Disordered Topological Insulators

Disordered Topological Insulators: A Non-Commutative Geometry Perspective

Emil Prodan Department of Physics, Yeshiva University, New York, NY 10016
July 12, 2019
Abstract

The progress in the field of topological insulators is impetuous, being sustained by a suite of exciting results on three fronts: experiment, theory and numerical simulation. Very often, the theoretical characterizations of these materials involve advance and abstract techniques from pure mathematics, leading to complex predictions which nowadays are tested by direct experimental observations. Many of these predictions have been already confirmed. What makes these materials topological is the robustness of their key properties against smooth deformations and onset of disorder. There is quite an extensive literature discussing the properties of clean topological insulators, but the literature on disordered topological insulators is limited. This review deals with strongly disordered topological insulators and covers some recent applications of a well established analytic theory based on the methods of Non-Commutative Geometry (NCG) and developed for the Integer Quantum Hall-Effect. Our main goal is to exemplify how this theory can be used to define topological invariants in the presence of strong disorder, other than the Chern number, and to discuss the physical properties protected by these invariants. Working with two explicit 2-dimensional models, one for a Chern insulator and one for a Quantum spin-Hall insulator, we first give an in-depth account of the key bulk properties of these topological insulators in the clean and disordered regimes. Extensive numerical simulations are employed here. A brisk but self-contained presentation of the non-commutative theory of the Chern number is given and a novel numerical technique to evaluate the non-commutative Chern number is presented. The non-commutative spin-Chern number is defined and the analytic theory together with the explicit calculation of the topological invariants in the presence of strong disorder are used to explain the key bulk properties seen in the numerical experiments presented in the first part of the review.

type:
Topical Review
pacs:
73.43.-f, 72.25.Hg, 73.61.Wp, 85.75.-d

1 Introduction

The field of topological insulators is progressing extremely fast on both theoretical and experimental fronts and in the past few years it attracted an unprecedented attention from the condensed matter community. This expedited but self-contained review is concerned with a less studied aspect of the field, namely the effect of strong disorder on topological materials. A material is called topological insulator if it behaves like an insulator when probed deep into the bulk and as a metal when probed near any edge or surface cut into the material [Kane:2006xu]. This behavior is not triggered by any externally applied field and, instead, it is an intrinsic property of the material. For a topological insulator, the metallic character of the edges or surfaces is robust against smooth deformations of the material, as long as the insulating character is maintained in the bulk of the material. Examples of topological insulators are the Chern and Quantum spin-Hall (QSH) insulators which will be extensively discussed later.

The idea behind this review was to bring the attention to a set of analytic tools developed for the Integer Quantum Hall-Effect (IQHE) by Bellissard and his collaborators in the late 1980’s, reviewed in the excellent paper from 1994 [BELLISSARD:1994xj]. As we shall see, these analytic tools can be applied quite directly to many classes of topological insulators [Prodan:2010cz], therefore providing a natural theoretical framework to analytically treat the effect of strong disorder in topological materials (for alternative approaches we point the reader to Refs. [HastingsJMP2010gy, Hastings2010by]). Is the work by Bellissard et al relevant to the field of topological insulators? We think it is more than ever, at both conceptual and practical levels.

It is important at the conceptual level because the main claim in the field of topological insulators is the robustness of the topological properties against disorder. This is a huge claim, holding a lot of promises as most of the newly envisioned applications are based on it. But to date, we are still missing the hard evidence for it, counting experiment, numerics and theory all together. The transport experiments on HgTe/CdTe quantum wells [Koenig:2007ko, RothSc2009cu], the only 2D QSH insulator discovered to date, consistently showed a decrease of the conductance with the increase of separation between the ohmic contacts. The quantization of the conductance, as predicted by theory, was observed only for short contact distances of less than 1 micron (one should be aware that HgTe and CdTe materials are extremely difficult to work with so it is hard to pinpoint the cause of this behavior). For 3D samples, angle resolved photoemission spectroscopy (ARPES) measurements for disordered surfaces [Xia2008fk, HsiehNature2009io, Xia2009xu] seem to indicate robust topological extended surface bands, but these conclusions need to be confirmed by transport measurements. The transport measurements in 3D topological samples have been notoriously difficult [TaskinPRB2009xu, CheckelskyPRL2009fy, PengNatMat2010gy, AnalytisPRB2010hv] because of a metallic bulk. Recent characterizations also showed that band bending near the crystal’s surface can trap conventional 2D electron states that coexist but also mix with the topological surface states [BianchiNatComm2010fy], making the experiments even harder. But most recently, high quality single crystals grown by molecular beam epitaxy have been reported [ChengPRL2010gj, ChenPRL2010gy, ZhangNatPhys2010vy, Wang2010lf] (see also the important new development in Ref. [SatoPRL2010yv]). When properly doped, these crystals display insulating bulk [Wang2010lf] and accurate transport measurements of the surface can be recorded. Unfortunately, the first such transport measurements indicate that the topological surface states are slightly localized (the localization is believed to be induced by inelastic scattering processes). Ref. [Wang2010lf] also reported ARPES measurements, which look very similar to the previously reported data [Xia2008fk, HsiehNature2009io, Xia2009xu], despite the fact that the surface states are localized. The character of the surface states has been also probed by STM measurements [RoushanNature2009vu, Yazdani:2010cz], which gave evidence that at short scales (the “field of view” of the STM is less than 1 m) the states are extended despite the presence of strong defects. Shubnikov-de Haas oscillation measurements, which allow one to map the Fermi surface (if any), gave an inconclusive picture so far, with few studies [QuScience2010gy, RenPRB2010vu] reporting clear signal coming from a metallic surface and another study [ButchPRB2010gl], done with ultra high-quality crystals, reporting no significant contributions from the surface, implying an extremely low surface conductivity. The conclusion here is that the experimental measurements are converging to a point where the robustness against disorder can be rigorously tested but we don’t have yet the experimental confirmation of this property.

The number of existing numerical simulations for disordered topological insulators is quite small compared with the number of simulations done in the 80’s and 90’s for the Anderson localization. Until recently, there was only one short numerical study [Onoda:2007xo] on the robustness of the bulk extended states in QSH insulators. This study used the transfer matrix analysis and was completed for small quasi-one dimensional samples. Recently, the transfer matrix analysis was repeated for much larger systems in Ref. [Yamakage2010xr], re-confirming the existence of robust extended bulk states against disorder in QSH insulators. This study seems to contain the most accurate computations to date. The transfer matrix analysis was also adopted in [Obuse:2007qo, Obuse:2008ff] for a representative network model in the symplectic class, with emphasis on the critical exponents at the mobility edges. The properties of the bulk states in a 2D QSH insulator were also probed in Ref. [Essin:2007ij] by computations of certain topological invariant in the presence of disorder (the systems considered in this study are extremely small). Same method was adopted in Ref. [GuoPRB2010fu] for a 3D QSH insulator with disorder (probably the first simulation in 3D; also very small). Ref. [Prodan2010ew] presented a level statistics analysis for Chern insulators and computations of the Chern number in the presence of disorder. The robustness of the edge states against disorder was studied in Refs. [LiPRL2009xi, GrothPRL2009xi, JiangPRB2009sf] by computing the Landauer conductance for a disordered ribbon connected to clean leads. All these three studies worked with conserving models and were limited to small systems (the length of the ribbons was about 200 lattice sites; this number was in the transfer matrix computations of Ref. [Yamakage2010xr]; we also want to mentioned that the theory of the edge states in conserving models with disorder is firmly established [Prodan:2009lo]). But despite all these fine numerical simulations, we are still lacking systematic studies that combine more than one method and where the numerical convergence is rigorously analyzed. From our experience, we can attest that, no matter how elaborate these numerical experiments are, there will always be a margin of doubt about the localization-delocalization issues, until an analytic proof will be available.

The non-commutative methods have also a great practical value. As pointed out in Ref. [HastingsJMP2010gy], the non-commutative formulas of the invariants can lead to extremely efficient numerical algorithms, allowing computations of the invariants in the presence of disorder for system sizes that are orders of magnitude larger that what was possible with more traditional methods [Loring2010vy, Prodan2010ew, Hastings2010by]. Developing accurate and efficient methods for mapping the phase diagram of disordered topological insulators is extremely important for practical applications especially that, as pointed out in Refs. [LiPRL2009xi, GrothPRL2009xi, Yamakage2010xr], the phase diagram of a topological insulator can be strongly deformed by the presence of disorder.

There are additional reasons for writing this review. By its very nature, the field of topological insulators can lead to an unprecedented cross fertilization between condensed matter physics and various fields in pure mathematics with tremendous potential benefits for both fields. We have already seen applications from classic Topology [Kane:2005np, Moore:2007ew, Hatsugai2010nj, Roy2010nj], Chern-Simons Theory [qi2008B], Conformal Field Theory, K-Theory [Kane:2005zw, kitaev:22], Random Matrix Theory and nonlinear -models [Schnyder:2008qy, Ryu:2010tq]. One hope is that we will see many more contributions of this sort from theoretical condensed matter physicists and from pure mathematicians. For this reason, we have tried to keep the presentation pedagogical and appealing to a wide audience, especially to the undergraduate and graduate students looking for good projects, to the theoretical condensed matter researchers who like to compute things explicitly, and to specialists in Non-Commutative Geometry looking for exciting applications of their field. The targeted audience is quite broad and choosing the style of the exposition was not easy. Our final choice might not satisfy all readers, but at least we want to let the readers know that a great deal of effort was spent on this issue.

Our discussion is restricted only to the bulk properties of topological insulators in two dimensions. Although the current and broadly accepted definition of a topological insulator highlights only the robust metallic character of the edge or surface states, every known topological insulator seems to display extended bulk states that are robust against disorder. This is an extraordinary behavior, especially in two dimensional models. When an edge or a surface is cut in a sample of topological insulator, the emerging edge/surface states seem to be connected to these bulk states. In fact, the edge/surface states can be viewed as these extended bulk states terminating at boundary. For this reason, understanding the bulk and the edge/surface properties of the topological insulators is equally important.

We will present several numerical experiments, involving straightforward applications of classic techniques such as level spacing statistics. We will follow the standard interpretation of the numerical outputs, which will show a clear difference between the behavior of a normal and a topological insulator in the presence of disorder. This together with a detailed introduction of two models of topological insulators will occupy the first part of the review. We also describe here how to define a robust spin-Chern invariant for non-conserving models (that is, systems that do not decouple into independent copies of Chern insulators). Several interesting questions will emerge, which will set the direction for the rest of the review.

The analytic part of the review presents a brisk account of the Non-Commutative Theory of the Chern number developed by Bellissard, van Elst and Schulz-Baldes [BELLISSARD:1994xj]. We have reworked certain parts to give the exposition a more “calculatoristic” flavor, so that condensed matter physicists who like to compute could easily follow the arguments. We have complemented the proofs with discussions and remarks, and tried to keep the reader informed at all times about where the calculation or the argument is heading and why do we need to go there. We summarize the arguments before each lengthy proof to provide more guidance. We decided to collect the important statements in Theorems, Lemmas and Propositions, something to the taste of the mathematicians but that could easily irritate other people. One reason for why we chose to do so was to alert the reader that these statements have a rigorous proof and that they can be applied with absolute confidence. Another reason was that, by doing so, we can state in one place the result and the conditions when the result is valid. The last part is especially important for our subject because our main goal is precisely to find the most general conditions that assures the quantization and invariance of the topological numbers.

The review includes a presentation of a numerical technique to evaluate the Chern [Prodan2010ew] and spin-Chern numbers in the presence of disorder. This technique steams directly from NCG Theory and allows one to compute the invariants for finite systems without imposing the traditional twisted boundary conditions. The finite size formulas converge exponentially fast to the thermodynamic limit, given by the NCG Theory. The technique allows us to compute the Chern and spin-Chern numbers for large lattice systems and large number of disorder configurations (at least one order of magnitude over what is currently available in the published literature).

The review has a section devoted entirely to applications. The class of Chern insulators was chosen as the “control case,” because they closely resemble the Integer Quantum Hall-Effect, already extremely well understood. In this case, the NCG Theory gives a full account of all the effects seen in the numerical experiments on Chern insulators, presented at the beginning of the review. Calculations of the Chern number will allow us to witness explicitly its quantization when the Fermi level is located inside the localized part of the energy spectrum, and the failure of such quantization when the Fermi level is located inside the delocalized part of the energy spectrum. For Quantum spin-Hall insulators, we define the non-commutative spin-Chern number, following Ref. [Prodan:2009oh], and discuss the conditions when its quantization occurs. Explicit calculations of the spin-Chern number indicate again quantization when the Fermi level is located in the localized part of the energy spectrum.

2 Topological insulators: A brief account

This will be a brief account, indeed. The reason we kept it short is because there are now several reviews surveying the evolution of the field and its current status, from both theoretical and experimental point of views [Qi2010, ZHassanRevModPhys2010du, Qi2010hg, Hasan2010by]. Nevertheless, through this brief account we want to let the readers know about the impetuous advances that are happening right now in the field of topological insulators.

It is probably a good idea to start from the Integer Quantum Hall Effect, (IQHE) which is now extremely well understood. Discovered at the beginning of the 1980’s [Klitzing:1980vh], IQHE revealed a truly spectacular manifestation of ordinary matter, displayed in the quantization of the Hall conductance and the emergence of dissipationless charge currents flowing around the edges of any finite IQHE sample. The intellectual activity spurred by this effect has led to some of the greatest leaps in condensed matter theory. Working with a clean periodic system and using Kubo’s formula for the Hall conductance , Thouless, Kohmoto, Nightingale and den Nijs made the famous connection between and a topological invariant now known as the TKNN invariant. Using general charge-pumping arguments, the Hall conductance was also linked to the classic Chern number (see Avron in Physics Today 2003). But it was already clear from the early works [Laughlin1981fe, Prange1981uc, Halperin1982fs, Joynt1984hc] that impurity states are essential for explaining the Hall plateaus seen in the IQHE experiments. The quest for an analytic theory of IQHE that includes the disorder has led Bellissard, van Elst and Schulz-Baldes [BELLISSARD:1994xj] to one of the most amazing applications of a new and exciting branch of mathematics called Non-Commutative Geometry [Connes:1994wk]. This work gives an explicit optimal condition that assures the quantization and invariance of the bulk Hall-conductance in the presence of strong disorder. Further homotopy arguments for quantization and invariance of the bulk Hall-conductance were developed in Refs. [Aizenman1998bf] and [Richter2001jg].

The progress about the edge physics of IQHE satarted with the works by Hatsugai, who established in 1993 a fundamental result [Hatsugai:1993cs, Hatsugai:1993jt] saying that the number of conducting edge channels, forming in an energy gap of the Landau Hamiltonian due to the presence of an edge, is equal to the total Chern number of the Landau levels below that gap. The technique developed by Hatsugai can deal only with clean systems, rational magnetic flux and homogeneous edges with Dirichlet boundary conditions. It was only about 10 years later when, using the methods of Non-Commutative Geometry, Kellendonk, Richter and Schulz-Baldes established [SchulzBaldes:2000p599, Kellendonk:2002p598, Kellendonk:2004p597] a new link between the bulk and edge theory, which ultimately allowed them to generalize Hatsugai’s statement to cases with weak random potentials, irrational magnetic flux and general boundary conditions. The equality between bulk and edge Hall conductance was also demonstrated by Elbau and Graf [Elbau:2002qf], soon after the publication of Ref. [Kellendonk:2002p598], this time using more traditional methods. Further progress was made in Ref. [Combes:2005qd], which treated continuous magnetic Schrödinger operators and potentials that can assume quite general forms, in particular, they can include strong disorder. A similar result was established for discrete Schrödinger operators in Ref. [Elgart:2005rc]. We mentioned that some of these ideas were formalized in an abstract setting in Ref. [Prodan:2009od] and applications to the edge states problem in topological insulators were given in Refs. [Prodan:2009mi, Prodan:2009lo].

The IQHE can be observed only in the presence of an externally applied magnetic field. In 1988, Haldane presented a model of a condensed matter phase that exhibits IQHE without the need of a macroscopic magnetic field [HALDANE:1988rh]. The systems that behave like the one described by Haldane are now called Chern insulators. The time-reversal symmetry in these systems is broken like in the IQHE, but it is broken by the presence of a net magnetic moment in each unit cell rather than by an external magnetic field, as it is the case for IQHE. As we shall see, the techniques developed for IQHE can be directly applied to Chern insulators, whose bulk and edge physics [Prodan:2009lo] is very well understood now. The Chern insulators were never found experimentally, even thought there is not one known physical reason for this not to happen one day. The Haldane model truly describes the first topological insulator, but because of the lack of the experimental evidence, the field of topological insulators took off many years after the work of Haldane.

The interest for a related phenomenon, the spin-Hall-Effect, was picking up in the mid 2000’s. The effect was predicted decades ago [Dyakonov1971sc, Dyakonov1972er], and says that a bar made of a semiconductor with strong spin-orbit interaction will display spin-polarized edge states when an electric charge current is forced through it. The effect was finally observed in 2004-2005 [Kato2004ht, Wunderlich2005hg]. The search for a quantized version of the spin-Hall-Effect started at the time when the results on the classical spin-Hall-Effect were making the news [Zhang2001vc, Murakami:2003zf, Murakami:2004tv, Murakami:2004dx, Sinova:2004ad, Culcer:2004zi, Wunderlich:2005io, Sheng:2005mg, Sheng:2005lj], but at that time nobody could imagine that there are samples displaying a quantized spin-Hall-Effect in the absence of externally applied fields. This changed after the discovery of graphene [Novoselov:2004oc, Novoselov:2005lo, Novoselov:2005rk, Zhang:2005mu], which inspired Kane and Mele to propose an explicit model [Kane:2005np] displaying topological edge modes carrying a net spin current around the edges. The emergence of the spin-carrying edge states is triggered solely by the intrinsic spin-orbit interaction and the flow of the spin current is protected by the time reversal symmetry of the model. All materials that are not magnetically ordered display the time-reversal symmetry, and there is a large number of materials with strong spin-orbit interaction that are not magnetically ordered. Therefore, the chance of observing the Quantum spin-Hall-Effect in real materials is quite high. The materials exhibiting this effect are now called Quantum spin-Hall (QSH) insulators and their hallmark is a dissipation-less spin current flowing along the edges of the samples, an effect due to the non-trivial topological properties of the bulk [Kane:2005zw].

The original calculations suggested that the newly discovered graphene could be a QSH insulator. Unfortunately, the spin-orbit interaction is very weak in graphene and that makes the experimental detection of the effect very difficult. Nevertheless, the race for the discovery of the first QSH insulator was on. HgTe/CdTe quantum wells were predicted to display QSH-Effect in 2006 [Bernevig:2006hl] and confirmed experimentally in the following year [Koenig:2007ko]. The first QSH insulator was discovered and a new field emerged [Kane:2006xu], that of topological insulators defined as materials that are insulators in the bulk but metallic along any edge or surface that is cut into the material. Unfortunately, the HgTe quantum wells remain the only two dimensional QSH insulators discovered to date. In three dimensions, the list of confirmed QSH insulators is quite impressive [Hsieh:2008vm, Kuga:2008hv, Shitade2008, Xia:2009cu, Hsieh:2009wq, Zhang:2009mm, Hor:2009zd, Chen:2009vs, Lin2010gh, Lin2010vb, Lin2010fg, Yan2010gl, Xiao2010kc, Xu2010ci, Xu2010jf, SatoPRL2010yv, KimPRB2010ru, Wang2010bu] and the experimental characterization of these materials is vigorously underway [koenig2008, hsieh2009a, hsieh2009b, taskin2009, gomes2009, checkelsky2009, alpichshev2009, tzhang2009, hor2009, hor2009a, EtoPRB2010xy, TaskinPRB2010vi, Nishide2010nj, Jenkins2010te, Zhao2010hf, Wray2010ce] (see also the references cited in the Introduction). Additional classes of topological insulators are expected to emerge in the future [Fu2010jy, Hughes2010gh, Turner2010cu].

3 Introduction to Chern Insulators

3.1 Chern insulators in the clean limit

Figure 1: The honeycomb lattice, its unit cell and generating vectors, together with some notations used in the main text.

A Chern insulator is a periodic band insulator with broken time reversal symmetry, with the distinct property of having a net charge current flowing around the edges of any finite sample. The time reversal symmetry is broken not by an externally applied magnetic field, but by some intrinsic property of the material, such as the occurrence of a net magnetic moment in each unit cell. In the following, we will use an explicit model to exemplify some of the most important features of these materials.

The first model of a Chern insulator was introduced by Haldane in 1988 [HALDANE:1988rh], who worked with the honeycomb lattice shown in Fig. 1. The honeycomb lattice can be viewed as a triangular lattice with two sites per unit cell (see the shaded region in Fig. 1). The two sites of the unit cell will be labeled by as in Fig. 1. The triangular lattice is generated by the vectors . An additional vector is shown in Fig. 1, which will play a certain role later. The Haldane model involves spinless electrons and assumes only one quantum state (orbital) per site, denoted by . The linear combinations of these states generate a Hilbert space , which is equipped with the scalar product . The system is assumed half-filled, which means there is one electron per unit cell .

Figure 2: (a) The bulk spectrum of Haldane Hamiltonian Eq. 1 ( and ) as function of . (b) The energy spectrum of the same Hamiltonian when restricted on an infinitely long ribbon with open boundary conditions at the two edges. The spectrum is represented as function of parallel to the ribbon’s edges. (c) The local density of states (see Eq. 6) of the ribbon, plotted as an intensity map in the plane of energy (vertical axis) and unit cell number along the red line shown in panel (d) (horizontal axis). Blue/red colors corresponds to low/high values. (d) Illustration of the ribbon used in the calculations shown in panels (b, c) and (f, g). The ribbon was 50 unit cells wide. (e-g) Same as (a-c) but for and .

In its simplest form, the Haldane’s Hamiltonian reads:

(1)

where with being the label attached to each site , depending on how is positioned in the unit cell (see Fig. 1), also known as the isospin in the condensed matter community. The single (double) angular parenthesis indicate that the sum over runs over all the lattice sites while the sum over is restricted to the first (second) near neighboring sites to . Notice that a second neighbor hopping always connect sites with same . If we view the honeycomb lattice as a triangular lattice with two sites per unit cell, then the Hamiltonian takes the form:

(2)

where now denotes the position of the unit cell in the triangular lattice and is the isospin labeling the two sites of a unit cell. The variable takes the values 0 and , and are the vectors shown in Fig. 1. We actually prefer this later form of the Hamiltonian, which will be used from now on. The Hamiltonian depends on the two parameters . We will omit the label “Chern” and use the simplified notation for the Hamiltonian of Eq. 1 throughout the current section.

In the absence of disorder, we can perform the Bloch decomposition using the isometric transformation from into a continuum direct sum of spaces:

(3)

where is the Brillouin torus = and:

(4)

Under this transformation, with:

(5)

We denote by the two eigenvalues of . The plot of as function of will be referred to as the bulk band structure of the model.

We now imagine the following numerical experiment. We let the computer pick random points in the plane and then perform a computation of the energy spectrum for an infinite sample (the bulk spectrum), a computation of the energy spectrum for a ribbon shaped sample and a computation of the local density of states (LDOS) for the ribbon. The experiment will reveal that, with probability one, the system is an insulator because the occupied states are separated by a finite energy gap from the un-occupied states, as it is exemplified in panels (a) and (e) of Fig. 2. The bulk band spectrum will not reveal major differences between various regions of the parameter plane. However, the calculations for the ribbon geometry will bring major qualitative differences. For some values such as =0.1 and =0, the energy spectrum for the ribbon geometry displays an insulating energy gap, while for values like =0 and =0.1 it doesn’t. Things become even more intriguing if we look at this spectrum as function of the momentum parallel to the direction of the ribbon. Examining panels (b) and (f) of Fig. 2, we see that, when =0 and =0.1, the spectrum displays two solitary energy bands crossing the bulk insulating gap. For =0.1 and =0, we can still see two solitary bands but they don’t cross the bulk insulating gap. If we let the computer run for a while, picking random points in the plane, it will slowly reveal that this plane splits into regions were the model displays bands that cross the insulating gap like in Fig. 2(b) and region where the insulating gap remains open like in Fig. 2(f). These regions are shown in Fig. 3.

Figure 3: The region of the parameter space where the model of Eq. 1 is in the Chern insulating phase and displays the topological edge bands.

It is instructive to also take a look at the maps of the local density of states (LDOS):

(6)

which will reveal the spatial distribution of the quantum states. The written above depends on 3 variables, the energy plus the two spatial coordinates, but for a homogeneous ribbon, like the one shown in Fig. 2(d), is independent of the coordinate parallel to the edge. Hence is only a function of energy and one lattice coordinate, chosen to be along the red line of Fig. 2(d), in which case we can display using an intensity map. Such maps are shown in Figs. 2(c) and (g). Here, one can see that, if the spatial coordinate is away from the edges of the ribbon, there are clear regions of practically zero density of states, regions that are perfectly aligned with the bulk gaps seen in the band spectra of Figs. 2(a) and (e). When the spatial coordinate approaches the edges of the ribbon, the LDOS inside the bulk gap starts to pick up appreciable values in Fig. 2(c). This part of the LDOS can come only from the two bands crossing the bulk insulating gap in Fig. 2(b). In other words, the quantum states associated with these two bands are localized near the edges of the ribbon and, for this reason, they are called edge bands. Since the slope of a band gives the group velocity of an electron wave-packet generated from that band, we can label the edge bands as right and left moving. A more detailed analysis of the LDOS will reveal that the right/left moving bands are localized on the lower/upper edges of the ribbon, respectively (the correspondence will switch if we change the sign of ). Of course, there is a hybridization between two edge bands and a tiny energy gap is opened at the apparent band crossing, but this hybridization becomes exponentially small as the width of the ribbon is increased. For the ribbon considered in Fig. 2, this hybridization can be practically ignored. In fact, if we keep one edge at the origin and send the other edge to infinity, that is, we consider a semi-infinite sample, we will observe just one edge band crossing the insulating bulk gap.

If is in the shaded region of Fig. 3, the ribbon is in a metallic state, while if in the non-shaded region the ribbon remains in an insulating state. The edge bands seen in Fig. 2(b) are called chiral because they connect the valance and the conduction states. Due to this feature and provided the bulk insulating gap remains open, those bands will not disappear when the Hamiltonian is deformed by either changing the existing coupling constants or by turning on additional interaction terms. For this reason, we can say that the metallic state of the ribbon is topologically protected. In the trivial case, the bands can totally disappear when additional terms are turned on, and what typically happens is that the bands sink into the bulk spectrum. When that happens, there will be little or no trace of edge spectrum in the LDOS.

As shown in Fig. 4, if we pick any point at the boundary of the shaded region of Fig. 3, we will find that the bulk insulating gap is reduced to zero. We can also see some very distinct features emerging, namely, conic points where the bulk bands touch. These singular points are called Dirac points and they are actually at the origin of the topological properties of the model. When (=0,=0) there are two Dirac points, while for any other point of the phase boundary there is just one Dirac point.

Figure 4: The bulk spectrum when is located at the boundary between the topological and normal insulating phases. (a) corresponds to ; (b) corresponds to .

3.2 The Chern number

We give here a brief and formal introduction of the Chern number. Let denote the projector onto the occupied spectrum:

(7)

where is a contour in the complex plane surrounding the occupied energy spectrum. Under the Bloch transformation, decomposes in a direct sum of projectors: , where is a finite matrix acting on . It is analytic of , except when is on the phase boundary. The Chern number is given by the following formula:

(8)

where “tr” means trace over the two dimensional space. The integrand in Eq. 8 is called the adiabatic curvature and the integral of Eq. 8 can be shown to take only integer values, provided the family of projectors are smooth of over the entire Brillouin torus. A plot of the adiabatic curvature is shown in Fig. 5 for the topological phase =0, =0.1 (panel a), and for the trivial phase =0.1, =0 (panel b). The plot was generated by computing on a mesh-grid of 150150 points and by approximating the -derivatives by the second-lowest order finite difference. Fig. 5 shows a distinct behavior of the curvature when the topological and trivial phases of the Haldane model are compared. In both cases, the curvature peaks near the split Dirac points, but in the topological phase the peaks have same signature, while in the normal phase the peaks have opposite signatures. Consequently, the curvature integrates to a non-zero value for the topological case, which is precisely 1, and to 0 for the normal case (plus/minus a small numerical error for both cases). A direct calculation will reveal that takes the value 0 inside the trivial phase and the values inside the topological phases as shown in Fig. 3.

Figure 5: Plots of the adiabatic curvature as function of , when (a) is in the topological insulator region and (b) normal insulator region. The top plots show the bulk spectrum and they have been aligned with the plots of the curvature in order to show that the peaks seen in the curvature plots occur near those points where the bulk bands tend to touch each other. The spikes become less prominent if the insulating gap is increased.

3.3 Chern insulators with disorder

In this section, we present several numerical experiments on the bulk of Chern insulators. They will reveal one of the flagship properties of these materials, manifested in the existence of spectral energy regions that contain delocalized states, even in the presence of strong disorder. Recall that we are dealing with a 2-dimensional model where, in general, the quantum states are localized in the presence of disorder [Abrahams:1979et]. We will work with the following random Hamiltonian:

(9)

where are randomly distributed amplitudes taking values in the interval . We can think of the index in as the collection of all ’s, which in turn can be regarded as a point in an infinite dimensional configuration space .

Figure 6: (Please rotate at 90 degrees.) Level statistics for the Chern insulator = (upper panels) and for the normal insulator = (lower panels) at disorder strengths =3, 5, 8 and 11. The main panels show the spectrum of for many disorder configurations. Level spacings were recorded from a small window around an energy . Shown in light blue is the variance of these level spacings ensembles, when was brushed over the spectrum of . The dotted line marks the value of 0.178 corresponding to the variance of ensemble. The small panels show histograms of the level spacings recorded at the marked energies. The histograms are compared with the (blue curve) and (red curve) distributions.

In the following experiments, we used a random number generator to build the of Eq. 9 on a lattice containing 3030 unit cells. Periodic conditions were imposed at the boundaries of the lattice. We diagonalized and placed its eigenvalues on a vertical axis, repeating the calculation times, every time updating the random potential. The result is a sequence of vertical sets containing the eigenvalues for each run, as illustrated in Fig. 6 for different disorder amplitudes . The level statistics was performed in the following way. We picked an arbitrary energy and, for each disorder configuration, we identified the unique and that satisfy: . Then we computed the level spacings: =, letting take consecutive values between and . We have experimented with =15 and the results were virtually the same. Fig. 6 was generated with =2, in which case, after repeating the procedure for all disorder configurations, we generated an ensemble of level spacings for each , level spacings that were subsequently normalized by their average. Each diagram to the right of the energy spectrum in Fig. 6 shows the distribution (histogram) of these level spacings. We picked several values for and we computed a histogram for each value. Imposed over the histograms are two continuous lines, one representing the Poisson distribution and another one representing the Wigner surmise for Gaussian Unitary Ensemble (GUE), . Imposed over the energy spectrum is the variance - of the level spacings recorded at a large (continuous) number of energies. We marked the theoretical value of 0.178 for the GUE variance by a dashed line in Fig. 6. The level statistics was performed for several disorder amplitudes: =3, 5, 8, 11, for both the topological phase, =0, =0.6, and for the normal phase =0.6, =0.

Let us focus on the topological case first, shown in panels (a)-(d). Inspecting the histograms and the variance in Fig. 6, one can see energy regions where the level distribution is Poisson, thus the states in these regions are very likely to be localized. But one can also see sharp energy regions where the histograms overlap quite well with the distribution and where the variance converges to the 0.178 value. These regions are very likely to contain delocalized states [Evangelou1996yc, Cuevas1998xc], which is quite remarkable since the disorder amplitude in all these panels is larger than the bandwidth of the clean energy bands seen in Fig. 2. One can also observe in Fig. 6 that, as the disorder amplitude is increased, the spectral regions supporting the extended states do not abruptly vanish and instead they drift towards each other until they meet and only then they disappear. This is the so called levitation and pair annihilation phenomenon, which is a general characteristic feature of the extended states carrying a non-zero topological number. This will be discussed in depth later. Based on our current observations, the phase diagram in the plane of a Chern insulator with fixed in the topological region should look like in Fig. 7. If we examine the normal insulator in Fig.6, panels (e)-(h), we see that the spectral regions containing delocalized states are completely absent. There is no levitation and annihilation in this case, and instead the extended states become localized the moment we turn the disorder on.

Figure 7: The phase diagram of a Chern insulator as inferred from the numerical calculations of Fig. 6.

While all that has been said so far about the Chern insulators is just an introduction, we already reached the core of our investigation: To establish that the topological property of the Chern insulators (to carry a non-zero Chern number) has highly unusual physical consequences, manifested in the existence of extended bulk states that resist localization even in the presence of strong disorder. One of our goals will be to demonstrate that the phase diagram in the plane can be derived analytically, using the methods of NCG.

4 Introduction to Quantum spin-Hall (QSH) insulators

4.1 QSH insulators in the clean limit

Figure 8: (a) The bulk spectrum of the Kane-Mele model of Eq. 10 ( and and ) as function of . (b) The energy spectrum of the same Hamiltonian when restricted to an infinitely long ribbon with open boundary conditions at the two edges. The spectrum is represented as function of parallel to the ribbon’s edges. (c) The local density of states (see Eq. 6) of the ribbon, plotted as an intensity map in the plane of energy (vertical axis) and number of unit cells (horizontal axis) along the red line shown in panel (d). Blue/red colors corresponds to low/high values. (d) Illustration of the ribbon used in the calculations shown in panels (b-c) and (f-g). The ribbon was 100 unit cells wide. (e-g) Same as (a-c) but for ( and and ).

The first model of a QSH insulator was introduced by Kane and Mele [Kane:2005np, Kane:2005zw], who worked on the same honeycomb lattice of Fig. 1, but considered also the spin degree of freedom. The Kane-Mele Hamiltonian reads:

(10)

Here, is the spin operator (, and ) and and the electron spin degrees of freedom, taking the values . The Hamiltonian acts on the Hilbert space spanned by the orthonormal basis , where is a site of the honeycomb lattice. The simple and double angular brackets below the sums in Eq. 10 have same meaning as before. Inside the second sum, represents the unique common nearest-neighbor of and and , are the displacements shown in Fig. 1. The underline on the vectors in Eq. 10 means normalization to unity. The model was built specifically for graphene and the three terms in Eq. 10 represent the nearest neighbor hopping, the second nearest neighbor hopping where the intrinsic spin-orbit coupling occurs, and the Rashba potential induced by the substrate supporting the graphene sheet or by an externally applied electric field. Half-filling will be assumed, that is, two electrons per unit cell, until it is specified otherwise. If we view the honeycomb lattice as a triangular lattice with two sites per unit cell, then the Hamiltonian takes the form:

(11)

where and the rest of the notation was already explained. As before, we prefer to work with this form of the Hamiltonian, which we will actually do from now on.

In the absence of disorder, we can perform the Bloch decomposition, given by the isometry from the Hilbert space into a continuum direct sum of spaces:

(12)

where

(13)

We have:

(14)

where was given in Eq. 5 and

(15)

As before, takes the values 0, and . We denote by the four eigenvalues of the Hamiltonian . The bulk band spectrum contains 4 bands, out of which 2 are occupied and 2 un-occupied (assuming a half-filled system).

The parameter space of the model is 3 dimensional . We will let again the computer choose random points in this parameter space and then instruct it to repeat the numerical experiments already discussed for the Chern insulators. Such experiment will reveal that, with probability one, the bulk system is an insulator (see panels (a) and (e) of Fig. 8). Again, by looking only at the bulk band spectrum, we will not be able to distinguish any major qualitative difference between different parts of the parameter space , but the calculation for the ribbon reveals again major qualitative differences. For some values such as , the energy spectrum for the ribbon geometry displays 4 distinct energy bands that cross the bulk insulating gap (see Fig. 8(b)). For other values such as , the spectrum still displays 4 distinct energy bands but they don’t cross the bulk insulating gap (see Fig. 8(f)). If we let the computer for a while to pick random points in the space, it will slowly reveal a distinct region were the model displays bands that cross the insulating gap like in Fig. 8(b) when restricted to the ribbon, and another region where the ribbon has an insulating gap like in Fig. 8(f).

The plot of the local density of states shown in Fig. 8(f) reveals that the two solitary bands marked by () are localized on the top edge and the bands marked by () are localized on the bottom edge of the ribbon. Since the ribbon was 100 units wide, there is practically no hybridization between the bands localized at different edges. The fact that each edge supports two bands steams from the time reversal symmetry of the model and the half-integer value of the spin. The time reversal operation is implemented by the anti-unitary operator:

(16)

where is the component of the spin and is the complex conjugation. The fact that has a distinct consequence in that if is an eigenvector of a time reversal symmetric Hamiltonian, than is also an eigenvector that is orthogonal to because:

(17)

The conclusion is that the spectrum of a time reversal symmetric spin Hamiltonian is always doubly degenerate, a phenomenon known as Krames’ degeneracy. For this reason, even when considering a semi-infinite sample with one edge, one will necessarily observe pairs of right and left moving bands. The Bloch edge Hamiltonian inherits the time-reversal symmetry at =0 and = points. At this points, the spectrum of the Bloch edge Hamiltonian is necessarily doubly degenerate, which means the edge band crossings occurring at =0 and = cannot be split by any deformation that preserves the time-reversal symmetry. Edge band crossings occurring at any other points can and are in general split by such deformations. Now, if the number of pairs of chiral edge bands is odd, like in the Kane-Mele model, then a simple exercise will show that one cannot open a gap in the edge band spectrum by performing all the allowed splittings of the edge band crossings. The situation is different if the number of pairs of chiral edge bands is even, in which case a gap can be opened, and generically will open under deformations that preserve the time-reversal symmetry. This leads to the celebrated topological classification of the time-reversal invariant insulators introduced by Kane and Mele [Kane:2005zw]. For our simple model, the conclusion is that the ribbon is in a protected metallic state.

Returning to our specific model, if is set to zero in Eq. 10, the spin up and spin down sectors are left invariant by the Hamiltonian, which is reduced to a direct sum of two copies of Haldane Hamiltonian with for =1 and for =1. Concentrating for a moment on the bottom edge of the ribbon in Fig. 8 and recalling our discussion from the previous section, one can see that the right/left moving edge bands belong to the =1 sectors. Therefore, the edge bands generate a spin flow, because one band carries a =1 spin in one direction and the other carries a =-1 spin but in the opposite direction. When is turned on, the spin sectors are no longer invariant under the action of the Hamiltonian and, as a consequence, the edge bands will acquire a finite opposite spin component, but still the picture remains practically the same.

4.2 The spin-Chern number for non-conserving models

Figure 9: The left panel shows the spectrum of . Its bands below the gap are highly entangled. The right panel shows the spectrum of . One can see that the occupied states have been disentangled into two widely separated bands with non-zero Chern numbers.

Time reversal invariant insulators have trivial Chern number. According to Refs. [Panati:2007sy, Brouder:2007p233], these systems are trivial from the general homotopy point of view. However, if one insists on preserving the time reversal symmetry, these insulators still display topological properties, as we’ve already seen. There is quite a variety of approaches when it comes to the classification of time reversal symmetric insulators [Volovik:1989eb, Moore:2007ew, Kane:2005zw, Prodan:2009oh, Fu:2006ka, Murakami:2006bw, Sugimoto:2006zr, Fu:2007ti, Fu:2007vs, Teo:2008zm, Sheng:2006nq, Sheng:2006na, Murakami:2007zd, Essin:2007ij, Fukui:2007dz, Fukui:2008mi, Qi:2008cg, Moore:2008ok, Subramaniam:2008tt, Obuse2008, Roy:2009cc, Roy:2009am, Gurarie2010by, RachelPRB2010vy, Ryu2010cu]. Here we discuss the topological properties of the QSH insulators using the spin-Chern number [Sheng:2006na]. While many of the proposed invariants can be, formally, extended to the strong disordered case, and this is done usually by employing finite samples and twisted boundary conditions, it is not clear at all that the quantization and invariance of these extensions survive in the thermodynamic limit when the system becomes gapless due to the strong disorder, like in Fig. 6(b) for example. Note that most of these extensions tacitly assume that a small spectral gap remain open. In reality, or better said, in the numerical simulations with strong disorder, the typical cases are those in which a multitude of eigenvalues cross the Fermi level, from below and above when the boundary conditions are twisted (hence one cannot avoid Fermi level crossings by moving the Fermi level). In this case, the projector onto the occupied states is no longer smooth when twisting the boundary conditions, and ad-hoc numerical solutions must be implemented to deal with this fact. This is a serious problem, because, even when the states are localized near the Fermi level, the number of states crossing the Fermi level can be very large and one needs a criterion that specifies when this can create a problem (in some sense this is what NCG gives us). Another weak point of these extensions is that their quantization can be probed only numerically and the algorithms are not efficient because one needs to repeat the diagonalization of the Hamiltonian of a whole sample for a large number of boundary conditions. The spin-Chern number is the only invariant to date that can be defined in the presence of strong disorder and directly in the thermodynamic limit and this is why we focus here exclusively on it.

Originally [Sheng:2006na], the spin-Chern invariant was defined for a large but finite, squarely shaped sample. Twisted boundary conditions were imposed at the boundaries:

(18)

and the projector onto the occupied states was calculated for each on the Brillouin torus. The spin-Chern number was then computed via Eq. 8. The same original work has shown, through impressive numerical calculations, that the spin-Chern number remains quantized and invariant when disorder is added, even after the insulating gap was completely filled with localized spectrum.

Figure 10: The left panel shows the energy gap of for various values of and and fixed at . The right panel shows the gap of . In the left panel, one can see a fine white line where the gap of is closed. This line delimitates the QSH phase. The gap of is positive inside the QSH phase.

The present discussion follows an alternative definition of the spin-Chern number [Prodan:2009oh], which is more convenient for analytic calculations. The idea is to split the occupied space into two or more sectors with non-trivial Chern numbers. For the Kane-Mele model, one can use the spectral properties of to achieve just that, where . Indeed, given the Bloch decomposition Eq. 14, one can easily compute the Bloch representation of the projector onto the occupied states and then form the matrix . In Fig. 9 we chose and plotted the energy spectrum of and the spectrum of . Looking at the energy spectrum, one can see the two energy bands below the insulating gap being highly entangled. For this reason, no topological invariant can be associated with the individual bands. In contradistinction, the bands in the spectrum of are separated by a seizable gap. As long as this gap and the gap in the energy spectrum remain open, a Chern number can be associated to each individual bands of . If we denote by the projector onto the upper/lower eigenvalue of , we can compute the corresponding Chern numbers via

(19)

Since the total Chern number is zero, is an even number and we can define the spin-Chern number as the integer:

(20)

In Fig. 10 we present the insulating energy gap of and the spectral gap of for fixed at 0.3 and and varied over a wide range. One can see in panel (a) the insulating energy gap closing along a certain line in the plane, line that delimitates the QSH phase. As one can see, the spectral gap of remains open for all and values inside the QSH phase. The picture remains for any other value of , showing that bands of are always separated by a finite gap and consequently the spin-Chern number is well defined. The spin-Chern number takes the values (depending on the sign of ) in the QSH region of the phase diagram, and in the trivial region of the phase diagram.

As a concluding remark, we mention Ref. [LiPRB2010uv] where analytic calculations of the spin-Chern number were carried out for a non-conserving model of a QSH insulator. These analytic calculations are interesting because they show that the Pfaffian function needed in the computation of the invariant [Kane:2005zw] and the integrand in Eq. 19 are closely related.

Figure 11: (Please rotate at 90 degrees.) Level statistics for the QSH insulator =0, =0.6 and =0.3 (upper panels) and for the normal insulator =0.6, =0 and =0.3 (lower panels) at disorder strengths =3, 5, 8 and 11. The main panels show the spectrum of for many disorder configurations. Level spacings were recorded from a small window around an energy . Shown in light blue is the variance of these level spacings ensembles, when was brushed over the spectrum of . The dotted line marks the value of 0.104 corresponding to the variance of ensemble. The small panels show histograms of the level spacings recorded at the marked energies. The histograms are compared with the (green curve) and (red curve) distributions.

4.3 QSH insulators with disorder

We consider here the Kane-Mele model with diagonal disorder:

(21)

We have repeated the numerical experiments presented in the previous section and the results are show in Fig. 11. The computed histograms are now compared with Wigner surmise for the symplectic case: . The variance of this distribution is 0.104. For the topological case, shown in panels (a-d), the numerical experiment reveals again the existence of energy regions where the histograms of the level spacings overlap quite well with and the numerically computed variance takes values extremely close to 0.104. These indicate again the existence of delocalized states [Evangelou1996yc], which persist even for large disorder amplitudes. The levitation and pair annihilation is still visible in the upper panels of Fig. 11. The regions of extended states are absent in the trivial case shown in panels (e-h). The levitation and annihilation is also absent in the trivial case.

There is one distinct difference between the Chern and QSH insulators, regarding their bulk properties. The spectral regions containing the extended states are reduced to a single point for the Chern insulators, while they remain of finite width for QSH insulators. Hence, our numerical experiments imply the phase diagram shown in Fig. 12 for the Kane-Mele model when and are varied.

We would like to end this section with a discussion of the relation between the bulk and edge topological properties of QSH insulators. It is now well established that the QSH insulators with odd spin-Chern number do present robust edge modes, while those with even spin-Chern number do not. This gives the connection between the spin-Chern number and the classification of the QSH insulators, which is based on the edge physics. For example, numerical experiments indicate that the Kane-Mele model displays extended edge states over the entire QSH phase drawn in Fig. 12. But the story does not end here. As we will argue in the following, the spin-Chern number protects a set of extended bulk states against disorder, like the ones revealed in Fig. 11, regardless of its parity. We have now gathered enough numerical evidence to announce here with confidence that explicit models with =2 [Shulman2010cy] or with =1 but broken time-reversal symmetry do posses robust extended bulk states in the presence of strong disorder, even though the edge spectrum displays a mobility gap.

5 The Chern invarint for disordered systems: The Non-Commutative Geometry approach of Bellissard, van Elst and Schulz-Baldes

The definition of the spin-Chern number given in the previous section is based on the Chern number. Consequently, the Non-Commutative Theory of the Chern number is relevant for both the Chern and Quantum spin-Hall insulators and will be presented in depth in this section.

We start by setting some basic notations. The symbol will denote the operator norm: , where the supremum is taken over all vectors of norm one in the underlying Hilbert space. We will often make reference to the space of bounded operators, , which is the linear space of all ’s for which the operator norm is finite. When we use the wording “continuous deformation” of an operator we mean variations of that operator that are continuous with respect to the operator norm. Several additional classes of operators and norms will be introduced later.

We restrict the discussion to a Hilbert space spanned by orthonormal vectors of the form: , where is a site of a 2D lattice and labels the orbitals associated with a particular site. We denote the projector onto these elementary states by and the projector onto the quantum states at a site by :

(22)

The theory will be developed for generic orthogonal projectors that act on this Hilbert space ( and ).

Figure 12: The phase diagram of a QSH insulator as inferred from the numerical calculations of Fig. 11.

5.1 The Fredholm class and the Index of a Fredholm operator

The Index is one of the main tools of modern topology and will be heavily invoked in the following, so we need to begin with the basics of the Index. Given a bounded linear operator , we define its null space as the linear space of its zero modes:

(23)

The Index of the operator is defined as the difference between the number of its zero modes and the number of the zero modes of its conjugate:

(24)

To have a meaning, at least one of the above null spaces must be of finite dimensionality. In fact, if we want the Index to be of any use, we must require that both null spaces have finite dimensions. We must also rule out the existence of the so called generalized wave-functions which obey , as it happens when 0 is in the continuum spectrum of . This can be done by restricting ourselves to operators for which the range is a closed space. So we alreay singled out a very special class of operators, called the Fredholm class, defined by the following properites [Gilkey1994tr]:

  • The Fredholm class contains all bounded operators with the property that , , and F^*missingH

    The Index, as defined by Eq. 24, takes finite integer values when evaluated on operators from the Fredholm class and it has no meaning when evaluated on operators outside of the Fredholm class. For this reason, we always need to make sure that the operators belong to the Fredholm class before we evaluate their Index. This is probably the appropriate moment to introduce another class of operators, which contains all operators for which the trace operation makes sense. This class is called the trace class and is defined as:

    • The trace class ^1A∑_i μ_i ¡ ∞{μ_i}AmissingAA^*S^1BAABAB

      The trace operation is finite when evaluated on operators from the trace class, and it can be computed as , with being any orthonormal basis in . The sum is independent of the chosen basis if is in the trace class. For outside the trace class, the sum can diverge, be oscillatory or change with the change of basis. For this reason, whenever we plan to compute a trace of an operator, we will make sure first that the operator is in the trace class, even if this sometimes can be more difficult than computing the trace itself.

      We now turn to the question of how to evaluate the Index. If the action of is known explicitly and is simple enough, the Index can be evaluated by using its very definition given in Eq. 24. But this is not the case in a large number of situations, in which case we must rely on more computationally friendly methods. One such method was derived by Fedosov and will be used here.

      Proposition 5.1 (Fedosov’s formula).

      If, for some finite positive integer , the operators and are in the trace class, then is in the Fredholm class and its Index can be computed as:

      (26)

      It is a fact that, if and are in trace class for some , then they are in the trace class for any other integer larger than . The computation will lead to the same Index, independent of which allowed we choose to work with. In practice, however, one tries to work with the smallest possible value for this .

      When the right part of Eq. 26 is evaluated, it often leads to explicit formulas that involve geometric data. In general, a successful and useful Index calculation, which is conditioned by the choice of the operator and the ability to compute the right hand side of Fedosov’s formula Eq. 26 (or whatever formula we choose to work with), leads to statements of the form:

      (27)

      where the integral involves objects with explicit geometric meaning, such as the curvature in the case of the Chern number. The equality written in Eq. 27 establishes that this integral is an integer, something that might be extremely difficult to see by just looking at the integral itself. In modern Topology, the left hand side of Eq. 27 is called the analytic Index, while the right hand side is called the geometric or topologic Index. The analytic Index provides the quantization (because from its very definition, the Index is an integer) and, as we shall see in a moment, also the topological invariance, while the geometric Index provides the geometric interpretation and an explicit way to compute the actual value of the Index. This is precisely the philosophy that will guide us throughout the present paper. The topological invariance follows from the following remarkable property of the Index [Gilkey1994tr, Murphy1994hf].

      Proposition 5.2.

      Suppose that the operator changes continuously with and for all ’s the operator stays in the Fredholm class. In this case, is well defined and takes the same integer value for all ’s.

      In other words, the Index of an operator remains unchanged under continuous deformations that keep the operator inside the Fredholm class. This principle not only gives a very general way to prove the topological invariance of the geometric Index, but also allows one to figure out the precise conditions that assures this invariance. For this one has to find out how far can be deformed and still remain inside the Fredholm class. This will be exemplified on explicit models, shortly. We end this section by listing some additional properties of the Index [Murphy1994hf].

      Proposition 5.3.

      Let , be any Fredholm operators, any compact operator and any unitary operator. Then:

      i) .

      ii) .

      iii)

      5.2 The Chern number as an analytic Index: The translational invariant case.

      Assume the existence of a translational invariant projector acting on the Hilbert space , with exponentially decaying matrix elements:

      (28)

      with fixed. Consider the Bloch transformation from into a continuum direct sum of complex spaces:

      (29)

      where are the single column matrices with entry 1 in row and zero everywhere else. The transformation takes the projector into a direct sum of projectors on : . The projectors are finite matrices acting on space and they are analytic of , a property that follows from Eq. 28. The following formula for the Chern number is familiar to most of the condensed matter theorists and we already mention it when we discussed the Chern insulators:

      (30)

      This formula packs geometric and physical content, since the integrand is precisely the adiabatic curvature, derived from the adiabatic connection, both having precise and deep physical meanings (see Avron in Physics Today 2003). The following result links with an analytic Index.

      Theorem 5.4.

      Let be the following unitary transformation:

      (31)

      Then, if Eq. 28 holds true, the operator is in the Fredholm class and:

      (32)

      Therefore, the Chern number is quantized and invariant to continuous deformations of the projector , as long as Eq. 28 holds true for some finite .

      The unitary transformation describes the effect of an infinitely thin quantum flux threaded through the lattice at the spatial point located at . Recalling our previous discussion, it is quite clear that the Chern number appears in Eq. 8 as the geometric Index corresponding to the analytic Index of ( is the orthogonal complement of ). With this relation established, the topological properties of the Chern number follow solely from the extremely general properties of the analytic Index. If is the projector onto the occupied states of an insulator, then the existence of the energy gap assures the exponential localization of this projector and we can safely conclude that the Chern number of the occupied states is quantized and invariant under continuous deformations of the insulator that keep the energy gap open. If the energy gap closes and opens again, then exits and re-enters the Fredholm class, in which case the Chern number can assume a different value. This concludes our short exercise showcasing the strength of the general strategy that we next apply to the disordered case. The above statement will then appear as a particular case of the disordered case, which will be treated in full detail.

      5.3 Differential and integral non-commutative calculus for the disordered case

      A disordered configuration will be labeled by the random variable , which takes values in a configuration space . We assume that the disordered configurations cannot be macroscopically discerned, therefore any laboratory measurement involves averaging over the random variable . The averaging is done with respect to a model dependent probability measure on . For example, for the white noise considered in the previous sections, (this formal expression can be given a rigorous meaning). We also consider homogeneous disorder, that is, various translations of a sample generate allowed disordered configurations of the original sample. Mathematically, this means that each lattice translation by a vector induces a map of onto itself and, for consistency, these maps must satisfy:

      (33)

      Moreover, the probability measure must be invariant with respect to these maps, that is, if is a subset of and , then = . Formally, we can write this as . We will also require that any subset of left invariant by the maps () have zero measure: . In other words, we assume that the measure is ergodic with respect to the flow .

      We now introduce the notion of a covariant family of bounded operators. A collection of bounded operators is called a covariant family if:

      (34)

      Here represents the implementation in the Hilbert space of the lattice translation by : . The covariant families form an algebra, since:

      (35)

      are also covariant families if and are covariant families. To make sense of infinite sums and infinite products within this algebra, one defines the following norm:

      (36)

      The infinite sums and products can be seen as limits, like for example:

      (37)

      The definition of the bounded covariant families, the algebraic operations and the norm defined above leads to a well defined algebra , which replaces the algebra of bounded translationally invariant operators.

      We are now in position to describe the transition from commutative to non-commutative calculus. In the translational invariant case, the Bloch decomposition allows us to view the operators as matrices, whose entries are functions defined over the Brillouin torus. The existence of this classical manifold allows one to define two operations over the space of translational invariant operators: a) the differentiation with respect to and b) the integration with respect to . Let us first describe how this operations extend to the disordered case and then comment on the notion of non-commutative Brillouin torus.

      The first observation is that, for the Bloch decomposition , we have

      (38)

      The second observation is that:

      (39)

      where is taken only over the sites included in a finite area. These two observations provides the clue on how to proceed. The partial derivatives of a covariant family are the covariant families () defined by:

      (40)

      Given that with probability one,

      (41)

      the integration over the Brillouin torus translates into the non-commutative version:

      (42)

      where is the projector onto the quantum states at the origin. To ease the notation, we will use to denote .

      The algebra together with the differentiation and integration defined above can be regarded as a non-commutative Brillouin torus [BELLISSARD:1994xj]. The terminology becomes clear if one remembers that the classic manifolds are in one to one correspondence with the commutative algebra of smooth functions defined over them, that is, one can reconstruct a manifold if he is given this algebra. The Brillouin torus can be recovered from the algebra of translationally invariant operators, whose Bloch decomposition leads to ordinary functions defined over the classic Brillouin torus. The algebra of translationally invariant operators was enlarged here to that of covariant families of bounded operators. No classic manifold can be recovered from this algebra, yet we can define a differential calculus for it.

      5.4 The Chern number as a analytic Index: The disordered case.

      Given the non-commutative rules of calculus, we can easily spell out the non-commutative version of the Chern number:

      (43)

      The main question now is if the non-commutative Chern number remains quantized and invariant to deformations, and under what exact conditions is this happening?

      Theorem 5.5.

      [Bellissard et al, 1994 [BELLISSARD:1994xj]] Consider a covariant family of projectors . If

      (44)

      then is, with probability one, in the Fredholm class. Moreover, with probability one, its Index is independent of and:

      (45)

      The above result gives the quantization of the Chern number in the presence of disorder but does not provide yet the invariance, which will be discussed later in the paper. We want to point out several things. First, one should notice that the condition written in Eq. 44 is extremely simple and transparent. It can be reformulated as:

      (46)

      and now one can see that can be thought as a localization length associated to . Second, the condition written in Eq. 44 is optimal, that is, Eq. 44 provides the most general condition that assures the quantization of the non-commutative Chern number. The remaining of the section is devoted to proving the Theorem 5.5.

      Lemma 5.6.

      Let be an orthogonal projection such that is in the trace class for some integer . Then is in the Fredholm class and

      (47)
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