle problem. This formula we discover implies what is fundamental in topological invariants is the continuity of the energy gap, rather than the continuity of states.

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