Download Primer on topological insulators

Primer on topological insulators
Alexander Altland and Lars Fritz
ii
Contents
1 What is a topological insulator?
1.1 What is an insulator anyway? . .
1.1.1 Setup . . . . . . . . . . .
1.1.2 Bloch theorem . . . . . .
1.1.3 Band structure . . . . . .
1.2 Topological insulators . . . . . .
1.2.1 Hall response . . . . . . .
1.3 Topology . . . . . . . . . . . . .
1.4 The emergence of Dirac Fermions
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2 The bulk-boundary-correspondence
2.1 Electrons in a magnetic field - the quantum Hall effect . . .
2.1.1 The classical Hall effect . . . . . . . . . . . . . . .
2.1.2 The quantum Hall effect . . . . . . . . . . . . . . .
2.1.3 Landau levels . . . . . . . . . . . . . . . . . . . . .
2.1.4 The occurrence of boundary modes . . . . . . . . .
2.2 Quantum Hall effect without external field: Chern insulator
2.2.1 Phase diagram of the Chern insulator . . . . . . . .
2.2.2 Boundary modes with open boundary conditions . .
2.2.3 Boundary modes: a continuum picture . . . . . . .
2.3 The quantum spin Hall insulator . . . . . . . . . . . . . . .
iii
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Chapter 1
What is a topological insulator?
These notes are meant to explain in the simplest possible terms what a ’topological insulator’
(TI) is. From contents and style of the presentation it should be clear that in no way can this
1
text replace a thorough introduction to the subject. All we aim for, is shedding some light
on the connections between the (seemingly unrelated?) terms ’insulator’ and ’topology’.
1.1
What is an insulator anyway?
As is suggested by the name, a ’topological insulator’ is a particular variant of a (crystalline)
’insulator’. So, we first have to understand what an insulator is. Physicists know this, of
course, so they may skip this section, or just briefly glance over it to absorb the notation used
later on.
Figure 1.1: Cartoon of a one dimensional crystalline solid
1
By now, several review-like papers on topological insulators are out on the market. If one of the authors
of these notes (a.a.) had to go to an island, just one paper on TI in the luggage allowed, it would be this one:
X.-L. Qi, T.L. Taylor, and S.-C. Zhang, Topological field theory of time-reversal invariant insulators, Phys.
Rev. B. 78, 195424 (2008). (Remarkably, that paper was written a year or so before the general hype on TIs
broke loose.)
1
2
CHAPTER 1. WHAT IS A TOPOLOGICAL INSULATOR?
1.1.1
Setup
Barring complications such as static disorder and particle interactions, we may think of an
insulator as a periodically extended quantum system. A one-dimensional cartoon of such a
system is shown in Fig. 1.1: imagine a one-dimensional quantum system of length L. For
convenience, and not inflicting significant physical consequences, we compactify our system,
i.e. we think of it as a ring, R, of circumference L (cf. Fig. 1.2, inset.) Now, assume that we
put a periodic potential onto our ring. Mathematically, this will be described by a function
V : R → R, x 7→ V (x), where V (x + a) = V (x) and a ≡ L/N defines the periodicity
interval dividing our ring into N 1 unit cells. Next, we assume our ring to be populated by
particles (’electrons’) of mass m. The quantum dynamics of these particles will be described
1 2
p̂ + V (x̂), where x̂ and p̂ are the coordinate and the position
by a Hamilton operator Ĥ ≡ 2m
2
operator, respectively, obeying canonical commutation relations [p̂, x̂] = −i. Representing
the momentum operator in a coordinate representation, p̂ → −i∂x , we may interpret Ĥ as
a linear operator in the Hilbert space HL ≡ L2 (R, C) of complex valued square integrable
functions on the ring.
1.1.2
Bloch theorem
The periodicity of the potential entails the reducibility of the Hamilton operator down to
effectively finite-dimensional operators acting in Hilbert spaces much simpler than the full HL .
To see this, notice that the periodicity of Ĥ entails the discrete symmetry
[Ĥ, T̂a ] = 0,
(1.1)
where T̂a is the translation operator by a, i.e. (T̂a f )(x) = f (x+a) for functions f ∈ HL . This
means that Ĥ and T̂a can be simultaneously diagonalized. A complete set of eigenfunctions
3
of T̂a is given by the Bloch functions,
ψk,n (x) ≡ eikx φk,n (x),
(1.2)
4
, 2 2π
, . . . , N 2π
= 2π
are N so-called Bloch momenta, φk,n (x) = φk,n (x + a)
where k = 2π
L
L
L
a
are functions periodic on the so-called unit cell [0, a], and, for each value of k, the index
n labels a complete set of functions {φk,n |n = 0, 1, 2, . . . } of the reduced Hilbert space
Ha ≡ L2 ([0, a], C).
The definition (1.2) implies an eigenvalue equation T̂a ψk,n = eika ψk,n and a set of reduced
1
(p̂ + k)2 + V : Ha → Ha acting in the eigenspaces Ha of the
Hamilton operators Ĥk ≡ 2m
translation operators T̂a . It is convenient to chose the functions φk,n to be eigenfunctions of
Ĥk (the set of all these eigenfunctions, {φk,n } is complete in Ha .)
2
Planck’s constant ~ = 1, throughout.
The option to represent the wave functions in this way, is stated by the Bloch theorem.
4
The quantization of Bloch momenta in units of 2π/L is required to ensure periodicity ψk,n (x + L) =
ψk,n (x).
3
1.1. WHAT IS AN INSULATOR ANYWAY?
1.1.3
3
Band structure
We have reduced the linear operator Ĥ (an
’infinite dimensional matrix’) to a one parameter family chain of N block matrices Ĥk .
Suppose we had managed to diagonalize the
operators Ĥk , to find a spectrum of eigenvalues {k,n } (along with the Bloch eigenfunctions {φkn }.) We then wind up with the situation depicted in Fig. 1.2, where we imagine
the spectrum {k,n } plotted for each value of k. In the limit L/a → ∞, the spacing between consecutive values of k, 2π/L shrinks to zero, and we may think of the assignment
[0, 2π/a] → R, k 7→ k,n as smooth functions, indexed by n. These function are known as
(energy) bands, and the entity of all bands defines the band structure of the solid. The
spectrum of the Hamiltonians Ĥk is unbounded from above, i.e. one might suspect that there
is an infinity of bands. This, however, is a consequence of our modeling of a lattice system (the
solid) in terms of a continuum Hamiltonian. At any rate, only a finite and small number of
bands carries physical significance, as we will show momentarily. Notice that the 2π/a periodicity of the Bloch ’phases’ exp(ika) means that we may think of the bands as mappings from
a circle ([0, 2π/a] + periodic boundary conditions) into the reals. For illustration, the figure
above shows the band structure of a realistic system (Bi2 Te2 Se(111) thin films) numerically
calculated by band structure methods.
Figure 1.2: Schematic band structure of one-dimensiona solid
We now invest a little bit of physics to show why only a small number of bands really matter,
and why the introduction of Bloch Hamiltonians Ĥk amounts to a massive simplification of
the problem. In a solid, single particle eigenstates ψk,n will be filled by fermions (no double
occupancy!). The ground state of the system is formed by filling up states successively,
beginning with the states of lowest energy, and ending at state no. N counted from below in
energy, where N is the number of particles. The energy of the highest occupied state is called
the Fermi energy. Now, that Fermi energy may lie somewhere within the image, [En− , En+ ]
of the band function for some value of n, or it may coincide with the maximum energy of a
band, EF = En+ . In the former case, we speak of a metal, in the latter of an insulator. To
4
CHAPTER 1. WHAT IS A TOPOLOGICAL INSULATOR?
understand the fundamental difference between the two cases, imagine our system exposed to
a ’small’ perturbation, such as an external electric field, a heat source, external radiation, or
similar. Formally, we may represent this perturbation by addition of some perturbing operator
δ Ĥ to the native Hamilton operator of the system. ’Small’ means that the eigenvalue of
δ Ĥ, i.e. the energies provided by the perturbation, are usually much smaller than the gaps
∆E between consecutive bands. In a metal, unoccupied states ∈ [En− , En+ ] right above the
Fermi energy are available (cf. Fig. 1.3, top.) This means that the energies deposited by the
perturbation suffice to change the occupation (the ’state’) of the system – a metal is responsive
to external perturbation. By contrast, the external energy does generally not suffice to change
the state of an insulator (for that would require energy ∼ ∆E.) The insulator remains inert,
5
which manifests itself in poor heat or electricity conduction properties.
Figure 1.3: Top: in a metal, the Fermi energy lies within a band. Small amounts of energy
may be deposited in the system to change its particle occupation – the system is responsive to
perturbation. Bottom: in an insulator, a minimum energy equal to the band gap, ∆E, between
consecutive bands has to be provided to change the state of the system. The insulator is largely
inert to perturbation.
In general, an insulator will contain N− occupied and N+ empty bands. The highest lying
occupied band is called valence band, while the lowest lying empty band goes by the name
valence band. From our discussion above, it should be evident that these two bands play
a dominant role in the physics of an insulating material, i.e. many physical properties of
insulators can be understood, focusing on the case N− = N+ = 1.
While the so-far discussion has been restricted to one spatial dimension, generalization to
higher dimensional systems is straightforward. In general, a crystalline system of dimension
d comes with d independent translational symmetries. Accordingly, we need to introduce d
Bloch momenta, k ≡ {ka }, a = 1, . . . , d and the bands become functions over the d-Torus,
6
Td = S 1 × . . . S 1 , i.e. n : Td → R, k 7→ n,k . The torus Td is an example of a d-dimensional
| {z }
d
Brillouin zone.
5
Insulators of anomalously small band gap are called semiconductors. As is indicated by the name, their
physical properties are intermediate between those of metals and insulators.
6
For lattices characterized by translational symmetries more complex than the one discussed here (triagonal,
or hexagonal lattices, for example) Brillouin zones more complex than a torus emerge.
1.2. TOPOLOGICAL INSULATORS
1.2
5
Topological insulators
Only very recently it has been appreciated that, their notorious inertia to perturbations notwithstanding, insulators can display rather interesting physical behavior. Broadly speaking, this
discovery rests on three observations:
. The band structure of some insulators encodes nontrivial topological structure.
. Every real life insulator comes with a surface.
. In cases where the bulk topology of the system is non-trivial, the surface actually shows
metallic behavior, and rather unusual metallic behavior at that.
In hindsight, it appears strange that it took so long to realize these points: three decades
ago, the quantum Hall insulator (sic) was discovered. Referring for a discussion of the
quantum Hall system to the second part of these notes, we here just mention that soon
after the experimental discovery of the quantum Hall effect it became understood that the
phenomenon rests on an interplay of bulk insulating behavior characterized by the presence of
a topological index and a metallic surface, and that these two features imply each other. For
some reason, it took nearly three decades to generalize this bulk/surface correspondence to
other system classes.
Of course, these remarks will raise more questions than they answer. In particular, one
will ask (i) what is meant by ’metallic surface’, and what the consequences of this metallic
behavior are, (ii) what we mean by ’nontrivial topology, and (iii) how (i) and (ii) are connected.
1.2.1
Hall response
The detailed discussion of the bulk-surface correspondence alluded to above will be the subject of
the second part of these notes. At this point, all we
need to know that it holds responsible for the nonvanishing of a class of physical observables generally known as Hall conductivities. Suppose you
subject a sample of the insulator under consideration to a ’voltage gradient’ in a direction we call the
x-direction. That gradient may refer to an electric
voltage drop across the sample, a ’thermal gradient’ (realized by heating one end of the system
and cooling the other), or a ’magnetic gradient’
(attaching opposite ends to systems of differing
magnetization.) A metallic system would respond
to this perturbation by the flow of currents, dominantly in a direction aligned with the voltage
gradient. The insulator doesn’t have this option, for otherwise it wouldn’t be an insulator.
However, some – topological – insulators respond by current flow, Iy , in a direction, y, transverse to x. Depending on the type of attached voltage, that current may be electric current,
6
CHAPTER 1. WHAT IS A TOPOLOGICAL INSULATOR?
heat current, or spin-current. The seeming self-contradiction of current flow in an insulating
system gets resolved by the observation that the current is carried by surface states forming
at the effectively metallic boundaries of the bulk insulating material (cf. inset of the figure.
Again, we refer to the second parts of the notes for a microscopic discussion of the formation
of metallic boundaries in a bulk insulating system.
The Hall conductance Gxy = Iy /Vx measures the ratio of applied current to driving
voltage. Both, current and voltage are related in non-universal (geometry dependent) ways to
the microscopic instances current density, jy , and electric field, Ex , underlying the transport
7
process. Physicists often prefer to work with the microscopic quantities, and to characterize
the transverse current flow by the Hall conductivity
σxy ≡
jy
.
Ex
(1.3)
8
Without proof we state that the Hall conductivity is given by
σxy =
ie2 1 X ha|p̂x |bihb|p̂y |ai
− (x ↔ y),
h Ωm2 <E
(a − b )2
a
(1.4)
F
b >EF
9
where {|ai} is a complete system of eigenstates of the bulk Hamilton operator Ĥ, a are the
eigen-energies, and Ω is the system volume.
7
The current density j = jy ω is a two form whose integral over a surface
R S perpendicular to the y direction
gives the current, Iy , (particles per unit time) flowing through S, Iy = S j, where ω is the area two-form of
S. Omitting the latter it is customary, to refer to the weight function jy as a current density. Similarly, the
electric field E = Ex dxRis a one-form whose line integral over a curve, γ, locally aligned in x-direction gives
the voltage drop, Vx = γ E. Again, the one-form dx is usually omitted when we talk about the electric field.
8
Eq. (1.4) is proven by computing the expectation value of the current density jy to first order in quantum
mechanical perturbation theory in an external field Ex . This type of perturbation theory goes under the name
linear response theory, and is discussed in advanced textbooks of quantum mechanics.
9
Remarkably, the Hall conductivity can be computed entirely from bulk properties of a fictitious system
subject to periodic boundary conditions (i.e. no boundaries), although the physical processes supporting the
Hall current take place at surfaces!
1.3. TOPOLOGY
1.3
7
Topology
In a breakthrough contribution, Thouless and collabora10
tors managed to show that the Hall conductivity σxy is a
topological invariant.
To understand how topology enters the stage, let us consider the simplest non-trivial setting, an insulator of dimension d = 2, with just one occupied and one empty band,
N+ = N− = 1. The reduced Hilbert space is then given by
Ha = C2 , the eigenvalue equation Ĥk |ψn (k)i = n (k)|ψn (k)i
defines a valence band ground state |ψ0 (k)i, and a conduction
band excited state |ψ1 (k)i along with their eigenvalues 0 (k) < 1 (k). All these constituents
are defined over the two torus T2 3 k (cf. the figure.)
Consider now the valence band ground states |ki ≡
|ψ0 (k)i, assumed to be normalized hk|ki = 1 for convenience. As usual in quantum mechanics, these states are
defined only up to a global phase, i.e. for any φ(k) ∈
[0, 2π], the state eiφ(k) |ki is an equally valid ground
state. We can thus think of the ground state in terms
of the equivalence class of Hilbert space states
[|ki] = {g|ki|g ∈ U(1)}.
Equivalently, we may say that the ground states defines
a U(1) principal bundle P . For each point, k ∈ T2 ,
the ’structure group’, U(1) acts by multiplication of a
reference ground state |ki with a phase g ∈ U(1) (see the figure), i.e. to k, we attach a ’fibre’
{g|ki|g ∈ U(1)} isomorphic to U(1).
The U(1) principal bundle P comes with a canonical connection a known as the Berry
connection. This connection is defined by the one-form
a(k) ≡ ihk|dki.
(1.5)
In a more explicit component-representation, a = aµ (k)dk µ , where the components aµ (k) =
ihk|∂µ ki = ihψ0 (k)|∂µ ψ0 (k)i ∈ R (why are the components aµ real-valued?), and ∂µ ≡ ∂k∂µ .
Physicists think of aµ (k) as a ’vector potential’. Under the action of the structure group (a
’change of gauge’) |ki → g(k)|ki ≡ exp(iφ(k))|ki, the connection form transforms as
a(k) → a(k) + ig(k)−1 dg(k) = a(k) − dφ(k),
which is the transformation behavior required for a connection form on an (abelian) principle
bundle. The components change aµ → aµ − ∂µ φ like the components of a vector potential,
as expected.
10
Q. Niu, D.J. Thouless, and Y.-S. Wu, Phys. Rev. B 31, 3372 (1985).
8
CHAPTER 1. WHAT IS A TOPOLOGICAL INSULATOR?
Notice that the Berry connection ’knows’ about the global structure of the ground state,
in particular it is aware of its ’topological structure’, as we are going to discuss next. We
define the Berry curvature, f = da = ihdk| ∧ dki, or
f = fµν dk µ ∧ dk ν ,
fµν = ih∂µ k|∂ν ki.
The integral of f over the torus defines the first Chern number of the system,
Z
1
f ∈ Z.
C1 ≡
2π T2
(1.6)
(1.7)
The integer valued Chern number is a topological signature of the ground state. Two ground
states |ki and |k0 i of two Hamilton operators Ĥ and Ĥ 0 , respectively, cannot be continuously
deformed into one another if they carry different Chern numbers. In this case, the two insulators
described by Ĥ and Ĥ 0 are topologically distinct. The only way to deform two systems
of different topological signature into one another is by closing the band gap, i.e. via an
intermediate metallic configuration. Insulators of non-vanishing Chern signature are called
topological insulators, all others are dismissively referred to as ’trivial insulators’.
Crucially, the first Chern number of the system determines the Hall conductivity as
σxy =
e2
C1 .
h
(1.8)
The importance of this relation can hardly be exaggerated. It states that the measurable ob11
servable ’Hall conductivity’ enjoys robust topological protection. Conversely, the observation
of a non-vanishing and universal Hall conductivity signifies the presence of a nontrivial ground
state.
INFO The proof of Eq. (1.8) is based on straightforward quantum mechanical perturbation
theory and we sketch it here for completeness for the case N+ = N− = 1. (Generalization to a
larger number of bands is straightforward.) We start by expressing Eq. (1.4) in a Bloch basis,
|ai = eik·x̂ |ψ0 (k)i and |bi = eik·x̂ |ψ1 (k)i for |ai and |bi in the valence band and conduction band,
12
resp., where x̂ = {x̂µ } is the two component position operator obeying canonical commutation
relations [x̂µ , p̂ν ] = iδνµ . Using that e−ik·x p̂eik·x = p̂ + k = m∂k Ĥk we represent Eq. (1.4) as
σxy =
hψ1 (k)|∂µ Ĥk |ψ0 (k)ihψ0 (k)|∂ν Ĥk |ψ1 (k)i
ie2 1 X
µν
.
h Ω
(1 − 0 )2
(1.9)
k
We next establish the connection to the Berry curvature. To this end, take a point k ∈ T2
and consider a vector q ∈ R2 ’tangent’ to the two-dimensional Brillouin zone. The C2 -valued
11
The quantization gets compromised only if the bulk spectral gap between valence and conduction band
closes, e.g. due to excessive amounts of disorder.
12
We use ’covariant’ (superscript) indices to indicate that the coordinates xµ are ’dual’ to kµ – maybe the
notation isn’t perfect.
1.3. TOPOLOGY
9
differential form |dki ≡ |dψ0 (k)i then evaluates as |dψ0 (k)i(q) = lim→0 −1 (ψ0 (k + q)i −
|ψ0 (k)i). The state |ψ0 (k + q)i solves the Schrödinger equation
Ĥk+q |ψ0 (k + q)i = 0,k+q |ψ0 (k + q)i.
It is straightforward to verify that to first order in Ĥk+q − Ĥk = ∂q Ĥk + O(2 ) this equation is
solved by
|ψ0 (k + q)i = |ψ0 (k)i + hψ1 (k)|∂q Ĥk |ψ0 (k)i
|ψ1 (k)i + O(2 ),
1 − 0
where ∂q Ĥk ≡ ∂k q=k Ĥk . From here, we readily obtain |dψ0 (k)i = hψ1 (k)|∂µ Ĥk |ψ0 (k)i |ψ1 (k)dk µ
and
f = ihdψ0 (k)| ∧ dψ0 (k)i =
hψ0 (k)|∂µ Hk |ψ1 (k)hψ1 (k)|∂ν Ĥk |ψ0 (k)i µ
dk ∧ dk ν .
(1 − 0 )2
Integrating this expression over T2 and converting the integral to a (Riemann) sum as
(2π)2 P
k we establish the equality of (1.9) and (1.8).
Ω
R
T2
→
Above, we discussed the particular case of a two-dimensional topological insulator, governed
by a Hamiltonian of no particular symmetries, other than the mandatory hermiticity (a class A
Hamiltonian in d = 2.) We found a Z-valued classification in terms of the first Chern number
computed for the Berry connection. Insulators in different dimensions and of different symmetry can be classified in different, if conceptually related ways. This includes Z2 classification,
Z-classifications in terms of higher Chern classes, or the absence of nontrivial topology. The
classification of the topological contents of insulators in according to symmetry and physical
dimension is the subject of the ’periodic table’ of topological insulators.
INFO Above, we have arrived at the classification of the d = 2 class A insulator by computing
Chern numbers. The same result can be obtained by homotopy theory. The two approaches
are surely related to each other, but the mathematics of this connection (Chern-Weyl theory) is
beyond the scope of the author.
Pick a value of k, say k = 0, compute the occupied and the empty eigenstates, |ψ0,0 i and |ψ0,1 i,
and identify them with orthonormalized basis vectors e0 and e1 , of C2 , respectively. The two
eigenstates at a general value k can be obtained by acting on the states e0,1 by a unitary transformation Uk ∈ U(2). Actually, we don’t need the full group U(2) to generate the transformation:
two transformation groups U(1) stabilizing the states e0 and e1 , respectively, (up to a phase) can
be divided out, i.e. transformations drawn from the manifold Gr(1, 2) ≡ U(2)/U(1) × U(1) suffice
13
to encode the information on the system of eigenstates. The manifold Gr(1, 2) is a particular
case of a Grassmannian.
The relevant information on the system’s band structure is encoded in the mapping T2 →
Gr(1, 2), k 7→ Uk assigning to each k the transformation to its eigenstates. An insulator is
’trivial’ if this transformation can be continuously deformed to a trivial (unit) transformation. In
other words, the information on the state-topology is contained in the homotopy classes of the
above mapping. Now, Gr(1, 2) ' S 2 , the two-sphere. The mappings (torus)→(two-sphere) can
be classified according to a Z-valued winding number, and we are back to the result above.
13
For systems containing a general number of bands, this generalizes to Gr(N− , N+ + N− ) = U(N+ +
N− )/U(N+ ) × U(N− ).
10
CHAPTER 1. WHAT IS A TOPOLOGICAL INSULATOR?
Figure 1.4: Band structure of the Hamiltonian defined by (1.10) and (1.12) in the vicinity of
the gap closing transition at m = 2. Left panel: r = 1.6, Center panel: r = 2, the critical
point, Right panel: r = 2.3.
1.4
The emergence of Dirac Fermions
Above we argued that the topological index of an insulator can only change via a bulk gap
14
closing transition. It is instructive to observe this phenomenon on a concrete example. We
consider the two-band Hamiltonian characterized by the Bloch Hamiltonians
Ĥk =
X
σi di (k),
(1.10)
i
where d(k) ≡ {di (k)} is a three-dimensional vector of real valued coefficients. The eigenvalues
15
of Ĥk are given by 0/1,k = ±|d(k). At zero Fermi energy, EF = 0, the system is an insulator,
provided |d(k)| =
6 0.
Now, let’s assume |d(k)| > 0, globally and define the map
φ :T2 → S 2 ,
k 7→ d̂(k) ≡
d(k)
.
|d(k)|
It is then not difficult to verify that the Chern number computes the number of times d̂(k)
winds around the sphere as
1
C1 =
4π
14
Z
1
φω=
4π
T2
∗
Z
2π
dk 1 dk 2 d̂(k) · (∂1 d̂(k) × ∂2 d̂(k)),
(1.11)
0
The situation may be more complicated in the presence of strong particle interactions. However, we won’t
address this point here.
15
In an insulator, the Fermi energy can be placed anywhere in between the valence and the conduction band.
It’s detailed position is of little relevance, as long as there aren’t any unoccupied states of energy < EF .
1.4. THE EMERGENCE OF DIRAC FERMIONS
11
where ω is the area two-form on the sphere. Now, let us take a look at the concrete represen16
tation


sin(k1 )
,
sin(k2 )
d(k) = 
(1.12)
r + cos(k1 ) + cos(k2 ),
where r is a real parameter. For large values |r| 1, the vector
d̂(k) is approximately aligned
in ±e3 -direction and does not cover the full sphere, i.e. C1 |r|1 = 0. Band closures occur at
the points r = 2, r = 0, and r = −2, where |d(k)| = 0 for k = (π, π), k = (0, π), (π, 0),
and k = (0, 0), resp. One may verify by direct calculation that the Chern number jumps as
r=−2
r=0
r=2
0 −→ 1 −→ −1 −→ 0 upon r crossing the sequence of critical points.
At a critical point, the Brillouin zone contains one or several ’hot spots’, where the gap
between the upper and the lower band closes in a linear manner. This is illustrated in Fig. 1.4
for m = −2. In the vicinity of the hot spot k = (0, 0) the Hamiltonian can be linearized as
Ĥ ' k1 σ1 + k2 σ2 + mσ3 ,
where m = r + 2. This is a two-dimensional Dirac Hamiltonian. Defining the linearized
vector d(k) = (k1 , k2 , m), the integral (1.11) may be computed in the linearized approximation, and one finds a jump − 21 → 21 as the ’Dirac mass’ m crosses from negative to positive
values. From this, we conclude that the regions of the Brillouin zone outside the linearization
domain k ' (0, 0) contribute 12 to the integral, so that the value of C1 is given by 12 − 12 = 0
and 21 + 21 = 1 for m < 0 and m > 0, respectively.
The take home message of this discussion is that
. Chern numbers change at bulk gap closing transitions.
. The vicinity of a transition point in the Brillouin zone can be approximated by an effective
Dirac Hamiltonian.
. While the value of the Chern number can only be obtained by an integral over the full
Brillouin zone, the integer jump of the topological invariant can be described within the
local Dirac approximation.
. You cannot tell whether an insulator is ’trivial’ or not just by staring at the band structure, nor by inspection of the mass term in a Dirac Hamiltonian. Only changes in the
topological index can be diagnosed in this way.
16
For a concrete lattice representation of this model Hamiltonian, see the second part of the notes.
12
CHAPTER 1. WHAT IS A TOPOLOGICAL INSULATOR?
Chapter 2
The bulk-boundary-correspondence
The second part of these lecture notes is devoted to the so-called bulk-boundary correspondence. The main objective of this chapter is to show that if a band structure of an insulator
encodes a nontrivial topology there are protected surface modes rendering the surface metallic. These surface modes are protected by the existence of a bulk excitation gap and cannot
be removed unless a symmetry is broken or the bulk gap is closed. Our main tool to show
these surface modes is in the form of toy models and exact solutions of systems with sharp
boundaries, so called open boundary conditions. This has to be contrasted from the first part
where only systems with periodic boundary conditions were considered.
2.1
Electrons in a magnetic field - the quantum Hall effect
The quantum Hall effect is the mother of topological order in electronic systems and as
such deserves a special spot in any discussion about topological order. It was experimentally
discovered in the beginning of the eighties and has not lost its appeal since then as one of the
most impressive manifestations of topology and the connection between physical observables
and topological invariants. We start with a review of the main experimental observations
and theoretical description of electrons subject to a magnetic field starting from a classical
description in terms of Drude’s transport theory all the way to a full quantum mechanical
treatment. The (quantum) Hall effect is observed in two dimensional electronic systems.
The standard setup is a sample such as depicted in Fig. 2.1(a) where a current I is driven
through the sample (injected in C1 and leaves the sample at C4 ) and voltages are measured
at the contacts C2 − C3 and C5 − C6 . In such a setup a two dimensional electron system is
subject to a perpendicular magnetic field. This magnetic field breaks time reversal symmetry
and invalidates momentum conservation. The presence of the magnetic field has spectacular
consequences which we now explore. We start with a discussion of classical physics and become
increasingly quantum mechanical.
13
14
CHAPTER 2. THE BULK-BOUNDARY-CORRESPONDENCE
(a)
I
C2
C1
C3
I
2D electron gas
C6
longitudinal
resistance
C5
C4
RH
Hall resistance
(b)
Hall
resistance
magnetic field B
Figure 2.1: (a) standard setup for a quantum Hall measurement. A two dimensional electron
gas is subject to a perpendicular magnetic field. A current I is driven through the sample.
The different contacts C1 → C6 allow to measure longitudinal and transversal (Hall) voltages,
which can be related to resistances. (b) Classical (low field) regime of the Hall effect. The
transverse resistance varies linearly with the magnetic field giving access to the nature of the
charge carriers as well as the density.
2.1.1
The classical Hall effect
As usual, when there is a quantum version of an effect, we tend to expect there also is a
classical version. The Hall effect does not deceive in that department and indeed in 1879
Hall showed that the transverse resistance RH of a thin metallic plate varies linearly with the
strength of the applied perpendicular magnetic field B [Fig. 2.1(b)],
RH =
B
,
qnel
(2.1)
where q is the charge of the charge carrier (q = −e for electrons in units of the elementary
charge) and nel is the charge carrier density in two dimensions. Intuitively, one may understand
the effect appealing to Newton’s law and the Lorentz force, which deflects the trajectory of a
charged particle thereby invalidating momentum conservation. This leads to a density gradient
which is built up between the two opposite sample sides leading to a potential difference which
can be measured for instance between contacts C5 and C3 , see Fig 2.1. Within this setup
the linear behavior is expected to hold at low fields. A measurement of the Hall resistance
in the classical low field regime still is a very valuable experimental tool, which, in contrast
to longitudinal resistance, is independent of the disorder and thus gives access to the nature
(electron vs. hole) and density of charge carriers, see discussion below.
A qualitative understanding of the classical Hall effect is achieved within Drude’s model of
diffusive transport in metals. One considers independent point like charge carriers subject to
Newton’s law
p
p
×B − ,
ṗ = −e E +
m
τ
2.1. ELECTRONS IN A MAGNETIC FIELD - THE QUANTUM HALL EFFECT
15
where E and B are the electric and magnetic fields, respectively. We consider transport of
negatively charged particles ( electrons with q = −e) with band mass m. The last term is
introduced phenomenologically and describes momentum relaxation processes due to scattering
of electrons from impurities (note that for B = E = ~0 we have p(t) = p0 e−t/τ ). The
characteristic relaxation time τ is called the scattering time. The macroscopic transport
characteristics can be obtained from the static solution of the equation of motion, ṗ = 0. For
two dimensional electrons where p = (px , py ) we find a set of equations
px
eB
py − ,
m
τ
eB
py
=
px −
,
m
τ
eEx = −
eEy
which describes the stationary solutions. We have chosen the magnetic field to be aligned with
the z-direction. The above expressions can more compactly be rewritten using a characteristic
frequency,
ωc =
eB
,
m
(2.2)
which is called cyclotron frequency. This quantity characterizes the cyclotronic motion of a
charged particle in a magnetic field due to the Lorentz force (it will receive a slightly different
but related meaning in the fully quantum mechanical solution). Introducing Drude’s result for
the longitudinal conductivity in absence of a magnetic field,
σ0 =
nel e2 τ
,
m
(2.3)
we end up with
px
py
σ0 Ex = −enel − enel ωc τ,
m
m
px
py
σ0 Ey = enel ωc τ − enel .
m
m
This can be related to the current density via
j = −enel
p
.
m
(2.4)
We immediately realize that the magnetic leads to a matrix structure connecting the current
and the electric field, E = ρ j, where ρ is the four component resistivity tensor
1
1
1
ωc τ
1
µB
−1
ρ=σ =
=
.
(2.5)
σ0 −ωc τ 1
σ0 −µB 1
In the last step we have introduced the mobility
µ=
eτ
.
m
(2.6)
16
CHAPTER 2. THE BULK-BOUNDARY-CORRESPONDENCE
which is another important characteristic of a metal. One can now readily read off the Hall
resistivity (the off-diagonal terms of the resistivity tensor ρ)
ρH =
eB
m
B
ωc τ
=
τ ×
=
.
2
σ0
m
nel e τ
enel
(2.7)
As mentioned before, τ cancels and one has direct access to nel and the sign of the charge
carrier. The conductivity tensor follows from the resistivity (2.5) by matrix inversion
σL −σH
−1
σ=ρ =
,
(2.8)
σH σL
with σL = σ0 /(1 + ωc2 τ 2 ) being the longitudinal conductivity and σH = σ0 ωc τ /(1 + ωc2 τ 2 )
the Hall/transverse conductivity. An interesting property of these expressions is revealed if we
consider the limit of a clean system, i.e., τ → ∞. In this case the resistivity and conductivity
tensors read
B 0
0 − enBel
enel
ρ=
and
σ=
,
(2.9)
enel
0
− enBel 0
B
respectively. In this limit it is obvious that the matrix structure is important: had we only
considered the longitudinal components, we would have come to the conclusion that the
(longitudinal) resistivity would vanish while the (longitudinal) conductivity is infinite. This,
however, is not true. It turns out that in the clean limit ωc τ → ∞ transport properties are
entirely governed, in the presence of a magnetic field, by the off-diagonal, i.e. transverse,
components of the conductivity/resistivity. This feature of quantum Hall systems is also
important when discussing the integer quantum Hall effect below.
2.1.2
The quantum Hall effect
The discovery of the integer quantum Hall effect (IQHE) by v. Klitzing, Dorda, and Pepper in
1980 was recognized with the Nobel Prize and, as mentioned before, introduced topological
concepts into the description of electronic systems. It was facilitated by enormous advances
in the ability to fabricate high mobility two dimensional electronic systems, which we will not
discuss within these lectures.
The integer quantum Hall occurs at low temperatures and high magnetic fields. It manifests
itself in a quantization of the Hall resistance, thereby invalidating the linear in B behavior
discussed previously. It reveals plateaus at particular values of the magnetic field (see Fig.
2.2). Inside the plateaus, the Hall resistance is given in terms of universal constants – it is
indeed a fraction of the inverse quantum of conductance G0 = e2 /h, and one observes
RH = G−1
0
1
,
n
(2.10)
in terms of an integer n. The plateau in the Hall resistance is accompanied by a vanishing
longitudinal resistance. This already gives us a first hint that we might actually consider an
2.1. ELECTRONS IN A MAGNETIC FIELD - THE QUANTUM HALL EFFECT
17
Figure 2.2: Typical signature of the quantum Hall effect (schematic drawing). Each plateau
in the Hall resistance is accompanied by a vanishing longitudinal resistance. The classical Hall
resistance is at the lower field limit and shows the linear behavior (note that the longitudinal
resistance in that regime is roughly constant, as suggested by Drude’s formula).
18
CHAPTER 2. THE BULK-BOUNDARY-CORRESPONDENCE
odd form of an insulator with a small remaining transverse conductivity. We will discuss at
length that this is in fact true and the only place where conduction is taking place is at the
boundaries.
The quantization of the Hall resistance (2.10) is a universal phenomenon, i.e., independent
of the particular properties of the sample, such as its geometry, the host materials used
to fabricate the 2D electron gas and, even more importantly, its impurity concentration or
distribution (in fact, disorder is even vital in establishing it). This universality is the dream
of any theoretical physicist and is the reason why – since 1990 – it is used as the resistance
standard,
RK−90 = h/e2 = 25 812.807 Ω,
(2.11)
which is also called the Klitzing constant. As suggested by the name, the understanding of
the quantum Hall effect requires a quantum mechanical description of the electrons in the two
dimensional system, which is what we attempt now.
2.1.3
Landau levels
We start with discussing a single electron whose movement is confined to a two dimensional
plane in an applied perpendicular magnetic field. In order to describe electrons in a magnetic
field, one needs to replace the momentum by its gauge-invariant form
p → Π = p + eA(r),
(2.12)
where A(r) is the vector potential that generates the magnetic field B = ∇ × A(r). Notice that since A(r) is not gauge invariant, neither is the momentum p, only the canonical
momentum Π is. Adding the gradient of an arbitrary differentiable function λ(r), A(r) →
A(r) + ∇λ(r), does not change the magnetic field. Indeed, the momentum transforms as
p → p − e∇λ(r) under a gauge transformation which compensates the transformed vector
1
potential, such that Π is gauge-invariant.
In the case of electrons on a lattice, this substitution is more tricky because of the presence
of several bands. Furthermore, the vector potential is unbound, even for a finite magnetic
field; this becomes clear if one chooses a particular gauge, such as e.g. the Landau gauge
AL (r) = B(−y, 0, 0), in which case the value of the vector potential may become as large as
B × Ly , where Ly is the macroscopic extension of the system in the y-direction. However,
one can shown that the substitution (2.12), the so-called Peierls’ substitution in the context
of electrons on a lattice, remains correct as long as the lattice spacing a is much smaller than
the magnetic length
r
~
lB =
,
(2.13)
eB
1
In a more mathematical language, the vector A = (A1 , A2 )T defines the components of a U(1) connection
form A = Aµ dxµ , not to be confused with the Berry connection discussed in the first part of the notes. The
curvature dA = µνρ B µ dxν ∧ dxρ defines the field strength, Π is the associated covariant derivative, and the
gauge λ defines an element exp(iλ) ∈ U(1) of the structure group.
2.1. ELECTRONS IN A MAGNETIC FIELD - THE QUANTUM HALL EFFECT
19
which is the fundamental length scale in the presence ofpa magnetic field. Because a is typically
an atomic scale (∼ 0.1 to 10 nm) and lB ' 26 nm/ B[T], this condition is fulfilled in all
atomic lattices for the magnetic fields, which may be achieved in today’s high-field laboratories
(∼ 45 T in the continuous regime and ∼ 80 T in the pulsed regime).
With the help of the (Peierls) substitution (2.12), one may thus immediately write down
the Hamiltonian for charged particles in a magnetic field if one knows the Hamiltonian in the
absence of a magnetic field,
H(p) → H(Π) = H(p + eA) = H B (p, r).
Notice that because of the spatial dependence of the vector potential, the resulting Hamiltonian
is no longer translation invariant, and the (gauge-dependent) momentum p is no longer a
conserved quantity. We will limit our discussion to Hamiltonians corresponding to parabolic
dispersion
HB =
1
[p + eA(r)]2
=
Π2x + Π2y .
2m
2m
(2.14)
It is straightforward to show within the Landau gauge, i.e., AL = −yB(1, 0, 0) that
[Πx , Πy ] = −i
~2
.
2
lB
(2.15)
This equation is the basic result of this section and one can convince oneself that it is indeed
gauge invariant by recalling, that the canonical momentum is gauge invariant. There are
two aspects which immediately follow from this consideration: (i) The components of the
gauge-invariant momentum Π are mutually conjugate in the same manner as x and px or y
and py . Remember that px generates the translations in the x-direction (and py those in the
y-direction). This is similar here: Πx generates a “boost” of the gauge-invariant momentum
in the y-direction, and similarly Πy one in the x-direction. (ii) As a consequence, one may
not diagonalize at the same time Πx and Πy , in contrast to the zero-field case, where the
arguments of the Hamiltonian, px and py , commute. This again is a manifestation of the fact
that magnetic field invalidates momentum conservation.
In order to solve the Hamiltonian (2.14) it is convenient to use the pair of conjugate
operators Πx and Πy to introduce ladder operators in the same manner as in the quantummechanical treatment of the one-dimensional harmonic oscillator. Remember from basic
quantum-mechanics that the ladder operators may be viewed as the complex position of the
one-dimensional oscillator in the phase space, which is spanned by the position (x-axis) and
the momentum (y-axis),
x
p
1
x
p
1
†
−i
and
ã = √
+i
,
ã = √
p0
p0
2 x0
2 x0
p
√
where x0 = ~/mω and p0 = ~mω are normalization constants in terms of the oscillator
frequency ω. The fact that the position x and the momentum p are conjugate variables and
20
CHAPTER 2. THE BULK-BOUNDARY-CORRESPONDENCE
the particular choice of the normalisation constants yields the commutation relation [ã, ㆠ] = 1
for the ladder operators.
In the case of the 2D electron in a magnetic field, the ladder operators play the role of a
complex gauge-invariant momentum (or velocity), and they read
lB
lB
and
a† = √ (Πx + iΠy ) ,
(2.16)
a = √ (Πx − iΠy )
2~
2~
where we have chosen the appropriate normalization such as to obtain the usual commutation
relation
[a, a† ] = 1.
(2.17)
Landau levels
In terms of the gauge-invariant momentum, the Hamiltonian (2.14) for non-relativistic electrons reads
1
Π2x + Π2y .
HB =
2m
The analogy with the one-dimensional harmonic oscillator is apparent if one notices that both
conjugate operators Πx and Πy occur in this expression in a quadratic form. If one replaces
these operators with the ladder operators, one obtains, with the help of the commutation
relation (2.17),
~2 †2
HB =
a + a† a + aa† + a2 − a†2 − a† a − aa† + a2
2
4mlB
~2
1
~2
†
†
†
=
a a + aa =
a a+
2
2
2mlB
mlB
2
1
= ~ωc a† a +
,
(2.18)
2
2
between the cyclotron frequency (2.2) and the
where we have used the relation ωc = ~/mlB
magnetic length (2.13) in the last step.
As in the case of the one-dimensional harmonic oscillator, the eigenvalues and eigenstates
of the Hamiltonian (2.18) are those of the number operator a† a, with a† a|ni = n|ni. The
ladder operators act on these states in the usual manner
√
√
and
a|ni = n|n − 1i,
(2.19)
a† |ni = n + 1|n + 1i
where the last equation is valid only for n > 0 – the action of a on the ground state |0i gives
zero,
a|0i = 0.
(2.20)
The energy levels of the two dimensional charged particles are therefore discrete and labelled
by the integer n,
1
n = ~ωc n +
.
(2.21)
2
Importantly, these levels are massively degenerate.
2.1. ELECTRONS IN A MAGNETIC FIELD - THE QUANTUM HALL EFFECT
21
Figure 2.3: Classical skipping-orbit picture: the boundary prevents electrons form completing
classical revolutions and introduces chiral channels due to bouncing of the sample boundary.
2.1.4
The occurrence of boundary modes
The purpose of this section is twofold: on the hand it is intended to convince the reader that a
quantum Hall system is an insulator. Secondly, we want to convince the reader that it is not!
Obviously, these two statements do not go together well and we will show that the boundary
provides the clue towards resolving this seeming contradiction.
The quantum Hall system as an insulator
We start our discussion with the energy levels given in Eq. (2.21). We see equally spaced
energy levels, which are massively degenerate. It turns out that the massive degeneracy can
easily be seen in the Landau gauge AL = −yB(1, 0, 0). In that gauge we can still us the
momentum kx as a good quantum number, meaning that the wave functions can be written
as Ψn,kx (x, y) ∝ eikx x φn,kx (y). The ’bulk’ band structure of the quantum Hall system would
thus be the one in Fig. 2.4(a), where we see completely flat, i.e., dispersionless bands. In
the case where the chemical potential lies in between Landau levels, the system is thus a true
’bulk’ insulator and one would not expect any sort of conductivity whatsoever.
The subtle question about why the chemical potential resides between Landau levels requires a discussion involving disorder which is beyond the scope of this lecture. For the moment
we just accept this as a given. We can now ask what happens if we consider boundaries. A
classical picture again gives us an important clue: classically in a magnetic field we expect
closed orbits. However, near the boundaries electrons cannot complete revolutions, which leads
to them bouncing along the boundaries. One can easily convince oneself that this introduces
a chirality into the system, see Fig. 2.3. This picture usually is referred to as the skipping orbit
picture and it is simply a consequence of the applied magnetic field and the boundary. We
can now ask how this classical picture gets modified if we consider the quantum mechanical
problem with boundaries. To that end we consider a bar with periodic boundary conditions
in x-direction and a harmonic confining potential V = 12 mω 2 y 2 in y-direction. This is a very
22
CHAPTER 2. THE BULK-BOUNDARY-CORRESPONDENCE
crude modeling of the boundary of a quantum Hall bar, but it has the appeal of being analytically solvable and showing the emergence of chiral edge channels. We leave the details of
the calculation to the interested reader and only sketch the key steps in the derivation. The
Hamiltonian reads
H=
1
1
Π2x + Π2y + mω 2 y 2
2m
2
(2.22)
and the key to its solution is that in Landau gauge and in the presence of the confining
harmonic potential we can still use plane wave solutions in x-direction. This allows to map
this problem to a simple harmonic oscillator, with energy levels given by
1
1 2
E (kx ) = ~ω̄c n +
+
k .
(2.23)
2
2m0 x
There we observe that the formerly flat Landau levels become dispersive, meaning one can
n
.
define a one dimensional group velocity as vn,kx = dE
dkx
Fig. 2.4(a) the right panel shows that the Landau levels as a function of the momentum
kx become dispersive if we subject the planar system to a harmonic confining potential in
y-direction. It is now very instructive to discuss the role of the chemical potential in this
situation.
We find that depending on where the chemical potential lies, the number of dispersive
modes crossing the chemical potential is different, see Fig. 2.4(b): more specifically, we see
that between the first and the second Landau level, there is one dispersive mode with a positive
group velocity, and another one with a negative group velocity. These modes correspond to
the chiral channels and one is running along one edge in x-direction with a positive velocity,
the other one with a negative group velocity along the other edge (not shown here, but later
for the lattice model). If we now change the chemical potential to lie between the second
and the third Landau level, we see that two channels add, again one with a positive and one
with a negative group velocity. Again these channels are confined to move along opposing
sample edges and thus are chiral. Upon iteration we see that for every Landau Level lying
below the chemical potential we obtain one chiral channel respectively at the opposing edges.
2
It is well known that for a dissipationless channel we obtain a conductance of G0 = eh , which
is a consequence of the contact resistance/conductance and no further dissipation within the
channel. This readily implies that the whole conductance/conductivity is thus simply given by
σ = nG0
(2.24)
where n simply denotes the number of Landau levels below the chemical potential. This picture
also gives us an idea why the quantum Hall effect is so robust against any kind of modification
of the system properties. Due to the spatial separation by the width of the sample W the
chiral channels are protected against backscattering since on the two respective edges there
is no open backscattering channel. This observation is at the root of the robustness of the
quantization. We will find that in the case of the quantum spin Hall insulator there will be
forward and backward moving channels on each edge, which are however protected against
2.1. ELECTRONS IN A MAGNETIC FIELD - THE QUANTUM HALL EFFECT
(a)
V (y)
E(kx )
23
E(kx )
µ
kx
(b)
kx
E(kx )
E(kx )
µ
µ
dE
vx =
<0
dkx
kx
dE
vx =
>0
dkx
dE
vx =
<0
dkx
kx
vx =
dE
>0
dkx
Figure 2.4: (a) in the homogeneous system in Landau gauge the Landau levels are perfectly
flat as a function of kx , relating to the notion of dispersion less levels. If a harmonic confining
potential is added the formerly flat bands become. (b) One observes that as the chemical
potential is located between Landau levels there always is the same number of dispersing
channels with positive (respectively negative) group velocity transversing the chemical potential
as there are Landau levels below the chemical potential. The channels with the same sign of
velocity are located at the same edge, while the opposing sign lives at the opposing edge.
24
CHAPTER 2. THE BULK-BOUNDARY-CORRESPONDENCE
backscattering by the presence of time reversal symmetry. In that case the protection is less
robust since local perturbations on each side can open backscattering channels.
We now show that the above picture of edge channels does not depend on boundary
conditions by solving a tight binding Hamiltonian on a lattice, which has periodic boundary
conditions in x-direction and open boundaries in y-direction. With this we mean that we
consider a finite sample in y-direction, while the sample is periodic in x-direction. We use
a nearest-neighbor hopping Hamiltonian on the square lattice with only s-wave hopping for
simplicity. It turns out there is a straightforward generalization of the Peierls’ substitution to
hopping Hamiltonians. We choose the following Hamiltonian in second quantization, again in
Landau gauge
H = −t
X
~
R
e
~y †
i φφ R
0
cR~ cR+x
~
−t
X
c†R~ cR+y
+ h.c. .
~
(2.25)
~
R
In the above formulation the operator cR~ denotes the annihilation operator of an electron,
t is the hopping amplitude which leads to the bandwidth of the band, φ0 = he is the flux
quantum, and φ is a measure of the magnetic field which pierces an elementary plaquette.
It is important to note at this stage that while the Hamiltonian is translationally invariant in
~
i φ R
x-direction, it is not in y-direction due to the presence of the Peierls factor, i.e., e φ0 y . This
implies that we cannot use Fourier transform to solve the Hamiltonian, which is in agreement
with the treatment of the quadratic dispersion. However, we can still use Fourier transform
in x-direction. For every value of kx we have to diagonalize a matrix of the dimension of the
system in y-direction. We have done this for a finite lattice of Ly = 800a, a being the lattice
constant and φ = 0.15. This is a good choice to address the so-called quantum Hall regime
for the following reason. In order to enclose a flux quantum an electron has to hop around 40
plaquets, which corresponds to a path much larger than the elementary lattice unit, but it is
still much shorter than the system size. In that sense we have the hierarchy a lB Ly .
The results are shown in Fig. 2.5. The main observation is as hinted at before, namely the
presence of boundary modes, which are spatially separated.
2.2
Quantum Hall effect without external field: Chern
insulator
The quantum Hall effect, as discussed above, happens for two dimensional electrons in a
transverse applied magnetic field. It appeared vital in our discussion that there are Landau
levels in the electronic structure. One can now ask the question, to which extend Landau levels
are vital in the Hall effect. Or, is it possible to formulate a tight binding model which shows
the Hall effect even in absence of Landau levels? In 1988 Haldane (Physical Review Letters 61,
2015 (1988)) managed to answer this question affirmatively in a paper whose value cannot be
overestimated for the understanding of topological insulators in general. In a sense explained
later it also provides the elementary building block of the quantum spin Hall insulator. The
model in Haldane’s version was originally formulated on the honeycomb lattice, but I will
2.2. QUANTUM HALL EFFECT WITHOUT EXTERNAL FIELD: CHERN INSULATOR 25
(a)
(b)
x
~ i+1
A
y
~i
A
d~l
d~l
i+y
i
(c)
right-mover
dE
vx =
>0
dkx
3
left-mover
dE
vx =
<0
dkx
2
1
-3
-2
-1
1
2
3
-1
Out[548]=
-2
y=0
y = Ly
2nd LL
1st LL
-3
Figure 2.5: (a) Square lattice tight binding problem subject to a magnetic flux through the
elementary plaquettes. We choose periodic boundary conditions in x-direction and finite system size (open boundaries in y-direction; (b) implementation of the flux in the tight-binding
Hamiltonian, Eq. (2.25)); (c) ’band structure’ in the finite system with Landau levels and
dispersive channels connecting Landau levels. The left panel shows the probability distribution
of finding an electron at a given y-value for the left- (blue) and right- mover (red). We see
the clear spatial separation of the two. The picture continues to hold if we increase the filling,
meaning the chemical potential.
transform it to a square lattice later for convenience. The two-band model is described by
the Hamiltonian (1.10), (1.12) discussed above. We saw that the different ’phases’ (here:
regimes characterized by different values of a topological index) are distinguished by a nonlocal property of the band structure. Specifically, the Hall conductivity was determined by the
Chern numbers (1.11).
2.2.1
Phase diagram of the Chern insulator
Despite the seeming simplicity of the above model it turns out to be rich. One finds that
the model has four phases which can be distinguished from their respective Chern number
introduced in the previous chapter. This is shown in Fig. 2.6 where for different values of the
mass parameter m the Chern number is shown. Note that in order to change the topological
index one has to close the bulk gap at some point. In the specific model this happens at
26
CHAPTER 2. THE BULK-BOUNDARY-CORRESPONDENCE
Figure 2.6: (a) Electronic excitation gap in the system as a function of the parameter m. For
|m| > 2 we have an ordinary insulator with Chern number n = 0, while for −2 < m < 0 we
have a Chern insulator with n = 1 and one with n = −1 for 0 < m < 2. (b) Plot of the Chern
number n as function of m. We clearly see that we have to pass a gapless point in order to
change the topological sector.
m = ±2 and m = 0. It is obvious from comparing the bulk band structure of the model for
m = 0.1 with m = −0.1 that there is no way from inspection to tell that there is such a
stark difference between the two cases. For clarity I have added a plot for the two situations
in Fig. 2.7.
2.2.2
Boundary modes with open boundary conditions
After having analyzed the bulk band structure of Haldane’s model we are now considering what
happens at the boundaries if we diagonalize the system with finite extension in y-direction and
periodic boundary conditions in x-direction. In order to do so we have to go back to the tightbinding formulation which for convenience we do on a simple square lattice with two orbitals
on each site. We start from the Hamilton operator, which in second quantization reads
X †
X † σz + iσx
† σz + iσy
ΨR+x
+
Ψ
Ψ
+
h.c.
+
m
ΨR~ ΨR~ .
(2.26)
H=
ΨR~
~
~
~
R
R+y
2
2
~
R
~
R
In a next step we Fourier transform the Hamiltonian in x-direction, but leave y intact. For
every value of kx we have a one dimensional hopping problem which reads
X
H (kx ) =
Ψ†y (kx ) (m + sin kx + cos kx ) Ψy (kx )
y
X
σz + iσy
†
+
Ψy (kx )
Ψy+1 (kx ) + h.c. .
2
y
(2.27)
This is now solved by exact diagonalization. We find that the spectrum we get is mainly a
projection of the bulk spectrum onto kx in the Brillouin zone which gets denser the longer the
2.2. QUANTUM HALL EFFECT WITHOUT EXTERNAL FIELD: CHERN INSULATOR 27
Figure 2.7: (a) Bulk spectrum as obtained from the diagonalization of a system with periodic
boundary conditions for m − 0.1; (b) band structure for m = −0.1. Despite there being no
visible difference the two situations can be distinguished by the Chern number as topological
index.
chain we diagonalize (the number of bands corresponds to the length of the chain). However,
this is not all we find. Instead we find that there are additional modes which traverse the bulk
gap and essential invalidate the picture of an insulator if 0 < |m| < 2, see Fig. 2.8. This
is the range in which we have Chern numbers, which are not zero. In Fig. 2.8 we observe
that the location of the crossing points of the edge states changes as we go from m = −0.1
to m = 0.1. Furthermore, the edge state with positive group velocity changes the edge,
and equivalently for the negative group velocity. This is consistent with the Chern number
changing from n = 1 to n = −1.
2.2.3
Boundary modes: a continuum picture
A simple picture of the occurrence of gapless surface modes at an interface between a Chern
insulator and vacuum can be gained in the framework of a toy model calculation. We model the
interface between a Chern insulator with a Chern number n 6= 0 and vacuum by an interface
between an insulator with n = 1 and one with n = 0, which is topologically equivalent to the
vacuum. In the above model we choose the vicinity of m = −2, where a phase transition from
n = 0 to n = 1 takes place. At m = −2 there is a single Dirac cone sitting at the Γ-point,
i.e., kx = ky = 0, see Fig. 2.9. If we expand the Hamilton operator around Γ in momentum
space in the vicinity of m=-2 we find
H(k) ≈ kx σx + ky σy + (m + 2)σz .
(2.28)
We parametrize m = −2 + δm, such that δm > 0 corresponds to the Chern insulator
with n = 1 and δm < 0 corresponds to n = 0. We thus assume a system has translational
invariance in x-direction, while in y-direction we have δm > 0 for y < 0 and δm < 0 for y > 0.
This mass profile mimics an interface between a Chern insulator and a trivial band insulator
and explicitly breaks translation invariance in y-direction. For that reason the Hamiltonian will
28
CHAPTER 2. THE BULK-BOUNDARY-CORRESPONDENCE
Figure 2.8: (a) Energy spectrum of the system with periodic boundary conditions in x-direction
and open boundary conditions in y-direction. We see that while for the bulk spectrum with
periodic boundary conditions we obtained a real insulator, see Fig. 2.7 the occurrence of
boundary modes, which traverse the bulk excitation gap. These modes, as before, live at the
boundary. (b) same for m = −0.1. The location of the crossing point has changed and also
one finds that the location of right- and left-movers has changed going from (a) to (b), in
agreement with the Chern number changing from −1 to 1.
Figure 2.9: At m = −2 there is a single Dirac cone sitting at the Γ-point (0, 0). If m ≈ −2
we can approximate the bulk theory by a low-energy theory of a single Dirac cone. The upper
right panel shows a possible mass profile δm(y) used in the toy model calculation.
2.2. QUANTUM HALL EFFECT WITHOUT EXTERNAL FIELD: CHERN INSULATOR 29
assume the following form
H (kx , y) = kx σx − i∂y σy + δm(y)σz
(2.29)
which we are now going to solve for the profile chosen (for our purposes it is unimportant
which function δm(y) is used, it only matters to take a profile which makes the aforementioned
sign change; for an example see Fig. 2.9 in the upper right panel). In the limit in which the
asymptotic value of limy→±∞ δm(y) = ∓δm0 is large compared to other scales we analyze
the following Hamiltonian
H (kx = 0, y) = −i∂y σy + δm(y)σz .
(2.30)
We are looking for a zero energy solution which implies we have to solve the equation
(−i∂y σy + δm(y)σz ) Φ(y) = 0
(2.31)
where for Φ(y) we make the ansatz Φ(y) ∝ φeiλ(y) , where φ is a two-component spinor.
Plugging in the ansatz and multiplying the equation with σz yields
−iλ0 (y)σx φ + δm(y)φ = 0 .
σx has two eigenvalues ±1 and eigenvectors
tors lead to two differential equations
√1 (1, 1), √1 (−1, 1).
2
2
∓iλ0 (y) + δm(y) = 0
(2.32)
The two different eigenvec-
(2.33)
Ry
Ry
0
0
which are solved by λ(y) = ∓i 0 δm(y 0 )dy 0 and thus Φ± (y) ∝ φ± e± 0 δm(y )dy . Normalizability of the wave function excludes the solution Φ− (it is unbound) leading to the following
allowed wave function
R
Ry
y
1
0
0
0 )dy 0
1
δm(y
e 0 δm(y )dy .
(2.34)
Φ+ (y) ∝ φ+ e 0
=√
1
2
We see that this solution describes a solution which is confined to the vicinity of y = 0 in lateral
direction. We now establish that there is a propagating solution in x-direction by projecting
the boundary Hamilton operator into the basis of the Φ+ which leads to a one dimensional
Hamiltonian given by
H 1d (kx ) = φT+ H (kx , y = 0) φ+ = kx
(2.35)
which describes a single chiral channel with positive group velocity moving into x-direction.
It is solved by a plane-wave eikx x and corresponds to the gapless mode with positive velocity
confined to the interface.
30
CHAPTER 2. THE BULK-BOUNDARY-CORRESPONDENCE
Figure 2.10: Schematic of a quantum spin Hall insulator. At the boundary with vacuum
two counter propagating modes with different spin quantum numbers appear. Backscattering
between the two is forbidden due to spin-orbit coupling. In a nutshell one can picture this
system as two copies of Haldane’s model on top of each other, where one model for spin up
with Chern number n = 1 is superimposed to one for spin down with Chern number n = −1.
2.3
The quantum spin Hall insulator
The quantum spin Hall insulator is a new class of insulator which is a cousin of the aforementioned Chern insulator. However, it is a system with intact time-reversal symmetry, which
implies without magnetic field (not only without net magnetic field, but entirely without magnetic field). It turns out the major player in that context is spin-orbit coupling. So far we have
discussed spinless fermions, but now spin-orbit coupling turns out to be a vital ingredient. The
first description was again carried out on the honeycomb lattice in the context of graphene,
and it was discussed in a paper by Kane et al. (Physical Review Letters 95, 146802 (2005)). In
experiment, however, this insulator turned out not to be accessible due to the ridiculously low
value of the spin-orbit coupling in graphene. Later on it was realized that in HgTe quantum
wells the physics of the quantum spin Hall insulator can be realized, which was experimentally
achieved by König et al. in 2007 (Science Vol. 318 no. 5851, pp. 766-770 ). In a nutshell,
the quantum Spin Hall insulator can be thought of as two copies of Haldane’s model: one for
electrons with spin up with Chern number n = 1 and another one for electrons with spin down
with Chern number n = −1. This is pictured in Fig. 2.10.
2.3. THE QUANTUM SPIN HALL INSULATOR
31
Acknowledgements
LF especially acknowledges M. O. Goerbig for giving the permission to use some of the material
including figures from his lecture notes (arXiv:0909.1998). He furthermore thanks E. Altman,
A. Rosch, and M. Sitte for collaborations on related works. Special thanks goes to M. Sitte
for carefully reading the notes and providing valuable comments and figures.