Topological insulators from the perspective of
non-commutative geometry and index theory
Hermann Schulz-Baldes
Department Mathematik, FAU Erlangen-Nürnberg, Germany
Abstract
Topological insulators are solid state systems of independent electrons for which the
Fermi level lies in a mobility gap, but the Fermi projection is nevertheless topologically
non-trivial, namely it cannot be deformed into that of a normal insulator. This non-trivial
topology is encoded in adequately defined invariants and implies the existence of surface
states that are not susceptible to Anderson localization. This non-technical review reports
on recent progress in the understanding of the underlying mathematical structures, with
a particular focus on index theory.
In solid state physics, one distinguishes between conductors and insulators. In conductors
the electrons at the Fermi level can move through the solid and lead to electric currents, while
in insulators there are either no such electrons due to a spectral gap or they are localized due
to destructive interferences and thus cannot move freely, and one then speaks of a mobility
gap due to Anderson localization. Whether a given solid is a conductor or an insulator can
be determined from a quantum mechanical one-particle Hamiltonian modeling the electrons
near the Fermi level. The parameters of this so-called tight-binding model are obtained as the
exchange integrals between the orbitals of the chemical compounds of the solid.
In the last decade, a multitude of both theoretical and experimental physics contributions
showed that insulators can be topological in the sense that the Fermi projections have nontrivial topological invariants (given in terms of winding numbers, Chern numbers, etc.). These
topological insulators or topological materials are distinct from conventional normal insulators,
and this difference manifests itself typically by non-trivial boundary physics. More precisely,
on the boundaries of a topological material there are delocalized surface modes that make
the insulator conducting after all. This makes such materials also interesting for technological
applications. The connection between non-trivial topology and these surface modes is called
the bulk-boundary correspondence. It is a very robust principle and also the main reason that
the field more recently attracted the interest of a several mathematical physicists. Indeed,
methods from K-theory, index theory and non-commutative geometry can be put to work in
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the models describing topological insulators. In some situations, the analysis even allows to
explain experimental facts that can be observed in a lab. This review starts out with a short
description of the physics and the underlying quantum mechanical models, and then offers an
(admittedly personal) overview of recent mathematical results on them. Hopefully, this allows
the interested reader to get a glimpse of the intrinsic beauty due to the materialization of deep
mathematical concepts in real world solid state physics systems.
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What is a topological insulator?
The title of this section and to some extent also its content may seem better fitted for a
physics than a mathematics review. Indeed, what now follows are a few pages outlining a
mathematicians simplistic view of the modern solid state physics background needed to describe
topological insulators. The focus will here be on systems of electrons in a solid. These are
microscopic particles, so there is no way of escaping quantum mechanics. On the other hand, in
a first approximation, the electrons are often described as independent non-interacting particles
only submitted to the Pauli exclusion principle, even though there are actually many of them
in a solid and they seem to interact strongly via the Coulomb interaction. What is usually
put forward as a justification is that screening plays an important role (from a distance, the
charge of an electron close to a charged nucleus doesn’t look so large any more) and that,
anyhow, a quasiparticle description of the excitations of an interacting electron system should
hold. On rigorous grounds, very little is known on how to derive the effective quantum theories
used below from first principle. A further standard simplification is the so-called tight-binding
approximation. This means that each atom in the solid offers the itinerant electron only a finite
number of bound states, say L of them. The single-electron quantum Hamiltonian H in the ddimensional tight-binding approximation is then a linear, self-adjoint and discrete Schrödinger
operator on the Hilbert space `2 (Zd ) ⊗ CL which is typically of the form
H = ∆B + W .
(1)
Here the operator ∆B is the kinetic part given by a discrete magnetic Laplacian with magnetic
field B and W is matrix-valued potential, namely an operator which commutes the position
operator X on `2 (Zd ), trivially extended to `2 (Zd ) ⊗ CL (for more details, see below). This
potential is, unlike in scattering theory, homogeneous in space. The easiest example of such
a homogeneous potential is a periodic potential, other classes are quasi-periodic and random
potentials. The latter are of particular relevance for the description of solid state systems, which
are rarely perfectly clean and may even be doped on purpose with impurities on purpose. There
is an interesting and important physical phenomenon linked to such random potentials, and
random quantum Hamiltonians in general: Anderson localization. On an intuitive level it
can be described as follows. The electron wave undergoes multiple scattering processes in the
random environment; all the scattering phases along the path have to be added up; this can lead
to destructive interferences, namely the electron can be trapped in the random environment.
This is by now known to be a very effective mechanism in dimensions 1 and 2 as well as for
states energetically close to a band edge, and in all these cases one speaks of dynamical strong
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Anderson localization, and an energy interval within this regime is called a mobility gap. Many
mathematical physicists have contributed to the mathematical understanding of this regime
[7, 1], but a proof for all energies in dimension d = 2 remains a very challenging issue. On
the other hand, in dimension d = 3 and higher, physicists predict the existence of a regime
of so-called weak localization in which the destructive interferences merely lead to a diffusive
motion, surely slower than a ballistic spreading, but also not localized in a strict sense. To
provide a rigorous proof of this is another challenging open question in mathematical physics.
Now, what does all this have to do with topological insulators?
As the electrons are Fermions, the zero temperature ground state for Fermi energy µ ∈ R is
characterized by the Fermi projection obtained via functional calculus from the Hamiltonian:
P = χ(H ≤ µ) ,
where χ is the indicator function. The system is then said to be an insulator if µ lies in a
mobility gap of the spectrum of H. A special case of this is that µ lies in a true gap with no
spectrum at all. Now there is something that makes the insulator topological. Let us attempt
to explain this in an example, namely that of a periodic quantum Hall system. Then the spatial
dimension is d = 2 and only one state per lattice site is needed (that is, L = 1). Furthermore
the uniform magnetic flux through a unit is supposed to be rational (in units of 2π). Then
the Fermi projection P below a gap of the spectrum can be diagonalized by a Bloch-Floquet
transform F:
Z ⊕
∗
FP F =
dk P (k) ,
T2
where T2 = R2 /2πZ2 is the two-dimensional Brillouin torus, P (k) is a projection depending
smoothly on k ∈ T2 and the symbol ⊕ indicates that we have a direct integral representation.
Such a projection is, via its range, naturally identified with a vector bundle over the torus. The
topological invariant is now the Chern number associated to this vector bundle. It is defined
as
Z
dk
Tr P (k)[∂k1 P (k), ∂k2 P (k)] .
(2)
Ch{1,2} (P ) =
T2 2πı
This index {1, 2} indicates that this invariant involves derivatives in directions 1 and 2. Even
though this is not obvious on first sight, it is not too difficult to check that Ch{1,2} (P ) is an
integer which is homotopy invariant under smooth changes of P (k), and, moreover, has the
additivity property Ch{1,2} (P + Q) = Ch{1,2} (P ) + Ch{1,2} (Q) for two orthogonal projections P
and Q, e.g. [6]. When this integer is non-vanishing, one says that the insulator is topological.
This non-trivial topology is then responsible for physical effects, in this case the quantum
Hall effect, namely the Hall conductance of the system takes an integer value in units of the
inverse Klitzing constant e2 /h . The reader may have noticed that up to now, Ch{1,2} (P ) was
defined merely for a periodic system for which the Bloch-Floquet transform could be applied.
Hence the system is topological, but it is not a disordered system. However, the quantum Hall
effect results only from the interplay between topology and Anderson localization. One of the
challenges for mathematical physicists in the 1980s was to develop a theory which could define
the Chern number also for disordered systems. This was achieved in the work of Bellissard
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[4, 5] and Avron, Seiler and Simon [3], a situation that will be covered by the theory below.
The Chern numbers are called the bulk invariants of the quantum Hall system as they are
calculated merely from a model on the entire two-dimensional plane. It was also clear to
physicists [9] that these bulk systems go along with the existence of so-called chiral edge states
when boundaries are introduced, e.g. by restricting the system to a half-plane and imposing
Dirichlet boundary conditions. From these edge states it is possible to calculate a boundary
invariant as an adequately defined winding number. Moreover, this winding number is equal to
the Chern number, a fact that is now commonly referred to as the bulk-boundary correspondence
(BBC). All this was put into a K-theoretic mathematical description in [12] which also applies
to disordered systems and will be described in some more detail below.
Quantum Hall systems are historically the first topological insulators, even though this
terminology was only used much latter. The breakthrough came on the theoretical side with
the work of Kane and Mele [10, 11]. They realized that systems of independent Fermions with
half-integer spin may have non-trivial topology even in presence of time-reversal symmetry
(TRS). Such a symmetry means that the Fermi projection satisfies
Str = e−ıπσ2 .
Str∗ P Str = P ,
(3)
a given complex
Here P = CP C denotes the complex conjugate of the Fermi projection w.r.t.
0 −ı
conjugation on Hilbert space (an anti-linear involution), and σ2 = ı 0 is the second Pauli
matrix acting on the spin degrees of freedom (recall that TRS is given by complex conjugation
and a rotation in spin space by 180◦ ) so that Str is a real unitary squaring to minus the identity for half-integer spin, which is often called an odd TRS. Now for such Fermi projections
one readily checks that the Chern number (2) vanishes. Hence the discovery [10] that there
are nevertheless two distinguishable classes of projections came as a surprise, and made solid
states richer by adding the new so-called quantum spin Hall phase. It is not described here
how Kane and Mele [11] constructed for periodic systems a Z2 -invariant distinguishing the new
phase from a conventional normal insulator. However, Section 7 defines such an invariant as
a Z2 -index of a Fredholm operator with a symmetry. The new Z2 -invariant is not as directly
related to an observable quantity (as the Chern number to the Hall conductance), it is nevertheless related to a physical phenomenon, namely via the BBC to the existence of quantum
mechanical boundary states. These states were, moreover, predicted to be remarkably stable
under perturbations [11]. The field then got a further boost when systems with such edge states
were discovered experimentally (see the references in [17]). In the last several years, several
groups [14, 15] observed experimentally that these delocalized edge states are also very stable
under the perturbation by very strong magnetic fields. This leads to serious doubts about the
theoretical interpretation – which after all is rooted in TRS – and indicates that theory may
not yet have reached its definite state, see Section 9. From a mathematical perspective, the
aim is first to distinguish and classify Fermi projections with TRS (3). This is achieved using
K-theory with symmetries which following Atiyah is called KR-theory. The second aim is
then to calculate these invariants and to analyze the physical effects that go along with these
invariants.
The TRS invokes complex conjugation. It can be odd as in (3), or even if Str2 = 1. There
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is another class of Fermionic systems for which the Fermi projection satisfies the similar, but
distinct invariance property:
∗
P Sph = 1 − P .
(4)
Sph
2
Here the Fermi level is µ = 0 and Sph is a real unitary which can either be even (Sph
= 1) or
2
odd (Sph = −1). Such a system is then said to have a particle-hole symmetry (PHS) because
the symmetry exchanges particles above the Fermi level with holes below the Fermi level. A
PHS naturally appears when a quadratic many-body Hamiltonian describing a superconductor
is written using the Bogoliubov-de Gennes (BdG) Hamiltonian H [2]. Again such systems can
be topological and may have non-trivial boundary modes, and it is interesting to distinguish
the possible phases by K-theoretic means and index theorems. Furthermore, by combining
TRS and PHS one obtains a so-called chiral symmetry (CHS) of the form
∗
Sch
P Sch = 1 − P .
(5)
Here Sch = Sph Str , but a phase may be added to make this symmetry always even. All combinations of TRS, PHS and CHS lead to 10 classes, the so-called Altland-Zirnbauer classes [2]
which also correspond nicely to the 10 classical groups of Weyl if they are classified by symmetries (see e.g. [2] or the appendix in [8]). Within each class and for every given dimension
d of physical space, there may or may not be different distinguishable topological phases [18].
A K-theoretic classification scheme for such phases was put forward by Kitaev [13], and this
will be elaborated on here. One further important element making the connection to operator
algebraic K-theory (which considers homotopy classes of both projections and unitaries) is the
following. Every projection having a CHS is encoded by a unitary operator U if one goes into
the eigenbasis of the (even) symmetry operator Sch :
1 1 U∗
1 0
,
Sch =
.
(6)
P =
0 −1
2 U 1
This U is also called the Fermi unitary of a chiral system and again there may be non-trivial
topology encoded in it. Let us briefly discuss the Su-Schrieffer-Heeger model [19] as a simple
one-dimensional example of such a situation (a detailed discussion of this model can be found
in [16]). Its Hamiltonian acting on `2 (Z) ⊗ C2 is of the form
S ∗ ⊗ (σ1 − ıσ2 ) + m 1 ⊗ σ2 ,
where S is the bilateral shift on `2 (Z), m ∈ R is a mass and σ1 = 01 10 and σ2 =
In the grading of the Pauli matrices,
0
S − ım
H =
.
S ∗ + ım
0
H =
1
2
S ⊗ (σ1 + ıσ2 ) +
1
2
0 −ı
ı 0
as above.
This Hamiltonian is off-diagonal which reflects the CHS Sch HSch = −H with Sch = σ3 = 01 −10 .
The Fermi projection P = χ(H ≤ 0) hence satisfies (5) and is of the form (6) as long as
m 6∈ {−1, 1}. One finds U = (S ∗ + ım)|S ∗ + ım|−1 . Upon discrete Fourier transform, it
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becomes obvious that this unitary has a winding number Ch{1} (U ) equal to 1 for m ∈ (−1, 1)
and no winding for |m| > 1. This is the robust topology of this model. If it is non-trivial it
goes along with the existence of chiral zero modes at the edges, notably bound states of the
half-line restriction of H with energy 0 which are also eigenvectors of Sch .
Before going on, let us briefly resume what makes out a topological insulator:
• d-dimensional disordered system of independent Fermions
• Fermi level in a gap or a mobility gap
• System is submitted to a combination of basic symmetries (TRS, PHS, CHS)
• non-trivial topology of bulk states (winding numbers, Chern numbers, etc.)
• Delocalized edge modes resulting from a bulk-boundary correspondence (BBC)
Let us note that there are other physical effects linked to the non-trivial topology in these
systems, e.g. topological bound states at point defects. Furthermore, topological states of
matter have been found in the physics literature in a great variety of other physical system:
Fermions with spacial symmetries (such as reflection, rotation), interacting Fermions, bosonic
systems (topological photonic crystals, topological magnons), and also in spin systems. Here
we focus only on non-interacting Fermions.
Acknowledgements: The author thanks his main collaborators Jean Bellissard, Johannes
Kellendonk and Emil Prodan as well as further coauthors Giuseppe De Nittis, Stefan Teufel,
Julian Grossmann, Carlos Villegas, Julio Cesar Avila, Alan Carey and John Phillips for inspiring intellectual input and persistence. This work is in part supported by the DFG.
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