Topological aspects of Aharonov-Bohm vortices in topological band



Topological aspects of Aharonov-Bohm vortices in topological band
Topological aspects of Aharonov-Bohm vortices
in topological band insulators
Robert-Jan Slager
supervisor: Prof.dr.J. Zaanen
Topological band insulators provide for a major research interest in presentday condensed matter physics. Their non-trivial topological order has many
consequences like metallic edge states, which are robust to impurities, and the
emergence of particles with fractional charge and statistics. Moreover, the topological characterization allows for a general classication of topological band
insulators by specifying the discrete symmetries and the spatial dimension.
In the broad sense, the work presented in this thesis is inspired by the 'quest'
for some additional structure to this classication, originating from the underlying lattice. Namely, we studied the response of a quantum spin Hall insulator
upon introducing a -ux vortex, which mimics the eect of a lattice dislocation
when the gap opens at the point in the Brillouin zone. Specically, we show
that the existence of a Kramers pair of zero-energy modes bound to the vortex
is a generic feature of topologically non-trivial phase of the m B model, describing the quantum spin Hall state. We analytically derive the explicit form
of these states, which shows their exponentially localized nature. Furthermore,
we demonstrate the correspondence of the zero-energy states to the metallic
surface states and analyze the resulting non-trivial quantum numbers. We then
conclude by placing these results in a more general context.
1 Introduction
1.1 Concept of the topological band insulator . . . . . . .
1.1.1 General remarks on topological band insulators
1.1.2 Historical perspective . . . . . . . . . . . . . .
1.1.3 Field theory perspective . . . . . . . . . . . . .
1.2 Overview of this thesis . . . . . . . . . . . . . . . . . .
2 Models of topological band insulators
2.1 Two dimensional quantum spin Hall eect . . .
2.1.1 Eective model . . . . . . . . . . . . . .
2.1.2 Corresponding tight-binding model . . .
2.1.3 Topological characterization . . . . . . .
2.1.4 Edge states . . . . . . . . . . . . . . . .
2.2 Three dimensional topological band insulators .
2.2.1 Eective model . . . . . . . . . . . . . .
2.2.2 Edge states and surface Hamiltonian . .
. 5
. 5
. 7
. 13
. 15
3 Topological eects of a -ux vortex in the quantum spin Hall
3.1 Zero-energy solutions in the presence of a ux vortex . . . . . .
3.1.1 Derivation of the zero-energy bulk states . . . . . . . . . .
3.1.2 Self-adjoint extension . . . . . . . . . . . . . . . . . . . .
3.2 Non-trivial quantum numbers of the zero mode . . . . . . . . . .
3.2.1 The Su-Schrieer-Heeger model . . . . . . . . . . . . . . .
3.2.2 Non-trivial quantum numbers of the zero-mode . . . . . .
3.3 Interpretation and context of results . . . . . . . . . . . . . . . .
3.3.1 Nature of the zero-energy states in the dierent phases of
the m B model . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Possible implications of the zero-mode for the classication of topological band insulators . . . . . . . . . . . . . 52
4 Conclusions
A Lauglin's argument
B Eective Hamiltonian of graphene
C Manifolds and vector bundles
D Adiabatic curvature and Chern numbers
E Discrete symmetries
F Anomalous quantum Hall eect of the massive Dirac equation 75
G The Aharonov-Bohm eect
H Atiyah-Singer theorem
Chapter 1
1.1 Concept of the topological band insulator
This chapter addresses the fundamental aspects of the topological band insulator.
Starting from some general notions, the theoretical and experimental advances
concerning the topological band insulator are sketched in historical perspective.
Additionally, some of the corresponding eld theory is highlighted. The chapter
concludes with an overview of the thesis.
1.1.1 General remarks on topological band insulators
Condensed-matter is concerned with the physical properties of macroscopic objects. One of the most important aspects is how order emerges from the interactions of the relevant elementary constituents, like ions and electrons. In many
instances the order is explained by means of symmetry breaking. For example,
in the formation of a crystal lattice the electrostatic repulsion of the ions results
in a periodic array, breaking continuous translation and rotation symmetries.
However, in order to describe the integer quantum Hall eect a completely different characterization based on topology is needed. Topology essentially is the
study of classes of objects invariant under smooth deformations, known as homotopies. A coee cup handle can, for instance, be smoothly deformed into a
torus, whereas for the two-sphere this is not possible (Figure (1.1)). Moreover,
it is possible to assign invariants to distinguish the resulting equivalence classes,
being the analogue of the order parameters associated with broken symmetries.
The notion of topology may be used to classify band structures. Namely,
when there is a well dened Brillouin zone one can readily map it into the Bloch
space of Hamiltonians, producing a band structure. Consequently, the way how
Figure 1.1: Illustration of the basic idea of topology. Topology essentially studies
maps representing smooth deformations. In this case the gure shows a heuristic
picture of such a homotopy between a coee cup and a torus.
these bands are knotted denes topological invariants characterizing dierent
equivalence classes of knotted spectra, that can continuously be deformed into
each other. Moreover, as long as the system is an insulator the topology of the
bands is protected. Hence, we conclude that there might be insulators that can
be distinguished from the ordinary ones by means of topology. The topological
insulator is essentially dened in that manner.
Figure 1.2: Intuitive picture of topological band insulators and the resultant
edge states [70]. Represent the trivial insulator with a simple loop (right) and
the topological band insulator by a trefoil knot (left). In order to pass from the
trivial regime to the non-trivial phase, the string has to be cut open (middle),
illustrating the gapless edge states.
An important consequence of the topological insulator is the existence of
metallic states localized at the boundary of the system. Indeed, they are protected by the topology[23]. This can be illustrated by a simple analogy; represent the non-trivial knotting of the bands in the case of the topological insulator
by a non-trivial knot and illustrate the ordinary insulator by a simple loop. In
1.1 Concept of the topological band insulator
order to pass from a topological insulating phase to a trivial one the string has
to be cut open, straightened out and glued back together to obtain the trivial
loop. The open ended string then represents the edge states providing for the
very signature of topology.
1.1.2 Historical perspective
The ourishing of topological notions in condensed matter physics originated
from the study of the integer quantum Hall eect(IQHE); it provides for one of
the fundamental examples of topological order.
Integer quantum Hall eect
The quantum Hall eect[4, 5] is realized in two dimensional electron systems
subjected to a suciently low temperature and high external magnetic eld.
It features chiral metallic states localized on the edge of an insulating sample, whose direction of propagation is determined by the sign of the external
magnetic eld. Moreover, the number of edge states is in one-to-one correspondence with the integer lling factor and therefore results in a quantized Hall
conductance (xy ).
Laughlin[18] was the rst to realize that given its complete universality the
quantization must be insensitive to continuous deformations of the sample geometry and deeply rooted in the gauge principle. Specically, he used the
periodicity emerging from the gauge invariance to dene an adiabatic quantum
pump; upon completion of one cycle the Hamiltonian is adiabatically mapped
on a gauge equivalent copy, leaving the total basis of single electron functions
unchanged. However, this does not mean that the individual basis states are
mapped onto themselves. Indeed, a simple calculation (Appendix A.1) shows
that during one cycle electrons are transferred to the edge of the sample.
Although Laughlin's argument is very intuitive, it is not conclusive due to
the wave nature of the electrons. The denite answer came from topology.
Thouless et al.[13][15] showed that the charge transported during the adiabatic
ux change can be related to the Kubo formula[30]. Moreover the resulting
expression can be recognized as an integer topological invariant; it is the rst
Chern class of a U(1) principle bre bundle on a torus, with the magnetic
Brillouin zone being the torus and the Bloch waves the corresponding bres[12,
14]. The resulting invariants are known as the rst Chern characters (Appendix
D) and basically measure the twisting of the bundles during one cycle. Hence,
the total Hall conductance is calculated by summing over all the rst Chern
1 corresponding with the occupied bands m
characters Cm
xy =
e2 X 1
C :
h m m
Although the IQHE can be characterized by topology, the topological nontrivial phase is driven by the external magnetic eld. The reason is that the
IQHE is intimately related to time-reversal symmetry(TRS) breaking. However, Haldane realized that a macroscopic magnetic eld is not required. He
proposed a theoretical model exhibiting an integer quantum Hall eect without
a net external magnetic eld.
Haldane model
The Haldane model[3] essentially is a two dimensional lattice model of a semimetal in a perpendicular external magnetic eld, with a local ux density that
obeys the full symmetry of the lattice and has a vanishing zero total ux through
one unit cell. Explicitly, Haldane used the eective model of the electrons of
graphene[1][2]. In the model the electron's spin is suppressed and the resulting
spectrum consists of two bands that touch at two inequivalent points, referred
to as Dirac points, in the hexagonal Brillouin zone (Appendix B).
Introduction of the magnetic eld breaks time-reversal symmetry, while there
are no Landau levels formed. Furthermore, the next nearest neighbor hoppings
attain a phase producing a time-reversal breaking mass term mT , that gaps
the energy spectrum. Subsequently, Haldane included an additional inversion
breaking mass mI by assuming dierent on-site energies for the two sublattices.
Explicit calculation of the rst Chern character then shows the existence of the
quantum Hall state when jmT j > jmI j, producing the phase diagram depicted
in Figure (3.11). Moreover upon crossing the critical lines jmT j = jmI j the
system exhibits chiral edge states; the two bands are inverted closing the gap at
one of the Dirac points producing two opposite metallic chiral movers, bound
to opposite surfaces of the system (Figure(3.11)).
1.1 Concept of the topological band insulator
Figure 1.3: Illustration showing the band spectrum of the Haldane model[3, 84].
Left panel: The phase diagram of the model in the mI mT plane. The eective
model suppresses spin and therefore has two bands(), of which the lower one( )
is completely lled. Calculating the Chern character results in the values as
depicted in the diagram. We notice that the total Chern number is conserved,
but that upon entering the nontrivial phase one Chern unit is exchanged between
the two bands.
Right panel: the band spectrum in the high symmetry directions for dierent
mI j and with K and K 0 refereing to the two Dirac points. In case of
values of j m
mI j = 1, displayed with the solid line, the bands are inverted
the critical value j m
around the Dirac point, producing a cone as is shown in the inset. The resulting
chiral movers join to form one chiral edge state circulating at the outer perimeter
of the quantum Hall insulator.
The Haldane model was signicant because it showed that in principle a
nite net external magnetic eld is not needed to induce a quantum Hall state.
Rather, local uctuations of the magnetic eld may lead to band inversion, resulting in a non-trivial phase.
Quantum spin Hall eect
More recently, it became clear that spin-orbit coupling[8] can provide for such
topological non-triviality, due to inversion of the band spectrum. Although
spin-orbit coupling does not break time reversal symmetry it can lead to a Hall
eect that involves the spin degree of freedom, known as the quantum spin Hall
eect. The spin Hall eect is very similar to the integer quantum Hall eect.
Indeed, essentially the quantum spin Hall state may be viewed as two time reversed quantum Hall states, producing an insulator exhibiting two time reversed
helical edge states.
A realistic model, based on the Haldane model, was proposed by Kane and
Mele[10]. In this model the spin is coupled to the hopping term, inducing a
Figure 1.4: Illustration showing the resulting energy bands obtained by Kane
and Mele[8, 10]. Specically, the spectrum was solved for an one dimensional
zigzag geometry.
The left panel shows two time reversed bands in the spin Hall state, whereas the
right panel depicts the same spectrum in the trivial insulating state. The nontrivial state exhibits two pairs of time reversed helical edge states traversing the
gap. The edge states can be shown to be localized at the surface of the sample.
Moreover the resultant helical movers circulating the sample are protected from
Inset: the phase diagram, with so , and R referring the coecients of the
spin-orbit, inversion and Rashba term respectively.
phase. As a result, two Haldane copies, with opposite Hall conductivity for
the the up and down spins, are produced. In addition, the model includes a
Rashba term, which breaks mirror symmetry about the plane and results in
non-conservation of spin in the perpendicular direction.
Solving the band spectrum, Kane and Mele showed the quantum spin Hall
state. Namely, for certain values of the parameters determining the relative
strength of the perturbation terms two time reversed pairs of states cross the
gap to the opposite cone, as shown in Figure (3.12). Further inspection shows
these states are indeed helical movers localized at the edge of the system.
Time reversal symmetry and Z2 invariants
As the quantum spin Hall eect respects time reversal symmetry the rst Chern
character is obviously zero. In addition, quantum spin Hall systems do not exhibit quantized spin Hall conductivity. So at rst sight it does not seem obvious
how to dene topological order. However, in this case the time reversal invariance plays an important role in the classication. It turns out that time
1.1 Concept of the topological band insulator
reversal symmetry imposes conditions on the classication of the Bloch waves
as bres of the torus dened by the Brillouin zone, leading to a Z2 invariant [12, 17, 51, 77]. The Z2 invariant indicates whether the system is in the trivial
state ( = 0) or in the quantum spin Hall phase ( = 1).
This can also be understood on more physical grounds[10, 22, 40, 41]. Recalling that time reversal (Appendix E) dictates
H (k)
= H ( k);
we conclude that at points in Brillouin zone satisfying H (k) = H ( k) the
Hamiltonian commutes with . Hence, these points host Kramers pairs, which
are protected by time reversal symmetry. In the quantum spin Hall state the
Kramers pairs switch partners an odd number of times, essentially leading to
an odd number of crossings of the Fermi surface. Hence, the spectrum cannot
be deformed continuously into a trivial insulator, leading to protection of the
surface states. This closely resembles the band inversion phenomenon in the
quantum Hall state of the Haldane model.
Three dimensional topological insulator
The notion of the two dimensional quantum spin Hall state was subsequently
generalized to three dimensions[40, 41, 51]. In three dimensions, general homotopy arguments show that the system is characterized by four Z2 invariants,
denoted by (0 ; 1 ; 2 ; 3 ). Physically, 0 amounts to whether the Fermi surface
encloses an odd number of Kramers degenerate Dirac points. When this is the
case (0 = 1) the system is referred to as a 'strong' topological insulator. This is
an intrinsically non-trivial state, just as its two-dimensional counterpart. However, one can also imagine stacking two dimensional quantum spin Hall states
on top of each other creating a three dimensional system. This is known as the
'weak' topological insulator ( = 0) and is not stable to disorder, meaning that
it has no protected surface states.
The remaining invariants dene a vector, reecting in which plane the Kramers
points switch pairs. In the case of the 'weak' spin Hall state this pertains to
the Miller indices corresponding to the stacking of the two dimensional planes,
whereas in the 'strong' case this refers to the position of the protected Dirac
cone in the Brillouin zone.
Experimental realizations
Although the Kane-Mele model provides for a theoretical model, the spin orbit
coupling in graphene is relatively weak, making it dicult to implement experimentally. However, Zhang et al.[45] predicted that a topological insulator
with a quantized charge conductance could be realized in a HgTe quantum well
HgTe quantum wells consist of a thin layer of HgTe sandwiched between two
pieces of CdTe. The resultant low energy description, referred to as the m B model, is given in terms of four bands of well-dened spin. This model will be
considered more specically, in the next chapter. The crucial property, however,
is that the mass term changes sign as a function of an external parameter, being
the thickness of the well. Specically, there is some critical thickness dc at which
the bands are inverted, giving rise to a change in the rst Chern character. The
resulting quantized charge conductance was indeed experimentally veried in
Subsequently, in 2008 angle-resolved photo emission spectroscopy (ARPES)
experiments identied the rst three dimensional topological insulator Bix Sb1 x [53].
The verication of Bix Sb1 x as a topological insulator initiated a search for
topological insulators made out of heavy elements, showing relatively strong
spin-orbit interactions and small band gaps. A year later this search culminated in the discovery of topological insulating phases in bismuth and antimony
alloys, like Bi2 Se3 , Bi2 Te3 and Sb2 Te3 [49][66][67]. These topological insulators
of the 'next generation' exhibit non-trivial behavior at higher temperatures.
Moreover, whereas the non-trivial band structure of Bix Sb1 x shows complex
surface states, they show a full rotational invariant single cone with one spin
per momentum state (Figure (1.5)). The simple structure of the surface states
triggered experiments focusing on the manipulation of the protected edge states,
which may nd their application in spintronic and magnetoelectric devices and
is subject of present day research.
1.1 Concept of the topological band insulator
Figure 1.5: Illustration of the fully rotational Dirac cone, as featured by the
'next generation' topological insulators. Unlike Bix Sb1 x , which is characterized
by (0 ; 1 ; 2 ; 3 ) = (1; 1; 1; 1) and therefore has rather complex surface states,
these materials show a Dirac cone with (0 ; 1 ; 2 ; 3 ) = (1; 0; 0; 0). In both
cases, however, the edge states are helical. Consequently, the spin orientation
rotates along with the momentum, as indicated by the arrows.
1.1.3 Field theory perspective
The key features regarding topological order may also be understood from a
eld theory perspective. Topological phases of matter are universally described
by topological eld theories, analogous to the way symmetry-breaking phases
are described by Landau-Ginzburg eld theories [35].
In order to make this somewhat more concrete, we rst consider the quantum
Hall eect. It turns out that a quantum Hall response results from coupling
a (2 + 1) dimensional massive Dirac equation to an external gauge eld A.
Explicitly, computing the linear part of the current-current correlation function
shows that the Hall conductance equals 2eh sign(mt ), in terms of the time reversal
breaking mass mt (Appendix F). Hence, the system suers an anomaly; even in
the limit of the mass going to zero, the resulting transverse conductance is nonzero. Further inspection shows that this essentially is the continuum version of
the Haldane model, which also exhibits anomalous chiral movers as mt changes
sign.[3] More importantly, after integrating out the massive fermions we indeed
nd a topological term governing the response[31][33][54]
Scs C
dtA @ A
; ; = f0; 1; 2g:
This eective action is known as the Chern-Simons term, and is manifestly topological due to the presence of the rst Chern character C 1 and the fact that it
does not depend on the metric. The Chern-Simons term has been well studied
in the 1980s in order to understand the details of axion electrodynamics[33][34].
Basically, it endows particles with a ux, allowing for excitations with fractional
charge and statistics. We will explicitly see that these notions meet with topological considerations in the Su-Schrieer-Heeger model[27], but it also becomes
apparent in the description of the fractional quantum Hall eect. This Hall effect is reminiscent of the integer Hall eect, but is characterized by a fractional
lling factor due to electron-electron interactions[32].
Generalizing the eld description of the IQHE to (4 + 1) dimensions subsequently allows for a natural denition of the eective eld theory governing the
time reversal invariant topological insulator, being two time reversed copies of
the time reversal breaking system. The analogue of Eq. (1.2) takes the form
Scs C 2
d4 x
dtA @ A @ A
; ; ; ; = f0; 1; 2g;
with C 2 the second Chern character, which plays the exact same role as the
rst Chern number[39]. More importantly, it can be shown that the action describing topological band insulators with time reversal symmetry in (2 + 1) and
(3+1) dimensions can be obtained from this fundamental model by a procedure
called dimensional reduction. Basically, dimensional reduction results in a family of parameterized Hamiltonians of reduced dimension, which are connected
due to the von Neumann-Wigner theorem[55]. Moreover, time reversal symmetry dictates that the dierence in second Chern character of any two continuous
interpolations of such descended Hamiltonians is even and hence denes two different classes, characterized by the parity of the second Chern number. Further
evaluation shows that when the second Chern number is even(odd), there exist
an odd(even) number of Dirac cones. Consequently, one obtains a Z2 invariant,
being exactly 0 in the three dimensional case and the usual Z2 invariant in the
two dimensional case.
1.2 Overview of this thesis
1.2 Overview of this thesis
Topological band insulators are characterized by topologically non-trivial maps
from the Brillouin zone into the Bloch space of Hamiltonians. This can be generalized through a mathematical structure producing a classication of topological insulators[57, 59]. Essentially, specifying Hamiltonians by time-reversal
symmetry, charge conjugation symmetry and chiral symmetry reveals ten distinct classes[60]. Moreover, using general homotopy arguments it is possible
to determine the number of topological distinct maps from the Brillouin zone
into any of the specic classes of Hamiltonians in d spatial dimensions. As a
result, one obtains a table indicating the topological invariant for each class of
dimension d (Table (1.2)).
AI 1 0
BD1 1 1
0 1
DIII -1 1
AII -1 0
CII -1 -1
0 -1
CI 1 -1
Table 1.1: Table showing the ten classes, as described by Atland and
Zirnbauer[60]. The classes are specied by the dimension d, time reversal symmetry , charge conjugation symmetry and chiral symmetry = . Specically, the table denotes the absence of these symmetries with 0 and shows, in the
presence of these symmetries, to which values 1 the corresponding operators
square to.
However, the question is whether this completely characterizes topological
band insulators. For instance, in three dimensions the table reveals a Z2 invariant. Indeed, only the strong topological band insulator has edge sates protected
by time reversal symmetry. Nevertheless, there are three additional invariants
that reect the position of the Dirac cone or the orientation of the constituent
two dimensional layers, in the strong or weak case respectively.
In this thesis we will address the question whether the classication of the
two dimensional topological band insulators can be 'dressed up' with some additional structure. Specically, we will study the response of the topological
insulator upon the introduction of a magnetic -ux in the system, which does
not break time reversal symmetry, and explore its analogy to the topological
defects originating from lattice dislocations.
Starting from a tight-binding model that encodes a topologically non-trivial
phase, we will derive the corresponding continuum theory, describing the lowenergy excitations in the vicinity of the -point in the Brillouin zone. We will
then show that the vortex, carrying the -ux, hosts a Kramers pair of localized zero-energy modes when the system is in the non-trivial phase, by solving
the corresponding Dirac equation. In addition, we will demonstrate that these
modes exhibit non-trivial spin and charge quantum numbers. Furthermore, we
will discuss the connection of the zero-modes in the core of the -ux vortex
and the ones bound to the dislocation found numerically when the gap opens
at the (; 0) and (0; ) points in the Brillouin zone. Finally, a possible interpretation of our results regarding the topological classication of topological band
insulators will be given.
Chapter 2
Models of topological band
In this chapter we introduce the concrete models relevant for the description of
the topological band insulators in two and three dimensions and discuss their
relation to band topology.
2.1 Two dimensional quantum spin Hall eect
In this section we consider the most realistic two dimensional model that exhibits a quantum spin Hall (QSH) state. This model, referred to as the m B
model, is based on the HgTe/CdTe quantum well structure. Indeed, at present
the HgTe/CdTe quantum well provides for the exclusive system displaying an
experimentally veried two dimensional QSH eect.
2.1.1 Eective model
HgTe/CdTe quantum wells consist of a thin layer of HgTe sandwiched between
two pieces of CdTe. Both these materials have a zinc-blende lattice structure,
which has tetrahedral (Td ) point group symmetry and rather looks like a diamond lattice with dierent atoms on the sublattices. Moreover, the bulk spectrum of both HgTe and CdTe features three bands close to the Fermi level
characterized by the quantum number J , resulting from spin-orbit coupling.
Specically, these include two J = 12 -bands, referred to as 6 and 7 , and a
J = 23 -band, labeled by 8 [61]. In addition, both materials show a band gap
with the minimum at the point in the Brillouin zone.
Models of topological band insulators
Figure 2.1: The gure shows the band spectra of HgTe and CdTe[45]. In CdTe
the bands show normal ordering, whereas in HgTe the 6 band is pushed below
the heavy- hole subband and the spin-orbit split-o 7 band. The remaining
light-hole subband becomes the conduction band, whereas the heavy hole subband
becomes the rst valence band. Due to the degeneracy of these two bands at the
points, HgTe is a semiconductor.
There is, however, a crucial dierence between the band spectra of HgTe
and CdTe. Whereas the barrier material shows normal band ordering, the 6
and 8 band are inverted in the well material (Figure (2.1)). It is this feature
that leads to a topological phase transition as function of the thickness d of the
well; for widths larger than a critical value dc the connement energy is low
and the band structure remains inverted, while reducing the width decreases
the connement energy resulting in 'normal' ordering of the bands when d < dc .
Concretely, neglecting the 7 band produces a basis of six atomic states per
unit cell
fj 6 ; mj = 12 i; j 6 ; mj = 12 i; j 8 ; mj = 21 i;
j 8 ; mj = 21 i; j 8 ; mj = 32 i; j 8 ; mj = 23 ig:
Subsequently, growing a quantum well in the z-direction leads to the formation of
spin up and spin down states of an electron-like(jE 1i), a heavy- hole-like(jH 1i)
2.1 Two dimensional quantum spin Hall eect
and a light-hole-like(jL1i) subband[61]. While the jL1i subband is separated
from the other two, the jE 1i and jH 1i indeed invert as function of d (Figure
Figure 2.2: Progression of the subbands formed in the quantum well as function
of the thickness of the well[61]. The bands jE 1i and jH 1i invert at a critical
thickness dc of the well, resulting in a topological phase transition.
To obtain an eective continuum model, one neglects the jL1i subband and
considers the two sets of Kramers partners jE1 i and jH1 i[62]. Adopting the
basis fjE1 +i; jH1 +i; jE1 i; jH1 ig, time reversal symmetry then dictates that
the corresponding Hamiltonian takes the form
He (kx ; ky ) = H (k) 0
H ( k)
Additionally, assuming inversion symmetry and axial rotation symmetry along
the growth axis it can be shown that H assumes the form[68][62] [45][46]
H (kx ; ky ) = (k)I22 + Akx x + Aky y + [m B (kx2 + ky2 )]z ;
where (k) = C D(kx2 + ky2 ) and A; m; B; C; D are expansion parameters, that
depend on the specic structure and are the standard Pauli matrices. We
notice that this model relates to the ideas of the previous chapter. Indeed, it
consists of two time reversed copies of a massive Dirac equation. Moreover,
Models of topological band insulators
the mass term is momentum-dependent and can therefore change sign. This
consequently leads to band inversion, resulting in a topological phase transition.
2.1.2 Corresponding tight-binding model
Having considered the low energy continuum model based on the band structure
of HgTe and CdTe, a simplied tight binding model for the jE 1i and jH 1i states
can be introduced[45, 46]. Basically, one considers a square lattice with the
four states (two orbital and two spin states) per unit cell. Taking the symmetry
properties of the basis states into account, a tight-binding approximation results
in a Hamiltonian that has the same form as dened in Eq. (2.2), but with
H (kx ; ky )t.b. =(k)I22 + A sin (kx )x + A sin (ky )y
+ [m 2B (2 cos (kx ) cos (ky ))]z :
In the above equation the lattice constant is taken to be unity and (k); A; B
and m are dened as in the continuum version. This model simply reduces to
one dened in Eq. (2.3), when expanded around the point. However, the tight
binding model has the advantage that it provides for a natural regularization of
the continuum model, which is only applicable for small values of the momentum, giving the dispersion relation for the entire Brillouin zone. Moreover, it
still captures all the symmetries and topology.
2.1.3 Topological characterization
The upper subblock of the continuum model and the tight-binding model are of
the general form
H (kx ; ky ) = d(k) + (k)I22 :
The corresponding rst Chern character (Appendix D), in terms of the unit
vector d^ (k) is found to be[38, 39]
@ d^
dkx dky d^ (
We notice that in this case the quantization is rather manifest; the integrand
is nothing but the Jacobian of a mapping of the map d^ (k) : T 2 ! S 2 . Hence
integration results in the image of the Brillouin zone on the two sphere, which
is a topological (Pontryagin) winding number of quantized value 4 , 2 Z.
Inserting the continuum expression of d^ (k) into Eq. (2.6) and regularizing the integral yields the well-known result that the Hall conductivity equals
2h sign(m), with m the mass of the system (Appendix F). The continuum model
2.1 Two dimensional quantum spin Hall eect
Figure 2.3: Illustration showing a square Brillouin zone with the corresponding
vector eld d^(k) for m = 2 and B = 1. The conguration displays a full
skyrmion, characterized by the winding number 1.
cannot, however, determine the Hall conductance of the complete system, as
there are also relevant contributions at higher momenta.[63] Rather, the model
leads to a well-dened change in rst Chern character C 1 = 1, when the bands
invert at the point. Hence, we consider the lattice regularized version. Direct
calculation[44] shows that the gap closes at (kx ; ky ) = (0; 0) when m=B = 0, at
(0; ) and (; 0) when m=B = 4 and at (; ) when m=B = 8. Moreover, the
resulting phases are characterized by the following Chern numbers
C1 = >
0> m
B or B > 8>
0< B <4 >
4< m
B <8
Additionally, the topological non-triviality can be visualized by plotting the unit
vector eld d^ (k). The conguration with winding number = 1, displaying a
full skyrmion located at the center of the Brillouin zone, is shown in Figure
Having determined the rst Chern characters of the upper block, we can subsequently characterize the complete system. As the continuum and tight binding
model consist of two time-reversed subblocks, which have opposite Chern number, the total Chern number is zero. However, the spin Hall conductance is
given by the dierence between the two blocks and is therefore nite. Hence,
for 0 < m=B < 4 and 0 < m=B < 8 the system is in the QSH phase. Indeed, as
the Chern number is directly related to the number of edge sates, we conclude
that the trivial and non-trivial phase dier by a pair of helical edge states.
Models of topological band insulators
2.1.4 Edge states
Let us analyze the edge-state spectrum in more detail. Specically, we calculate
the surface surface states for the simplest case of a straight edge, although the
argument can easily be generalized to a zigzag edge for instance[47].
Neglecting the (k)-term, as this merely imposes a shift in energy, Eq. (2.4)
can be rewritten in terms of the usual creation cyk and annihilation ck operators
H(kx ; ky )t.b. =
[A sin(kx )
1 + A sin(k
2 + M (k) 0 ]cy c ;
k k
where M (k) = m 2B (2 cos (kx ) cos (ky )) and 0 ; 1 ; 2 obey the Cliord
algebra f ; g = 2; , with ; = 0; 1; 2. Specically, 0 = z 0 ; 1 =
x z and 2 = y 0 , with acting in the spin space and acting in the
space spanned by the orbitals jE 1i and jH 1i.
To nd the edge states, we introduce periodic boundary conditions along the
x-direction and edges perpendicular to the y-axis located at y = 0; L. Therefore,
ky is not a good quantum number anymore and must be transformed back to
real space
exp (ikx j )ckx ;j :
ck =
L j
Consequently, the Hamiltonian becomes
(M^ (kx )cykx ;j ckx ;j + T cykx ;j ckx ;j +1 + T y cykx ;j +1 ckx ;j );
M^ (kx ) = A sin(kx ) 1 2B [2 m=2B cos(kx )](kx ) 0 ;
A sin(kx ) 1 M~ (kx ) 0 ;
2 + B 0:
T = iA
In order to obtain the surface states we have to solve the corresponding
Schrodinger equation for zero energy. Sice we seek localized solutions, we infer
the following ansatz[64, 65]
(j ) = j ;
with a constant vector. In addition, we expect surface states at kx = 0,
signaling the transition at . Hence, we obtain the following equation
[ ( 1 ) 2 + (B + B 1 + M~ (0)) 0 ] = 0;
or equivalently
iA 1
) 0 2 = (B + B 1 + M~ (0)] :
2.2 Three dimensional topological band insulators
The solutions are then readily obtained by taking an eigenvector of i 0 2 ,
which has eigenvalues 1. Consider the case i 0 2 = , for which Eq. (2.15)
reduces to an algebraic equation that is readily solved
M~ (0) M~ 2 (0) + A2 4B 2
A + 2B
We notice that when i 0 2 =
we obtain the same equation, but with and 1 interchanged. Therefore, the full solution is
(j ) =
[(k j+ + k j ) k;+ + (k +j + k j ) k; ];
with k; being the two eigenvectors with eigenvalue 1 and ; ; ; 2 C.
Moreover, as the k; are independent, there are indeed localized time reversal
invariant solutions at kx = 0 if j+ j < 1 ^ j j < 1 (taking ; = 0) or
j+ j > 1 ^ j j > 1 (taking ; = 0), which is precisely the case in the
non-trivial regime 0 < m=B < 4. In addition, the derivation can readily be
generalized to the kx 6= 0 case, as [ 0 2 ; 1 ] = 0. This results in solutions
similar to Eq. (2.16), but with modied mass M~ .[47]
We conclude that the eective continuum and corresponding tight binding
model, which describe the HgTe/CdTe quantum well, capture all the relevant
physics and related topology. As a matter of fact, we can show analytically that
there are gapless edge states, which signal a non-trivial topological characterization.
2.2 Three dimensional topological band insulators
In this section, we consider the eective model for the three dimensional topological band insulator[48][49]. This model is motivated in the same way as the
eective model for the two dimensional spin Hall eect. Specically, it is based
on the properties of the Bi2 Se3 system, although being equally applicable to the
reminiscent materials Bi2 Te3 and Sb2 Te3 . These materials exhibit an experimentally veried topologically non-trivial state, featuring a single Dirac cone at
the surface[66][67].
2.2.1 Eective model
The crystal unit cell in Bi2 Se3 , Bi2 Te3 and Sb2 Te3 consists of a ve-atom basis
aligned in a rhombohedral structure, which is characterized by the space group
D35d . Concretely, Bi2 Se3 may be viewed as a stacking of ve-atom layers, consisting of two pairs of equivalent Bi and Se atoms and an inequivalent Se atom
Models of topological band insulators
(Figure (2.4)). Within these quintuple layers the individual layers show strong
atomic coupling, whereas among two dierent quintuples there is much weaker
coupling of the van der Waals type.
Figure 2.4: Illustration showing the crystal structure of Bi2 Se3 and Bi2 Te3 ,
consisting of atomic layers arranged in a triangular lattice[69]. The planes are
shifted with respect to each other, resulting in a A B C A B C stacking. Moreover, the crystal features quintuples of strongly coupled layers,
giving rise to a unit cell consisting of ve atoms.
To obtain the band spectrum one only considers the dynamically relevant
p-orbitals, i.e the outmost atomic shells of Bi and Se, and focusses on one
quintuple layer. Neglecting the spin, the three orbitals = px ; py ; pz , of the
ve atoms within the unit cell, then hybridize into fteen states near the
point. These include six states jB i; jB0 i resulting from the two Bi atoms,
another six jS i; jS0 i corresponding with the two equivalent Se atoms and three
hybridized states jS~ i associated with the remaining Se atom. These states can
then conveniently be combined into bonding and anti-bonding states of denite
parity jP 1 i = p1 [jB i jB0 i]
jP 2 i = p [jS i jS0 i]:
As usual, the bonding states have lower energy than the anti-bonding states.
Therefore, the jP 1+ i and jP 2 i states are the energy levels closest to the Fermi
2.2 Three dimensional topological band insulators
surface (Figure (2.5)). Moreover, taking into account the the layered structure
in the z -direction ensures that the pz orbital is split from the degenerate px ; py
orbitals. Explicit calculation[48] subsequently shows that this crystal eld splitting results in the pz orbitals jP 1+z i; jP 2z i being the two states nearest to the
Fermi level (Figure(2.5)). Hence, in the low energy limit, the valence band is
dominated by jP 2z i orbital and the conduction band mainly consists of the
jP 1+z i state.
Figure 2.5: Schematic picture of the relevant p orbitals in Bi2 Se3 [49]. Specifically. the gure shows how the bands evolve due to the crystal eld splitting
and the eect of spin-orbit coupling; starting from the hybridized orbitals of Bi
and Se, rst the crystal eld splitting is introduced. Subsequently, the diagram
shows the eect of additionally increasing the relative strength of the spin-orbit
Analogously to the two dimensional spin Hall eect, the crucial feature leading to a topological non-trivial state is band inversion. Whereas for the quantum
well the band inversion depends on the width of the well, this role is taken by
the spin-orbit coupling in the three dimensional case. Indeed, spin-orbit coupling lowers the energy levels of the spin up jP 1+z ; "i and spin down jP 1+z ; #i
states, while the energy levels of the jP 2z ; " (#)i states are pushed up (Figure
(2.5)). Consequently, when the relative strength of the spin-orbit coupling
term exceeds some critical value c the two bands invert, driving the system
into a topologically non-trivial phase. Numerical calculations show that spinorbit coupling is indeed strong enough to induce this band inversion; at values
of of about sixty percent of its realistic value, level crossing at the point
Models of topological band insulators
already occurs.
The eective model, focussing on the two Kramers partners jP 1+z ; " (#)i
and jP 2z ; " (#)i, is again determined using the theory of invariants[68] or
k p theory[62]; keeping terms up to order k2 , the Hamiltonian can be written down by requiring time reversal symmetry, inversion symmetry and threefold rotation symmetry, originating from the lattice. Specically, in the basis
fjP 1+z ; "i; jP 2z ; "i; jP 1+z ; #i; jP 2z ; #ig the Hamiltonian takes the form
M~ (k)
B A1 kz
He (kx ; ky ; kz ) = (k)I44 + B
@ 0
A2 k+
A1 kz
M (k) A2 k
A2 k+ M~ (k)
A1 kz
A2 k
0 C
C ; (2.18)
A1 kz C
M (k)
with M~ (k) = m B1 kz2 B2 (kx2 + ky2 ), (k) = C + D1 kz2 + D2 (kx2 + ky2 ), and
k = kx iky . We notice that this Hamiltonian is an obvious generalization of
the two dimensional Hamiltonian (2.3) and indeed has a band inverted regime
around k = 0, when m; B1 ; B2 are strictly positive.
2.2.2 Edge states and surface Hamiltonian
The topological phase is characterized by surface states. Let us therefore derive
them explicitly. Specically, consider the Hamiltonian in Eq. (2.18) dened for
the system in a half plane geometry, i.e z > 0. Neglecting the unimportant
(k)-term, we rewrite the Hamiltonian as
He (kx ; ky ; kz ) = A2 kx 1 + A2 ky 2 + A1 kz 3 + M~ (k) 0 ;
where the matrices obey the Cliord algebra and are specied according to
the above basis; 0 = z 0 ; 1 = x x , 2 = x y and 3 = x z ,
with acting in the spin space and in the space spanned by the orbitals
jP 1+z i; jP 2z i. Due to the open boundary conditions in the z-direction, kz is not
a good quantum number anymore and should therefore be transformed back to
real space, which amounts to replacing kz by [email protected] . We expect surface states of
zero energy and with kx ; ky = 0, resulting from the topological phase transition
at the point. Consequently, we are confronted with the following equation for
the edge states
Hedge ( [email protected] )(z) = [ iA1 3 @z + 0 (m + B1 @z2 )](z) = 0:
2.2 Three dimensional topological band insulators
As this equation is diagonal in spin space, we look for time reversed solutions
# (z ) and " (z ) of the form
# (z ) =
(z )
" (z ) =
(z )
Anticipating localized edge states, we subsequently write
(z ) = ez ;
resulting in
[ iA1 x + (m + B1 2 )z ] = 0:
Hence, multiplying Eq. (2.23) with z shows that must satisfy y = and subsequently reduces the dierential equation to an algebraic one, whose
solutions in the case of + are given by
A1 4mB + A21
Moreover, in the case we obtain the same algebraic solutions, but with
opposite sign. Therefore, the full solution is given by
+ z + e z )
0 (z ) = (e
+ z + e z )
+ + (e
; ; 2 C: (2.25)
Consequently, the normalization condition Re[ ] < 0 (taking ; = 0) or
Re[ ] > 0 (taking ; = 0) can only be satised when the band inversion
condition mB1 > 0 is satised, showing that the non-trivial phase is indeed
characterized by the existence of surface states. Additionally, the surface state
can completely be determined using that Re[a;b ] is strictly negative (positive)
when A1 =B1 is strictly positive (negative).
To conclude this chapter we notice that we can explicitly obtain an surface Hamiltonian, by simply projecting the original Hamiltonian dened in Eq.
(2.18) onto the subspace of the normalizable surface states j# (z )i; j" (z )i.
The result to the leading order in k is
A2 (kx iky )
H(kx ; ky ) =
A2 (kx + iky )
This Hamiltonian correctly captures the experimentally observed edge states[66]
in, for example, Bi2 Se3 .
Chapter 3
Topological eects of a
-ux vortex in the
quantum spin Hall phase
In this chapter we present the main results of this thesis. First, we analytically
show that the introduction of a -ux vortex in the continuum m B model,
when in the non-trivial phase, results in a Kramers pair of zero-energy states,
bound to this vortex. Subsequently, we show that these zero-modes carry nontrivial charge or spin quantum number depending on their occupation. We then
conclude this chapter by discussing the interpretation of these results.
3.1 Zero-energy solutions in the presence of a ux vortex
In this section we model the topological band insulator with the continuum
m B model and derive the explicit form of the zero-energy modes bound to a
magnetic -ux. In addition, we demonstrate that these solutions do not exist
in the absence of the vortex.
3.1.1 Derivation of the zero-energy bulk states
Consider the model Hamiltonian as dened in Eq. (2.4), neglecting the energy
shift due to (k) and with the expansion parameter A set to unity. Furthermore,
let us assume that m; B > 0 and that the system is in the non-trivial phase with
0 < m=B < 4, meaning that there is a protected Dirac cone at the -point in
Topological eects of a -ux vortex in the quantum spin Hall phase
the Brillouin zone. In order to focus on the long-wavelength properties in the
vicinity of this point, we expand the Hamiltonian to obtain
He (k) = H (k) 0
H ( k)
H (k) = kx x + ky y + [m B (kx2 + ky2 )]z :
Subsequently, we consider the eect of a magnetic -ux vortex in the system.
In the lattice model the ux can readily be incorporated using a Peierls
substitution, which amounts to a minimal coupling in its continuum counterpart.
Specically, Eq.(3.2) takes the form
H (k; A) = (kx +Ax )x +(ky +Ay )y +mz B [(kx +Ax )2 +(ky +Ay )2 ]z ; (3.3)
A(x) = 2(yxe2x ++ yx2e)y
representing the magnetic -ux vortex (Appendix G). We notice that the spindown electrons are coupled to the ux with opposite sign, as a consequence
of time-reversal symmetry.
To nd the corresponding eigenfunctions, we Fourier transform the resulting
Hamiltonian back to real space. Since time reversal symmetry ensures that
solutions come in the form of Kramers pairs
# (x; y) =
" (x; y) =
(x; y)
(x; y)
we obtain the following equation
j =x;y
[j @j + iz [m + B (@j2 + 2iAj @j
A2j )]] (x; y) = " (x; y):
More explicitly, introducing polar coordinates (r; ') and components (r; ')> =
(u(r; '); v(r; ')), this results in the following equation for the zero-energy solutions
m + B (4 + ri2 @' 41r2 )
ei' ( [email protected] + @' )
e i' ( [email protected] @' )
u(r; ')
= 0; (3.7)
m B (4 + ri2 @' 41r2 )
v(r; ')
in terms of 4 [email protected] + 1r @r + r12 @'2 j. The structure of the above equation allows
for an ansatz that separates the ' and r dependence
(r; ') =
cl l (r; ');
3.1 Zero-energy solutions in the presence of a ux vortex
eil' ul (r)
l (r; ') =
ei(l+1)' vl+1 (r)
l 2 Z ; 2 C:
As a result, we are confronted with a pair of coupled dierential equations for
ul (r) and vl+1 (r)
4~ l+ 21 ul (r) = i(@r + (l +r 2 ) )vl+1 (r)
~ l+ 3 vl+1 (r) = i(@r
l + 21
)ul (r);
4~ n m + B (@r2 + 1r @r nr2 ) m + B 4n :
A key observation regarding these equations is that
(@r +
) = 4n
)(@n +
) = 4n+1 :
~ n ; @r + nr ] is readily veried to be
Hence, the commutator [4
~ n ; @r + n ] = [4n ; @r + n ]
n 1
= B [@r +
; @r + ](@r + )
B (2n 1)
(@r + ):
Using Eq. (3.13) and Eq. (3.11), we can decouple Eq.(3.10) to obtain a single
equation for ul (r)
4~ l+ 23 4~ l+ 12 ul (r) = i(@r + l +r 2 )4~ l+ 23 vl+1 (r) + i[4~ l+ 23 ; @r + l +r 2 ]vl+1 (r)
= 4l+ 12 ul (r)
B (2l + 2) ~
4l+ 21 ul (r):
Similarly, identifying n with (l + 12 ) in Eq. (3.13) results in the corresponding
equation for vl+1 (r)
4~ l+ 12 4~ l+ 32 vl+1 (r) = i (@r l +r 2 )4~ l+ 12 ul (r) i [4~ l+ 12 ; @r l +r 2 ]ul (r)
B (2l + 2) ~
= 4l+ 32 vl+1 (r) +
4l+ 32 vl+1 (r):
Topological eects of a -ux vortex in the quantum spin Hall phase
More importantly, we can factorize these equations in a rather elegant form.
Initially, we establish
4~ a 4~ b =m2 + mB ([email protected] + 2r @r a r+2 b )+
(a2 4)b2
a2 + b2 + 1 2 3b2 + 1 a2
): (3.16)
B 2 (@r4 + @r3
Hence, applying this relation in opposite directions we verify that
1 2
4~ l+ 23 4~ l+ 12 + B (2rl 2+ 2) 4~ l+ 12 =m2 + mB ([email protected] + 2r @r 2(l +r2 2 ) )+
2(l + 21 )2 + 1 2
@r )+
B 2 (@r4 + @r3
2(l + 21 )2 + 1
((l + 12 )2 4)(l + 12 )2
~ l+ 1
3 2
4~ l+ 21 4~ l+ 32 B (2rl 2+ 2) 4~ l+ 32 =m2 + mB ([email protected] + 2r @r 2(l +r2 2 ) )+
2(l + 23 )2 + 1 2
@r )+
B 2 (@r4 + @r3
2(l + 23 )2 + 1
((l + 32 )2 4)(l + 32 )2
~ l+ 3 :
Consequently, we nd the following equations for ul (r) and vl+1 (r)
m2 ul (r) + (2mB 1) 4l+ 12 ul (r) + B 2 42l+ 12 ul (r) = 0
m2 vl+ 32 (r) + (2mB 1) 4l+ 23 vl (r) + B 2 42l+ 32 vl+1 (r) = 0
This result may also be obtained by noting that if the spinor in Eq. (3.9) is
an eigenstate with zero eigenvalue of the Hamiltonian (3.2), then it is also an
eigenstate with the same eigenvalue of the square of this Hamiltonian. One
readily veries
He (k; A)2 = B 2 (k~ 2 )2 + (1 2MB )k~ 2 + M 2 ;
with k~ k + A. Acting on the angular part of the spinor with the operator k~ 2
then yields Eq. (3.19).
We conclude that Eq. (3.19) implies that ul (r) and vl+1 (r) are eigenfunctions
of 4l+ 21 and 4l+ 32 , respectively. Moreover, they must have positive eigenvalues
2 since the operator k~ 2 when acting on a function with angular momentum
3.1 Zero-energy solutions in the presence of a ux vortex
l is equal to 4l+1=2 , and the eigenstates of the operator k~ 2 with a negative
eigenvalue are localized; k~ 2 ul (r) = 2 ul (r). Therefore, solving
(R2 + (l + )2 )]ul (R) = 0
3 2
[R @R + [email protected] (R + (l + ) )]ul+1 (R) = 0;
with R = r, reduces the dierential (3.19) equation to an algebraic one in
terms of 2 , whose solutions are easily obtained
[R2 @R2 + [email protected]
1 2MB 1 4MB
2B 2
Dierential equations of the type (3.21) are known as modied Bessel equations and have been well studied[74]. Specically, the general equation
2 =
[R2 @R2 + [email protected]
(R2 + 2 )] (R) = 0
has a linearly independent basis of solutions consisting of I (R) and K (R),
( 14 R2 )k
1 X
I (R) = ( 2 R)
k=0 k ! ( + k + 1)
I (R) I (R)
K (R) = lim
! 2
sin ()
The asymptotic behaviour of these functions is shown in Table (3.1.1).
Limiting forms of the modied Bessel functions
Small arguments x ! 0
I (x) ( 12 x)jj = (jj + 1)
I (x) ( 2 x) = ( + 1)
2= Z; 2 R
K (x) 2 ()( 2 x)
Large arguments x ! 1
I (x) pe2xx
K (x) 2x e x
Table 3.1: Table showing the limiting forms of the modied Bessel functions for
small and large arguments.[74]
Topological eects of a -ux vortex in the quantum spin Hall phase
The I (R) are exponentially growing functions and are therefore not normalizable. Henceforth, we concentrate on the exponentially decaying solutions
K (R), in terms of the positive solutions . Moreover using
@r Il (r) = Il+1 (r) +
and the fundamental relation for l
I (r)
r l
(m + B2 )2 = 2 ;
we readily verify that the relative phase equals i. Recalling Eq. (3.9), we thus
l (r; ') = l;
with l;+ ; l;+
eil' Kl+ 21 ( r)
eil' Kl+ 21 (+ r)
; (3.28)
iei(l+1)' Kl+ 23 ( r)
iei(l+1)' Kl+ 23 (+ r)
2 C and referring to the two positive solutions of Eq. (3.27)
1 1 4mB
The zero-energy solutions to the free Hamiltonian (3.1) can easily be determined as well. Specically, setting the vector potential A equal to zero and
imposing solutions (3.8), results in the same equations, but with l shifted by a
4~ l ul (r) = i(@r + (l +r 1) )vl+1 (r)
)u (r):
r l
Therefore, neglecting normalization conditions for the moment, the full solution
is given in terms of the modied Bessel functions of integer order
~ l+1 vl+1 (r) = i(@r
(r; ') =
eil' Il (j r)
eil' Kl (j r)
; (3.31)
iei(l+1)' Il+1 (j r)
iei(l+1)' Kl+1 (j r)
where l;j ; l;j 2 C and j are the four roots of Eq. (3.27).
As a result, we only nd square-integrable zero-energy solutions in presence
of the -ux vortex. Indeed, using Table (3.1.1), we deduce that there are
no normalizable solutions in the absence of the vortex, due to the behaviour
of the functions Kl (x) (Il (x)) at the origin (innity). However, introducing
the -ux shifts the momentum l of the solutions by a half, resulting in two
3.1 Zero-energy solutions in the presence of a ux vortex
square integrable zero-energy solutions in the zero momentum channel, being
the localized l = 1 solutions (3.28). Additionally, as
sinh (r);
these solutions take a particularly simple form. Specically, we should
distinguish two regimes of parameters, 0 < mB < 1=4 and mB > 1=4, for
which the argument of the square-root is positive and negative, respectively. In
the rst case the solutions simply reduce to
I 12 (r) =
cosh (r);
I 12 (r) =
e i'
e r
(r; ') = q
21 r
with > 0, of course. When mB > 1=4, up to a normalization constant, these
solutions can rewritten as
sin e r B cos cos r m
(r; ') =
e r B cos sin r m
sin B
+ (r; ') =
e i'
e i'
where we used to denote the solutions and introduced
= arctan
j1 4MB j:
Consequently, we can conclude that the Hamiltonian (3.1) in presence of the
magnetic -ux possesses zero-energy modes (3.5), with
(r; ') = (r; ') + +
+ (r; ');
in the entire range of parameters m and B for which the system is in the
topologically non-trivially phase, 0 < m=B < 4. In particular, when 4mB < 1
the zero-energy states are purely exponentially localized, while for 4mB > 1 the
exponentially localized solutions have an oscillatory part given by the angle .
We notice also that in the regime 0 < mB < 1=4, there are two characteristic
length scales associated with the midgap modes, 1 . Of course, after
a short-distance regularization is imposed, only a certain linear combination of
the two states survives. The physical interpretation of the two length scales
depends on the form of the superposition of the state after the regularization
has been imposed, as it may be easily seen from the form of the states (3.33). In
the regime mB > 1=4, the zero-energy states are characterized by two lengthp
scales, one of which is a localization length loc B=m, while the other scale
Topological eects of a -ux vortex in the quantum spin Hall phase
loc = sin characterizes the oscillations of the exponentially decaying state.
Therefore, the appearance of the zero-energy states bound to a -ux vortex
is a generic feature of the Hamiltonian (3.3) describing the quantum spin Hall
system. Furthermore, in the vortex-free system (3.1), the penetration depth of
the gapless edge modes of the system are given by exactly the same expression
as the localization length for the zero-energy modes bound to the -ux vortex;
an analogous derivation to the one in subsection 2.2.2 reveals surface states
# (x; ky = 0) =
(x; ky = 0)
" (x; ky = 0) =
ix @x + z [m + [email protected] ] (x; ky = 0) = 0;
(x; ky = 0)
in case of an edge perpendicular to the x-axis. Therefore the solutions are of
the form (3.37)
(x; ky = 0) = + '+ e x + where
' e x ;
1 1 4mB
2 C:
y ' = ' ;
This shows that the inverse penetration depth indeed agrees with the inverse
localization length (3.29) of the zero mode. Hence, the bulk-boundary correspondence may be probed by inserting a -ux vortex in the quantum spin Hall
We notice, nonetheless, that dene an overcomplete basis. This is a
consequence of the fact that the Hamiltonian (3.3) is not self-adjoint due to
the singularity of the vortex vector potential (3.4) at the origin. In order to
overcome this apparent inconsistency, the gauge potential has to be regularized. Essentially, the regularization xes the relation between and + , the
corresponding procedure of which is discussed in the subsequent subsection.
It is therefore a natural step to incorporate the lattice symmetries as being
reected in the distribution of signs of the Paan expression (??); altough the
i are not particularly important, the relative signs and how they are connected
convey extra information, enriching the classication based on the cube (square).
It is obvious that i related by symmetry group of the lattice are equivalent, as
can be veried by insertion of the symmetry operation U in Eq (??). As a result,
it is straightfoward to determine the dierent topological phases. Essentially,
one considers the momenta in the Brillouin zone at which the Hamiltonian
commutes with #, which by the very consideration of lattice symmetries may
comprise more than the usual four points.
3.1 Zero-energy solutions in the presence of a ux vortex
3.1.2 Self-adjoint extension
Singular potentials like the vortex gauge potential (3.4) usually result in overcomplete bases of solutions[71, 72]. Basically, the corresponding Hamiltonian
operator is not automatically self-adjoint on its natural domain and henceforth
the solutions should be regularized. Typically, such operators admit more than
one self-adjoint domain, resulting in a family of solutions. We will, however,
begin with a specic regularization, based on a matching procedure, and then
show how this corresponds to particular choice of parameters of the family of
Figure 3.1: Illustration of the matching procedure. Consider the vortex with
the ux concentrated in a thin annulus of radius R and thickness t R. The
zero-energy solutions of the vortex-free Hamilton inside the annulus are matched
with the zero-energy solutions of the Hamiltonian in presence of the -ux vortex
outside the annulus. Subsequently shrinking the radius R then yields a specic
linear combination of the zero-energy states, which is regular at the origin.
Let us rst consider the Hamiltonian in the range of parameters 0 < mB <
1=4. A possible regularization is then provided by considering the vortex with
the ux concentrated in a thin annulus of a nite radius R. Outside the annulus
the zero-energy states are then still given by Eq. (3.33), whereas inside the
annulus due to the niteness of R this momentum channel hosts zero energy
solutions of the form
< (r; ') = C1
with C1 ; C2
e i' I 1 (+ r)
iI0 (+ r)
+ C2
e i' I 1 ( r)
iI0 ( r)
2 C and as dened in Eq. (3.22). Matching the solutions at
Topological eects of a -ux vortex in the quantum spin Hall phase
r = R, we then obtain
e i' I 1 (+ R)
e i' I 1 ( R)
+ C2
iI0 (+ R)
iI0 ( R)
e + R e i'
e R e i'
+ ~ + p
~ p
where the numerical factors are absorbed in ~ 2 C. Subsequently, we take the
limit R ! 0. As I 1 ( r) linearly tend to zero, the upper component equation
(3.42) therefore ensures that
resulting in
0 = ~ lim
e R
+ ~ + lim
e + R
~ = ~ + :
Hence, we indeed nd a xed relation between the coecients ~ , which then
species C1 = C2 by the lower component of Eq. (3.42). Moreover, up to a
normalization constant, we are nally confronted with a concrete form of the
zero mode (3.36) bound to the vortex
(r; ') =
e + r
e i'
When mB > 1=4, the derivation is almost identical. The left hand side of
Eq. (3.42) stays unaltered, whereas the right hand side is given in terms of the
spinors (3.34). However, since
e R B cos sin(R m
B sin )
= 0;
the upper component of the resulting equation simply states that equals
zero. Therefore, the zero-energy solution is, in this case, just the normalized
version of + (3.34), which essentially pertains to same linear combination in
terms of the general solutions (3.33).
Although the regularization above results in concrete solutions to the problem, we can consider the self-adjoint extension of the corresponding Hamiltonian
(3.3) in a more general manner by specifying the proper Hilbert space. In what
follows, we shall illustrate this.
Recall that the solutions to the m B model (3.3) in the presence of a -ux
vortex come in the form of Kramers pairs
# (r; ') =
(r; ')
" (x; y) =
(r; ')
3.1 Zero-energy solutions in the presence of a ux vortex
with (r; ') a linear combination of eigenfunctions
eil' ul (r)
l (r; ') =
iei(l+1)' vl+1 (r)
of angular momentum l 2 Z. Due to the singular nature of the gauge potential
(3.4) we focus on the radial part of the Hamiltonian
Hrl (r) =
1 2
m + B [@r2 + 1r @r + (l+r22 ) ]
i(@r l+r 2 )
i(@r + l+r 2 )
1 2 ; (3.49)
m B [@r2 + 1r @r + (l+r22 ) ]
acting in the subspace of angular momentum l. Moreover, rescaling the basis
states by l (r; ') = p1r ~ l (r; '), reduces this Hamiltonian operator to
m + B [@r2 + l(lr+1)
2 ]
i(@r l+1
H~ rl (r) =
i(@r + l+1
r )
l+2) ] :
m B [@r2 + (l+1)(
The hermiticity requirement then imposes the following condition
(H~ rl )y j l i
dr l (r) H~ rl (r) l (r)
0 = hl j H~ rl
dr l (r) H~ rl (r)l (r)
= B fw~ @r u~ (@r w~ )~u x~ @r v~ + (@r x~ )~vgj1
i fw~ v~ + x~ u~gj1
~ 3 @r ~ (@r ~ )3 ~ ) i
~ 1 ~ (r = 0);
= B (
on the arbitrary wave functions l (r; ') and l (r; '), which vanish at innity
and take the form dened in Eq.(3.48) in terms of the radial components
l (r) !
wl (r)
xl+1 (r)
l (r ) and
ul (r)
vl+1 (r)
~ :
The above requirement (3.51) results in a continuous family of restrictions
on the behavior of square-integrable functions at the origin, the corresponding
parametrization of which is obtained by exploiting its linearity. Namely, we
dene two linear surjective maps 1 , 2 from the domain D(H~ y ), being arbitrary
functions, onto their value at the boundary. These operators are dened by Eq.
B (~ 3 @r ~ (@r ~ )3 ~ ) i~ 1 ~ (0) h 2 ~ ; 1 ~ i
In particular, we may choose the following explicit form
~ = B3 @r ~ (0) i 1 ~ (0);
2 = (0):
h 1 ~ ; 2 ~ i: (3.53)
Topological eects of a -ux vortex in the quantum spin Hall phase
We notice that the boundary space indeed becomes Hb = C2 , as any vector in
this space is the image of some wavefunction under i .
Generally, a necessary and sucient condition for an operator H to have a
self-adjoint extension is that its Cayley transform[72][73] W (H) has a unitary
extension. Specically, it can be shown[71] that there is a one-to-one correspondence between the unitary maps U 2 U (Hb ) and the self adjoint extensions of
H, resulting in a general condition on the wavefunctions, parameterized by U ,
such that (3.51) is satised:
D(H~ U ) = f j(U 1b )
+ i(U + 1b )
= 0g:
In the case at hand Hb = C2 and hence U 2 SU (2), which amounts to a
real-valued unit vector m relative to the usual quaternion basis q = (0 ; i)
consisting of the Pauli matrices , i.e.
= f0; 1; 2; 3g:
m q ;
Consequently, the condition (3.55) results in
q g @r ( r 0 )r=0 = f
q g r
0 r=0 ;
> = ( im3 ; im2 ; im1 ; i(m0 1));
1 m0
> = ( 1 + i(1 + m0 );
+ im1 ; 3 + im2 ;
+ im3 ):
Moreover, since
0 r=0
= i(CC11++CC22 )
@r ( r 0 )r=0 =
C1 +C2 +
i(C1 +C2 + )
; (3.59)
with the positive solutions of Eq. (3.27) and C1 ; C2 2 C, we obtain a
relation between C1 and C2 as function of the particular self-adjoint extension
m . Clearly, m parameterizes the hypersphere S 3 and can correspondingly be
rewritten in terms of angular coordinates
m = (cos ; sin cos ; sin sin cos '; sin sin sin ');
where 2 [0; ], 2 [0; ], and ' 2 [0; 2). Given the parameters m and B ,
Eq. (3.57) thus results in a parameterized family of restrictions on the solutions,
specifying the self-adjoint domain.
We notice that det( q ) = 2 = 2(mo 1). Let us therefore consider
the case m0 = 1 explicitly. Setting m0 to unity immediately implies m = 0 and
3.1 Zero-energy solutions in the presence of a ux vortex
hence results in 0 = i0 ~ (0) = i0 r ( 0). This then yields C1 = C2 , being
the exact same relation as found above (3.44). We therefore conclude that the
regularization resulting from the matching procedure indeed pertains to specic
choice of m , verifying the intuitive approach.
Topological eects of a -ux vortex in the quantum spin Hall phase
3.2 Non-trivial quantum numbers of the zero
One of the most remarkable consequences of topological order is the emergence
of particles with fractional charge and statistics[33, 34, 39]. In this section we
show that the localized zero-energy mode, as found in the previous subsections,
has non-trivial charge or spin quantum number, which is intimately tied to the
topologically non-trivial nature of the quantum spin Hall phase. In order to
place these considerations in the right context, however, it is benecial to take
a step back and consider the well-studied Su-Schrieer-Heeger (SSH) model.
Indeed, the properties regarding the relevant quantum numbers of the solutions
of this model are essentially directly applicable to the zero mode (3.36).
3.2.1 The Su-Schrieer-Heeger model
The Su-Schrieer-Heeger model actually is nothing but a description of a one
dimensional physical system, namely polyacetylene. Nonetheless, it turns out
that the resulting model has all the necessary ingredients for fermion number
Figure 3.2: Structure of the polymer trans-polyacetylene[26]. However, the
equally spaced chain is energetically unstable and therefore the bonds displace
by a small distance in opposite fashion. As result, there are two groundstates
associated with the polymer, as shown in the bottom two gures.
Consider a ring of (trans)polyacetylene consisting of N monomers of mass M
(Figure (3.2)). Assuming that the relevant degree of freedom is the displacement
u of the monomers, a Peierls tight-binding model including electron-phonon
3.2 Non-trivial quantum numbers of the zero mode
coupling to the lowest order is readily obtained
H = He + He
ph + Hph ;
He =
He h = Hph =
t0 (cyj +1 cj + cyj cj +1 )
uj )(cyj +1 cj + cyj cj +1 )
(uj +1
1 X 2 KX
u_ +
2M j
2 j j
uj +1 ):
In the above denotes the coupling constant, K refers to the spring constant
and j denotes the sites. Additionally, the spin is suppressed and we note that
the hopping term, containing the creation operators cj , actually describes the
hopping of electrons like in the graphene system[1].
It is known that the equally spaced chain is energetically unstable. Consequently the bonds shift by a distance in alternate fashion producing two
ground states or vacua (Figure (3.2)), denoted by the A phase and B phase.
This is captured by the tight-binding model (3.61), as we will see below.
Let us now consider the Hamiltonian for a xed conguration fuj g. Specifically, assume uj = ( 1)j uj , which leads to
(t0 + ( 1)j 2au)(cyj +1 cj + cyj cj +1 ) + 2NKu2 :
Obviously, the unit cell of the system doubles. We then set the new lattice
constant to unity and constrain the Brillouin zone to [ 2 ; 2 ]. Furthermore, we
k = (N ) 2
k = i(N ) 2
exp ( ikj )cj
exp ( ikj )( 1)j cj ;
being the usual valence and conduction band operators, respectively. Accordingly, the Hamiltonian becomes
"k (ky k
ky k ) + k (ky k
k = 4au sin(k)
ky k ) + 2NKu2 ;
"k = 2t0 cos(k):
Topological eects of a -ux vortex in the quantum spin Hall phase
In order to diagonalize the Hamiltonian we dene the following transformation
that preserves the Fermi statistics
k = wk k vk k
ok = vk k + wk k
jwk j2 + jvk j2 = 1:
Consequently the Hamiltonian takes the form
"~k (ky k
oyk ok );
"~k = ("2k + 2k )1=2 ;
wk = ( (1 + "="~k ))1=2 ;
vk = ( (1 "="~k ))1=2 sgn(k );
wk vk = k =(2~"k ):
Figure 3.3: The energy bands resulting from the tight-binding approximation of
the polyacetylene system. The bands are displayed in the reduced zone scheme.
Moreover, the gure shows the linearization of the bands around the Fermi
energy, resulting in a spin-degenerate continuum model with a linear energymomentum dispersion. The cut o momentum k0 is chosen in such way that it
reproduces the bandwidth t, vf ko = 2t.
The above computations and the resulting gap are exactly similar to the
mean-eld theory of conventional superconductivity and its quasiparticle spectrum[24].
Correspondingly, the ground state energy per site "0 (u) can be calculated in the
same way. Considering the case of half lling, the lower band (Figure (3.3)) is
3.2 Non-trivial quantum numbers of the zero mode
lled and the upper one is empty. Accounting for the spin, we obtain
"0 (u) = (Kt20 z 2 )=(22 )
(4to =)
Z =2
dk(1 (1 z 2 ) sin2 (k))1=2 ;
in terms of
z = (2u)=(t0 ):
Evaluating the above in the limit of small z u, we then nd
"0 (u) = (Kt20 z 2 )=(22 ) (4to =)(1 + (1=2)z 2 [ln(4=z ) 1=2]:
As the logarithm term dominates, we nd that indeed there are two vacua
associated with spontaneous symmetry breaking. Concretely, the resulting potential(Figure (3.4)) has a double well, reminiscent of spontaneous symmetry
breaking in 4 theory. Moreover, just like 1 + 1 dimensional 4 theory is known
to have solitons, this model also exhibits domain wall solutions. Introducing an
order parameter
j = ( 1)j uj ;
which takes values hj i = in the A/B phase, numerical calculations of Su,
Schrieer and Heeger [25, 26] show that there is a phonon eld conguration
that approaches the A(B) phase when j ! 1. Specically, this solution is
given by
j tanh((j j0 )= );
where jo refers to the center of the soliton and is the characteristic length
given by the ratio of the band width and the bad gap.
In order to capture the relevant low-energy physics, the bands can be linearized around the Fermi energy (Figure (3.3)). This procedure then results in
two spin-degenerate branches() with dispersion
" ~vf (k kf );
and corresponding two component wave function (x) = ( + (x); (x))> .
Moreover, as the dimerization has wave number 2kf (as a consequence of the
doubling of the unit cell), it couples the two branches. Consequently, we obtain
the following real space Hamiltonian with the real valued order parameter (x)
representing the dimerization
H = (x)> (Sz ( [email protected] ) + (x)Sx )(x);
where the units of vf and ~ are set to unity.
Topological eects of a -ux vortex in the quantum spin Hall phase
Figure 3.4: Sketch of the total energy of the polyacetylene system as function
of the mean amplitude of displacement u. The two stable minima at correspond to the A and B phase. The double minimum is very reminiscent of
spontaneous symmetry breaking associated with 4 -theory, which is known to
have time-independent solitons in (1+1) dimensions.
At rst sight there appears a problem as the spectrum is unbounded from
below. One could imagine introducing an innite full Dirac sea, using the interpretation of standard eld theory. This phenomenon is actually not surprising
as the linear approximation is only valid within a part of the band width, which
then denes a natural cuto.
In order to solve Eq. (3.81), we need to know the order parameters (x).
Generically, (x) is determined self-consistently, by minimizing the energy of
the system. However, we are interested in the non-trivial background, i.e the
soliton, which we know to be a solution. Even after introducing the soliton in
the Hamiltonian, this system can be solved analytically[28, 29]. However, let us
concentrate on the consequences concerning the quantum numbers.
First we notice that the Hamiltonian looks like an ordinary massive Dirac
equation and that it anticommutes with Sy , which pertains to presence of charge
conjugation symmetry. Therefore the energy spectrum is symmetric in both
a trivial(vacuum) and non-trivial (soliton) background. Moreover, the AtiyahSinger index theorem (Appendix H) guarantees that there is a normalizable zero
energy solution 0 in the center of the gapped energy spectrum in the presence
of the nontrivial background. Including the spin this state can accommodate
0; 1 or 2 electrons.
Subsequently, denoting the solution in presence of a soliton by s (x) and
the solution of the vacuum sector by (x), we consider the density n of the
3.2 Non-trivial quantum numbers of the zero mode
solitonic ground state relative to the vacuum. As second quantizing the theory
means lling up all the states with negative energy, n is given by
Z 0
dE (js (x)j2 j(x)j2 );
where the integration actually means integrating and summing over all negative
states. We can readily calculate n by considering the completeness relation
Z 0
Z 0
Z 1
Z 1
0 = j 0 j2 +
dE js (x)j2
dE j( x)j2 +
dE js (x)j2
dE j( x)j2
n (x) =
0 j + 2(
= 1 + 2(
Z 0
Z 0
Z 0
dE js (x)j2
dE js (x)j2
Z 0
dE j( x)j2 )
dE j( x)j2 );
where we exploited the conjugation symmetry and the fact that the zero energy
solution is normalized. Using that the fermion number of the non trivial ground
state is given by
Nf = dxn (x);
we thus conclude that the nontrivial vacuum carries fermion number 21 when the
zero mode is occupied and 12 when it is empty. Hence including the spin the
fermion number can be either 0, with corresponding spin Sz = 21 , or 1, with
Sz = 0 . Concretely, in the case of the polyacetylene ring, a soliton and antisoliton are obviously created in pairs. The valence band experiences a depletion
of one state for each pair. Additionally, by charge conjugation symmetry, the
conduction band then also loses a total of one state. These states form two
pairs located at midgap when the solitons are separated by large distances as
compared to the characteristic length.
To conclude this subsection, we notice that we can rephrase these ideas
into an explicit formula in terms of a general traceless Hermitian operator Q,
which commutes with the Hamiltonian and hence gives rise to a proper quantum
number. Assuming spectral symmetry then means that there is an operator T
that anticommutes with the Hamiltonian, relating the states with positive and
the negative energy. Subsequently, we consider
q(x) =
E 2R
E (x)QE (x);
where the sum has the interpretation of an integral when the spectrum is continuous and fE (x)g is the complete set of eigenstates of the Hamiltonian with
energy E . Clearly, if the summation is preformed over the complete set, we
Topological eects of a -ux vortex in the quantum spin Hall phase
obtain q(x) = 0. If R only includes the occupied states, however, the summation results in the ground-state average of a physical observable hq(x)i. Hence,
subtracting half of the vanishing integral results in
hq(x)i = 21 @
E (x)QE (x)
E (x)QE (x)A ;
This form makes it evident that if there exists a unitary operator that anticommutes with the Hamiltonian, then for any Q that commutes with Hamiltonian,
the only states that can contribute to hq(x)i are those of zero-energy. Specifying Q to the operator representing charge and the z -component of the spin,
we observe that this exactly pertains to the characterization of the quantum
numbers, as described above.
3.2.2 Non-trivial quantum numbers of the zero-mode
The considerations of the previous section are directly applicable to the zero
modes (3.36), localized at the core of the magnetic -ux vortex, previously
found in the m B model. Essentially, the Hamiltonian (3.1) has similar spectral
symmetries and an analogous Kramers pair of midgap states in the presence of
the vortex in the non-trivial regime.
To make this specic, we rewrite the Hamiltonian as
H (kx ; ky ) = i0 j (kj + Aj ) + (M
B (k + A)2 )0 ;
where the sum over j = 1; 2 is implied and are the four-dimensional gammamatrices, obeying the Cliord algebra. Concretely, these include 0 = 3 0 ,
1 = 2 3 , and 2 = 1 0 , with the Pauli matrices acting in the
spin space and the Pauli matrices acting in the orbital space spanned by
jE1 i and jH1 i. The vector potential A is given by Eq. (3.4) or equals zero,
corresponding to presence and absence of the vortex, respectively. We notice
that time-reversal symmetry, with the unitary part T = 1 5 = 0 i2 , is
explicitly ensured, because the vector potential carries the ux equal , and
thus it is invariant under A ! A. Moreover, there are unitary matrices
3 = 2 2 and 5 = 2 1 that anticommute with the gamma-matrices
. Therefore, the Hamiltonian anticommutes with the matrices 3 i0 3
and 5 i0 5 . The resulting chiral symmetry, which is the product of timereversal symmetry and charge conjugation symmetry, then makes the spectral
symmetry apparent.
3.2 Non-trivial quantum numbers of the zero mode
Using Eq. (3.86), we conclude that in the case of a quantum spin Hall
insulator threaded by a -ux, depending on the occupation of a pair of zeroenergy states, there are four possibilities for the ground state quantum numbers.
Namely, when both states are occupied or empty, according to the above expression, the charge is +e or e and the spin quantum number is zero. On the
other hand, when one of the states is occupied, the spin quantum number Sz is
+1=2 or 1=2, while the charge is zero (Figure (3.5)). Hence, the above model
essentially accounts for a two dimensional realization of spin-charge separation,
which is intimately tied to the topologically non-trivial nature of the quantum
spin Hall state[75, 76].
Figure 3.5: Schematic representation of the spin-charge fractionalization. The
gure shows a spin degenerate spectrum that has spectral symmetry. As a result,
the groundstate quantum numbers are essentially determined by the occupation
of the zero-modes. Filling up this midgap state with spin up and down electrons,
as indicated by the arrows, results in the charge Q and spin S quantum numbers
as shown.
Topological eects of a -ux vortex in the quantum spin Hall phase
3.3 Interpretation and context of results
The results of the previous two subsections relate to a more fundamental question regarding the classication of topological band insulators. Specically, we
are interested in some additional structure resulting from the structure of the
lattice, which may be probed by lattice dislocations. In order to give a deeper
interpretation of our results, we will qualitatively elaborate on these considerations and show in what way the -ux vortex problem, as discussed above,
pertains to a simplied related problem. We will begin the discussion by analyzing the zero-energy solutions in dierent phases of the system, characterized
by the parameters m and B . In particular, we show that the boundary condition, resulting from the regularization, in combination with the normalization
requirement leads to an existence condition on the wavefunctions that can only
be satised if the band inversion condition mB > 0 is satised.
3.3.1 Nature of the zero-energy states in the dierent
phases of the m B model
We already noticed the correspondence between the Kramers pair of zero-energy
states bound to the vortex and the surface states, characterizing the topological non-trivial sates with the Dirac cone situated the point in the Brillouin
zone. Essentially, the form of these solutions is similar and as a result the arguments, that ensure the existence of surface states only when the band inversion
condition is satised, also apply to the zero mode.
Starting from the original model (3.1), we observe that the subsequent
derivations in subsection 3.1.1 are applicable for any value of m and B . As
a result, we conclude that there are only normalizable zero-energy bulk states
in the zero momentum (l = 1) channel in the presence of the -ux vortex.
The only dierence, however, is that the overall signs of the roots
1 1 4mB
= = 2B
associated with the algebraic equation (3.27),
(m + B2 )2 = 2 ;
depend on the sign of the parameters m and B . Accordingly, there is a subtlety
involving the phase of the solutions (3.28), since actually is a function of
the roots . Indeed, inserting solutions
u 1 (r)e i'
e i'
e r
(r; ') =
v0 (r)
21 r
3.3 Interpretation and context of results
into the fundamental equation (3.10),
fm + B (@r2 + 1r @r 41r2 )gu 1 (r) = i(@r + 21r )v0 (r)
)gvl+1 (r) = i(@r + )ul (r);
fm + B (@r2 + @r
we obtain
m + B ( )2
Consequently, Eq. (3.36) generalizes into the following form
(r; ') = e r + + e + r + + e r + + e+ r ;
e i'
where =
and all the numerical factors are absorbed in ; 2 C.
The explicit zero mode then depends on the sign of m and B . Specically,
normalizabilty determines the non-zero pair of coecients, the relation of which
is subsequently xed by the regularization condition
lim (r; ') = 0:
In the case of m and B being strictly positive this pertains to the solutions
(3.36), as obtained above. When m and B are both strictly negative, the normalization condition forces to be zero. Requiring regularity at the origin
then yields a zero mode with = + , which is of the same form as in Eq.
(3.36). In contrast, if the band inversion condition mB > 0 is not satised, all
coecients vanish. Explicitly, let us rst suppose that m is positive. A priori,
the non-zero coecients are and + . However, as Eq. (3.94) imposes
0 = + +
0 =
+ ;
we see that indeed all coecients are zero. Moreover, in the other case where
m is negative and B is positive, we can simply replace and + by + and
and apply the exact same argument.
We conclude that the zero mode can only exist if the band inversion condition
is satised. This criterion is essentially all the topological encoding present
in the continuum m B model[45], showing that the zero modes are closely
related with the topology of the quantum spin Hall state. Nonetheless, a precise
mathematical statement in the form of an analogues index theorem applicable
to the complete M B model would be highly desirable in this context.
Topological eects of a -ux vortex in the quantum spin Hall phase
3.3.2 Possible implications of the zero-mode for the classication of topological band insulators
Let us now discuss the relation of the -ux vortex to dislocations. Crystal lattice dislocations are imperfections of the crystalline order, resulting in point and
line defects in two and three dimensions, respectively. These crystal defects can
be understood using the so-called Volterra construction, which is based on the
idea that internal stresses can arise from a relative misalignment in the crystal
and may therefore be described in that manner. Explicitly, in three dimensions
one begins with a perfect crystal and chooses a plane u that terminates at the
curve R(), parameterizing the line where the defect is to be produced. In the
two dimensional case, a line v, ending at the point where the defect is to be
created, is considered. Subsequently, the crystal at one side of the plane u, or
line v, is displaced by a lattice vector called the Burgers vector b, due to the
addition or removal of atoms. The Volterra construction then concludes by 'gluing' the two sides together again. As a result the crystal pieces t back along
the whole "Volterra" cut except at the line u or point defect v, producing the
desired line or point defect, respectively. We notice that in three dimensions
the Burgers vector b can be chosen parallel or perpendicular to the line defect
R(), which then produces so-called screw or edge dislocations. In contrast, the
two dimensional case only features edge-type dislocations (Figure (3.6)).
Lattice dislocations can readily be included in numerical tight-binding calculations by adding extra atoms, to the underlying lattice, according to the
Volterra construction. To study the eect in the corresponding continuum theory we consider a closed loop, but then drawn onto the original perfect lattice.
As a result, one nds a non-closure of the loop, which equals precisely the
Burgers vector b. Moreover, this pertains to loops of arbitrary size and hence
the eect on the wavefunctions is global and long-ranged. Therefore, the defect may be modeled by imposing non-trivial boundary conditions on the Bloch
wavefunctions. Specically, the spinor is translated by the Burgers vector to
maintain single-valuedness, which eectively results in the multiplication by a
phase factor.
Let us make this more concrete for two dimensional models resulting from
a continuum approximation around a Dirac cone with momentum K, e.g. the
Hamiltonian (3.1), with K = 0, or the eective description of graphene, with
K = 43 ex (Appendix B). The eect of the dislocation, being the translation
by the Burgers vector b, results in the multiplication of the wavefunction by
a phase factor eiKb . Moreover, this condition can equivalently be imposed
by the introduction of a time reversal invariant U (1) gauge eld, using the
3.3 Interpretation and context of results
Aharonov-Bohm argument[78]. Hence, we conclude that the dislocation acts as
a Aharonov-Bohm pseudo-magnetic ux vortex located at the core of the defect,
which shrinks to a point in the continuum limit, making the connection to the
above apparent. Indeed, when
K b = mod 2
the gauge eld A(x) is exactly given by Eq. (3.4)
A(x) = 2(yxe2x ++ yx2e)y ;
as can easily be veried by integrating Eq. (3.97) over a disk. Basically, -ux
is special in the sense that it is the only possible value of magnetic ux that
does not break time reversal symmetry.
We notice, nevertheless, that K has to be nite in order for the dislocation
to have an eect. Therefore, a -ux vortex in a topological insulator, with the
bandgap opening at the -point in the Brillouin zone, can only result from a
magnetic eld. However, for the range of parameters 4 < m=B < 8, the band
gap opens at K = (0; ) and K = (; 0) and in that case K b = , therefore
giving rise to an eective topological frustration analogous to the -ux vortex.
Expansion around these points should then pertain to a similar eective theory,
making the interpretation of the gauge potential being the consequence of a
dislocation valid.
A major concern is the existence of the zero-mode bounded to the defect in
the non-trivial phase. Its non-trivial quantum numbers and existence condition
seems to suggest that the zero-energy state is intimately tied to the non-trivial
toplogy[75, 76]. Also, numerical results using the lattice regularized m B
model (2.4) suggest that the system indeed only features a zero-energy mode
when the system is in the non-trivial phase with the Dirac cones located at the
(0; ) and (; 0), for 4 < m=B < 8. This then implies that the topological phase
with the cones located at the -point is dierent from the one with the cone
situated at (0; ) and (; 0), indicating some additional structure. However, a
proper topological characterization, which will give the conclusive verication,
is currently being investigated.
Topological eects of a -ux vortex in the quantum spin Hall phase
Figure 3.6: Edge and screw dislocations in a cubic lattice
Left panel: Edge dislocation. The dislocation line R() is situated in the middle
of the front view and runs into the paper. The Burgers vector b is indicated with
an arrow. The Volterra construction is obvious; start with the perfect crystal
and choose a plane u perpendicular to b beginning at the bottom plane and
terminating at R(). Subsequently, removing one row of atoms at the lower
side of the plane u and then 'gluing' the crystal back together results in the
dislocation line R().
Right panel: Screw dislocation. The Volterra construction is in principle the
same, with the exception that in this case the Burgers vector b runs parallel to
the pane u, which cuts the cubic crystal horizontally from the right side to the
middle of the crystal. Translating the lower side of the crystal by one crystal
unit in the b direction results in the screw dislocation.
Chapter 4
In conclusion, we have shown that a Kramers pair of zero-modes bound to a
magnetic -ux vortex are a generic feature of the m B model in the topologically non-trivial phase. We have analytically found these zero-energy states
for the continuum Hamiltonian of the model in the presence of the -ux in
the entire range of parameters describing a topologically non-trivial phase with
the bandgap opening at the point. The explicit form of the modes regular at
the origin results from a particular regularization of the vector potential, but
in general the form of the solution depends on the short-distance regularization
making the domain of the Hamiltonian self-adjoint.
The explicit form of the modes reveals that they are exponentially localized
at the vortex core. In essence, the zero-energy states are in correspondence with
the surface states characterizing the topological insulating phase. Indeed, the
corresponding localization length of the zero modes exactly agrees with the penetration depth of the edge states. Moreover, normalization and regularization
conditions can be applied in the same manner, leading to the same existence
condition for both the surface states and the zero modes. Hence, the bulkboundary correspondence may be probed by inserting the a -ux vortex in the
quantum spin Hall system.
Furthermore, the topologically non-trivial nature of the two-dimensional
quantum spin Hall phase results in non-trivial charge and spin quantum numbers for the midgap states, which depend on the occupation of the pair of
zero-energy states. Namely, when both states are occupied or empty the spin
quantum number S is zero and the system's charge Q equates to +e or e
respectively, whereas ling up one of the states gives rise to Sz = 1=2 and
Q = 0. Therefore, the studied model accounts for a two dimensional realization
of spin-charge separation.
Finally, regarding the quest for additional structure in the classication
scheme of topological band insulators, there is clearly a lot to be done. Specically, we are interested in the precise mathematical formulation of the nature of
the zero modes in terms of topology. If the zero-mode is intimately tied to the
topological non-triviality of topological band insulators, this would mean that
the introduction of a dislocation results in a dierent response for topological
insulating systems with a cone situated at a nite momentum as opposed to
those with a cone at located at the point in the Brillouin zone. Basically,
lattice dislocation impose a phase shift on the wavefunctions, which is equal
to the inner product of the Burgers vector b and the momentum of the wavefunction. As a result, only wavefunctions with nite momentum are eected by
the dislocation. In the continuum approximation this eect can be modeled by
pseudo-magnetic Aharonov-Bohm uxes, like the -ux in our model, that thus
only attain a nite value if the Dirac cone expanded around is located at nite
momentum K. Particularly, when
K b = mod 2
and the expansion around the cone results in the same eective theory as the
continuum m B model, we see that our calculations imply a zero-mode response. This then pertains to some additional structure to the two-dimensional
topological insulator, depending on the position of the Dirac cone in the Brillouin zone. Although, the results shown in this thesis and preliminary numerical
simulations on the tight-binding version of the m B model look promising,
the precise formulation applicable to general models is still an open question.
Appendix A
Lauglin's argument
In 1980 von Klitzing[4] experimentally discovered the integer quantum Hall
eect. In order to theoretically describe this phenomenon, Laughlin considered a
quantum gas conned to a metallic loop (Figure (A.1)), pierced by a a relatively
strong uniform magnetic eld B normal to its surface and with an electric eld
across the ribbon.
Figure A.1: Illustration of the gedankenexperiment proposed by Laughlin[18].
The gure shows a metallic loop, which is pierced by a uniform magnetic eld
and has a voltage drop V along the ribbon.
Suppressing spin and interactions for the moment and assuming that the
temperature is low enough in order to imply quantum coherence, the Schrodinger
equation for an electron with charge e and mass m is readily obtained
( ~2 @y2 + ( [email protected] eBy)2 ) (x; y) + eE0 y (x; y) = " (x; y);
with the vector potential A = ( yB; 0; 0)> . Using the fact that p^x commutes
with the Hamiltonian, we take solutions of the form (x; y) = exp (ikx)n (y
y0 ), where
m ~k E0
1 ~k E0
y0 = (
) (
eB m B
!c m B
Lauglin's argument
and n is the solution of a resultant harmonic oscillator-like equation
~2 @ 2 + m !2 ) = (n + 1 )! ;
2m y 2 c n
2 c n
n 2 Z;
resulting in the well-known Landau levels for the corresponding energies
1 E
"n;k = (n + )!c + eE0 y0 + m( 0 )2 :
2 B
Laughlin then claried the IQHE using an analogy to a quantum pump by
introducing a ux threading the loop. According to the Aharonov-Bohm
principle, the system is gauge invariant under ux changes of integral multiples
of the elementary quantum ux 0 = he . Therefore, adding a multiple of 0
denes the cycle of a pump. Specically, the increment in vector potential
(Aex ) shifts the center of the solutions; because p^x still commutes with the
Hamiltonian, the extra term only imposes a new y0 to complete the square
y0 ! y0
Furthermore, the energies are still given by equation (A.4) only in terms of the
new y0 and therefore depend linearly on the ux change.
Figure A.2: Energy spectrum of the Laughlin set up[83]. As a result of the
magnetic eld B , Landau levels are formed. The conning potential pushes the
energy levels up at the edges. The inset shown the spectral ow of the edge
states, resulting from the increase of the ux threading the loop.
To complete the argument, imagine the Fermi level( "f ) to be located between two Landau levels. The conning potential pushes the energy levels up
at the edges (Figure (A.2)), implying as many Fermi energy states (edge states)
as occupied Landau levels( ). Increasing the ux adiabatically by a multiple
of 0 over a period t induces an electromotive force along the loop, driving a
current that transfers charge across the ribbon. Particularly, as the system is
adiabatically mapped onto itself, the total basis of single electron functions is
left unchanged. However, this does not mean that the individual basis states are
mapped onto themselves. As the localized states far from the energy level can
not transverse the gap all the dynamics must be contained in the gapless edge
states. Specically, an integral number of states( ) at the inner edge are pushed
below the energy surface and the same number of states at the outer edge sink
below the surface. This non-invariance of individual states upon returning to
a gauge equivalent copy of the Hamiltonian is known as spectral ow. More
importantly, the above argument allows us to calculate xy explicitly. Using
with 2 Z, and
we conclude that
Idt = e;
V dt = (
)dt = ;
xy = :
This then shows the quantized nature of the transverse conductivity.
Appendix B
Eective Hamiltonian of
In this appendix we consider the low-energy eective Hamiltonian of graphene,
which pertains to a reminiscent two-dimensional massless quantum eld theory.
Graphene is a planar layer of carbon atoms arranged in a hexagonal lattice
as a result of sp2 hybridization. The atoms are linked by covalent bonds, known
as bonds, whereas the electrons in the orbitals, not involved in the bonds,
are delocalized over the structure. The hopping of these electrons accounts for
the transport properties.
Figure B.1: Honey comb lattice of graphene consisting of the triangular lattice
A (black sites) and the triangular lattice B (white sites). The vectors 1 ; 2 and
3 connecting the nearest neighbours are indicated with arrows.
The resulting honeycomb lattice may be viewed as two sublattices, denoted
by A and B, or equivalently as a triangular Bravais lattice with a two-atom
basis (Figure (B.1)). This triangular Bravais lattice, with lattice constant a, is
Eective Hamiltonian of graphene
spanned by two basis vectors
a2 = a( 12 ex + 21 3ey );
a1 = aex ;
whereas the vectors that connect the A and B sublattices, and consequently the
nearest neighbours, are given by
1 = 2pa 3 (
ey ); 2 = 2pa 3 ( 3ex ey ); 3 = pa3 ey :
In order to exploit the translational symmetry it is convenient to determine the
reciprocal basis vectors k1 and k2 . Using the dening condition ai ki = 2i;j ,
we nd
k1 = p43a ( 23 ex 12 ey ); k2 = p43a ey :
The corresponding rst Brillouin is bounded by the planes bisecting the vectors
to the nearest reciprocal lattice points, resulting in the original hexagon in real
space rotated by 2 . We notice that the hexagon only has two inequivalent
corners, due to lack of inversion symmetry around the lattice.
A simple model that accounts for the low-energy properties of graphene, is
obtained using a tight-binding approximation including only nearest-neighbor(n.n.)
hopping between the two sublattices[1]. As the electrons form energy bands
far away from the energy surface[2], the relevant orbitals are the orbitals oriented normal to the plane. Denoting the corresponding creation operators for
y and y , referring to an atom of
such an orbital on atom i with spin by i
the A and B sublattice respectively, the Hamiltonian takes the form
H= t
i;j =n:n:
y + y );
j i
where t is the nearest-neighbour hopping parameter. Subsequently, performing
a Fourier transform, Eq. (B.4) becomes
H^ (k) =
ky ; ky !
H^ (k) k ;
(k ) = t
exp(ik m )
Hence, the corresponding eigenvalues are readily obtained
"k = j (k)j = t 3 + 4 cos( x ) cos( py ) + 2 cos(akx ):
2 3
The previous equation shows that the energy spectrum has two bands, which
touch at the six corners of the Brillouin zone. Restricting to the rst Brillouin
zone we nd two inequivalent points in reciprocal space that equate "k to zero
k+ = 43a ex ;
k = 43a 3ex :
The points dened above are known as 'Dirac points' and are situated at the
corner of the Brillouin zone. The corresponding continuum theory is then obtained by expanding the Hamiltonian up to linear order around the two Dirac
points, resulting in
Heff (q) = vf (x qx y qy );
with indicating the Dirac points and
p3atq ~(k k ). Furthermore i refer
to the usual Pauli matrices and vf = 2~ , i.e. the Fermi velocity. Accordingly,
the eective Hamiltonian has the following energy dispersion relation
"(q ) = vf jq j;
= 1
After performing a rotation around the x-axis, the above can be rewritten as
H eff (q) = vf (z i )qi
where i are also Pauli matrices and the sum over i = fx; yg is implied. Specically, the valley isospin z indicates the two Dirac points and the Pauli matrices
x ; y act in the sublattice space. The four component eigenstates, corresponding with the eigenvalues " = vf jqj, are then given by
1 B
= 2B
@ exp ( i 2 ) A
exp (i 2 )
exp ( i 2 )
exp (i 2 )C
+ = p1 B
= arctan ( x ):
We notice that equation(B.9) can easily be transformed back to real space.
Using the notion of valley and sublattice isospin, this results in
H eff (r) = ~vf 0 @ ;
2 f1; 2g;
where the gamma matrices include 0 = 0 z , 1 = z y and 2 = 0 x
and obey the Cliord algebra. Hence, we deduce that the eective Hamiltonian
around the Dirac points looks exactly like the well-known massless Dirac Hamiltonian with c replaced by vf and with 'spin' referring to the sublattice on which
the component acts.
Appendix C
Manifolds and vector
In this appendix we review the basics of manifolds and corresponding structures
and consider some of the beauty of this mathematical machinery. We will be
rather intuitive, the reader is therefore referred to some proper mathematical
literature for a rigorous treatment of this subject.
Manifolds are topological spaces that locally look like Rn , but not necessarily
so globally. Therefore manifolds are basically generalizations of surfaces of arbitrary dimension, on which one can dene local coordinates and calculus. These
patches are then sewn together smoothly to form the manifold. More concretely
a dierentiable manifold M is dened by the following characteristics;
Denition 1 M is an n-dimensional dierentiable manifold if
M is a topological space, i.e. some space on which one can dene sensibly
connectedness, convergence and continuity[12]
M is provided with a family of pairs f(Ui ; hi )g of open subsets Ui , with
[i Ui = M , and hi local homeomorphisms from Ui to some open subset
Vi 2 Rn . These open subsets with their corresponding homeomorphisms
are known as charts, while the whole family is called an Atlas.
the transition maps hi hj 1 from hj (Ui \ Uj ) to hi (Ui \ Uj ) are innity
dierentiable, when Ui \ Uj 6= ?.
One can associate a vector space at every point p 2 M , e.g the tangent space
Manifolds and vector bundles
Tp M . Hence we can consider
T Ui =
Tp M:
As Ui and Tp M are homeomorphic to Rn this pertains to a 2n-dimensional
manifold, composed of a direct product Ui Rn . More generally, this leads to
the denition of a bre bunde[12], which basically is a dierentiable manifold
which can be composed into two dierentiable manifolds called the bre (Tp M
from above) and the base manifold. When the bre is a vector space, this
structure is known as a vector bundle.
Additionally the manifold may be endowed with dierential forms. Specifically an r-form is an asymmetric tensor of the type (0; r). Locally using the
charts, the r-form ! can be written as
! = !1 ;2 :::r dx1 ^ dx2 ^ :::dx1 ;
where !1 ;2 :::r is antisymmetric, ^ refers to the standard wedge product and
the summation is implied by the Einstein convention. Correspondingly, we
can dene the space r of all r forms. Clearly, for a m-dimensional manifold
there are mr possibilities to chose a set (1 ; 2 :::r ) out of (1; 2:::m), hence the
dimensionality of r equates to mr . Moreover, we can dene a map d : r !
r+1 by
1 @!1 ;2 :::r d! =
dx ^ dx1 ^ dx2 ^ :::dx1
r! @x
Due to the antisymmetry it is obvious that d2 = 0, hence we obtain that the
image of r under d is contained in the kernel of d
r+1 . The former are known
as exact forms, whereas the latter are know as closed forms. Exact forms are
obviously closed, whereas the reverse in not always true. The following lemma
gives an insight for the reversed case
Poincare's lemma. If a coordinate neighborhood is contractible to a point
p0 2 U then any closed form on U is also exact.
We can then dene the rth Rham cohomology, by looking at the global
exactness of closed forms
closed r-forms
H (M )r =
exact r-forms
The magic is that the above knows all about topology. Actually d is the coboundary operator, i.e the dual of the boundary operator in homology. Very
loosely speaking, one considers r dimensional cell complexes of oriented r dimensional subspaces embedded in the manifold. The guiding principle is that a
solid triangle diers from an open one is the existence of a loop that is not the
boundary of a surface in latter one. Correspondingly, one can dene a nilpotent
boundary operator, being the analogue of equation(C.4). This then denes the
set of closed r dimensional cells that are not the boundary of a r +1 dimensional
Eq. (C.4) basically encodes the topology of the manifold it allows for the
denition of topological invariants. The most famous one is of course the Euler
characteristic, which is dened as
( 1)r br (M )
br (M ) = dim(H (M )r )
where br (M ) are the so-called Bettti numbers. Finally, we can understand
so-called Chern classes. Chern classes are characteristic classes subsets of the
cohomology classes of the base space, and measure the twisting of the bundles
when local pieces are patched together. Specically, one can dene connections,
on the corresponding bre bundles. Of course this cannot depend on the specic
choice of connection and this indeed turns out to be the case. In the case of a
complex manifold with complex vector bundle, the corresponding eld strength
or curvature then allows for the denition of characteristic classes called Chern
classes associated with complex vector bundles, producing topological invariants
called the Chern characters, which are the subject of the subsequent appendix.
Appendix D
Adiabatic curvature and
Chern numbers
In this Appendix, we review how topological invariants arise in the relevant
physical context.
Adiabatic curvature
The rst concept we need is adiabatic curvature or Berry curvature resulting
from a connection[11]. Berry noticed that the phase acquired due to adiabatic
evolution of the wave functions has a relation with geometry; the angular mismatch can be thought of as a result of local intrinsic curvature.
To make this concrete we consider a Hilbert space parameterized by R = R ,
= 1:::M , with corresponding Hamiltonian and normalized eigenfunctions
H (R)jm(R)i = "m (R)jm(R)i:
We evaluate the connection in the usual way; consider an eigenstates at two
innitesimally close points, dene
hm(R)jm(R + R)i
exp (iAm (R )) =
jhm(R)jm(R + R)ij
and expand it to rst order. Canceling R on both sides and using the normalization condition of the wave function, yields the Berry connection
iAm = hm(R)j
Subsequently we can easily calculate the accumulated phase upon completing
a closed contour C in the parameter space, resulting in the Berry phase
exp (i
Am dR ):
Adiabatic curvature and Chern numbers
As the eigenstates jm(R)i are only dened up to a phase factor, i.e they are
representatives of a ray, there is a local U(1) gauge freedom
jm(R)i ! exp (im (R));
Am (R)
m (R)
! Am (R) + @@R
The relation with electrodynamics is obvious. We can make this even more
explicit by rewriting equation (D.4) using the theorem of Stokes
exp (i
Am dR ) = exp (i
dR ^ dR fm (R))
@Am (R) @Am (R)
is the gauge eld tensor, known as the Berry curvature.
fm (R) =
@A = C;
Interplay with topology
In order to understand the interplay with topology we consider the GaussBonnet
theorem which relates the geometry of surfaces to topology. It states that
KdA = 2(M );
where M is a manifold (AppendixC) a without boundary, K is the local curvature and (M ) is the Euler characteristic. In the present case, (M ) equates
to 2 2g, with g the genus which essentially counts the number of 'handles'.
The right hand side is manifestly quantized and topological in the sense of the
manifold. Therefore this can be viewed as a topological index theorem.
Intuitively, one would expect an analogue formula for adiabatic curvature;
consider an abstract surface without boundary (e.g. a two dimensional Brillouin zone or a parameter space of angular parameters) with well dened Berry
curvature and a closed contour. Subsequently, one can integrate the Berry curvature over the area enclosed by the loop or over the area outside the loop.
Due to single-valuedness the outcome in both cases must dier by an integer
multiple of 2. Shrinking this loop to zero one concludes that the integral of the
Berry curvature over the closed surface must also yield an integer multiple of
2, which indeed turns out to be the case. Chern generalized the Gauss-Bonnet
theorem by looking at so-called Chern classes of vector bundles of smooth manifolds. These characteristic classes are associated with topological invariants
called Chern characters or Chern numbers[12]. It essentially implies that the
integral of the Berry curvature (the rst Chern class) over a surface without
boundary yields an integer. Most importantly, in the case that the parameter
space in equation (D.1) is the two dimensional reciprocal space of a complete
translational invariant crystal, the above allows us to write
fmxy (k)dkx dky = 2Cm
1 is the rst Chern
where the integral is over the rst Brillouin zone and Cm
character. This essentially pertains to the winding number of the Berry phase,
picked up by a state when it completes the contour in the Brillouin zone.
Appendix E
Discrete symmetries
Symmetries play a crucial role in physics. In quantum mechanics symmetries
are represented represented by unitary and antiunitary operators.
An operator A acting on Hilbert space H is said to be antiunitary if A is
unitary, i.e. jhAjA ij = jhj ij, and antilinear, i.e.
A(aji + bj i) = a Aji + b Aj i);
ji; j i 2 H;
a; b 2 C
It is then obvious that an antiunitary A operator can be represented by A = UK ,
with K as the complex conjugation operator that squares to unity and U a unitary operator. The correspondence to symmetries is made exact by the following
fundamental theorem[21] due to Wigner.
Wigner's Theorem. Bijections on a Hilbert space H that preserve transition probabilities are implemented by either unitary or antiunitary operators
An important example is the time reversal operator . Time reversal amounts
X0 = X
P0 = P
0 = S0 = S;
where the accent indicates the time reversed operators. Applying on the
commutator of the position and momentum operator yields
i~ 1 = [X; P ] 1
= (X 1 )(P 1 ) (P 1 )(X 1 )
= (X 0 P 0 P 0 X 0 )
= (XP P X )
= i~ :
Discrete symmetries
Henceforth cannot be unitary and must therefore be antiunitary. Moreover,
Eq. (E.1) results in the following condition on a time reversal invariant Hamiltonian
H (k; r) 1 = H ( k; r):
We can now readily nd the explicit form of the unitary part of the time reversal
symmetry operator for electrons. The complex conjugation operator reverses the
momentum, whereas for the spin we get
j "i = UK j "i = j #i:
Adopting spherical coordinates
cos( 2 )
j "i =
exp(i) sin( 2 )
j #i = exp( i)sin( 2 )
cos( 2 )
then leads to the conclusion that = exp( i~ y ). Hence, the unitary part U
equals iy
We observe that squares to minus unity. This gives rise to what is known
as Kramers theorem, stating that the corresponding energy spectrum of a time
reversal invariant Hamiltonian of an odd number of electrons should be doubly
degenerate. This is readily understood remembering that time reversal symmetry ips the spin producing an orthogonal state.
Additionally, particle hole symmetry and chiral symmetry can be evaluated
on exactly the same footing. The result is that particle-hole symmetry is also
represented by an antiunitary matrix with corresponding expression
H (k; r)
= H ( k; r);
whereas chiral symmetry is expressed by means of an unitary operator , satisfying
H (k; r) 1 = H ( k; r):
Appendix F
Anomalous quantum Hall
eect of the massive Dirac
In this Appendix, we show that a massive Dirac equation can describe an anomalous quantum Hall response. Additionally, we will motivate that the eective
theory of this response is governed by the Chern-Simons term.
Consider the massive Dirac Hamiltonian in units of ~ = e = 1
H (kx ; ky ) = (kx + Ax )Sx + (ky + Ay )Sy + mSz ;
or equivalently, using standard terminology, the Lagrangian density
L = ([email protected]= A= m) ;
with > 0 and , 2 f0; 1; 2g, the two dimensional Dirac matrices, which
include 0 = Sz ; 1 = Sy ; 2 = Sx .
In order to obtain the linear response we need the rst order radiative correction to the vacuum polarization , being the analogue of the Kubo formula.
Using the standard fermion propagators this results in
d3 k
= i
k= m k= + =q m
Now due to the general properties of the gamma matrices it is clear that the
structure of is
= e + o
q q e 2
= (g
) (q ) + q o (q2 );
Anomalous quantum Hall eect of the massive Dirac equation
where the e and o refer to even and odd under the parity operation. By explicit
computation it is easily veried that
o (q2 ) =
4 3
d3 k
m2 ][(k + q)2
m2 ]
ie2 2m + q2
p ):
4 q2 2m
Consequently, the transverse conductivity xy is dened as[31, 33]
xy =
@ 1
jq2 =0 :
3! q
By construction only the parity breaking term survives. Specically substituting equation(F.6) into the equation above yields
xy =
We note that this pertains to xy = 2eh sgn(m) in physical SI units. Moreover,
in the parity invariant limit, m ! 0, the parity breaking term survives. Hence
we are confronted with an anomaly.
More importantly, we can introduce an eective action that describes the
response of equationF.1. Integrating out the massive fermions, it is given by
dxdy dtA @ A
; ; = f0; 1; 2g:
This action is known as the Chern-Simons action and is manifestly topological,
as it does not depend on a metric. Additionally, it is well dened as a gauge
change transformation changes the Lagrangian by a total derivative.
Generically, introducing a factor 2 to account for the suppressed spin, and
a factor C1 , corresponding to the rst Chern character of a QHE system, we
Scs =
Scs =
C1 Z dxdy Z dtA @ A ;
; ; = f0; 1; 2g:
Lcs = j it is easily veried that indeed this action leads to a conserved
Using A
j = 1 @ A ;
which implies the desired response equations
ji = xy
ij Ej ;
(B ) 0 = xy B:
Where in the above equation the Latin indices run over the spatial components
0 refers to the ground state charge density.
Appendix G
The Aharonov-Bohm eect
The Aharonov-Bohm[78] eect essentially pertains to the coupling of charged
particles to the vector potential. In classical physics one concentrates on the
electric and magnetic eld and hence considers the scalar and vector potential
as auxiliary objects. However, in quantum mechanics this is radically changed.
Indeed, even in regions where the electric and magnetic eld equate to zero a
nite vector potential has an observable eect.
Figure G.1: Schematic representation of the experiment, in which the AharonovBohm eect can be observed. Electrons, coming from a source S pass through
the slits and produce an interference pattern on the screen on the right. The
experiment also features an innitely long solenoid producing a nite magnetic
eld B inside the solenoid, but zero magnetic eld outside it. The gauge potential
is, however, nite outside the solenoid.
Suppose we consider the double slit experiment for a beam of electrons (Figure (G.1)). Subsequently, we introduce a solenoid with ux and innitesimally
small radius on the origin. Hence the vector potential is given by
The Aharonov-Bohm eect
A(r) = 2(xy2+ y2 ) ex + 2(xx2+ y2 ) ey :
This vector potential results in zero magnetic eld but nite vector potential
outside the origin. Classically this would have no eect, but quantum mechanically this results in the following Hamiltonian
e 2
( i~
@x c
where we used the standard minimal coupling procedure and neglected potential
terms. The result is a phase shift for the dierent paths H=
A(r0 ) dr0 :
hc Moreover, as the two paths add to a path enclosing the total ux, the two paths
acquire a relative phase dierence of
hc (G.4)
The result is some cross term accounting for interference of the paths, which
depends on the vector potential, even though the associated elds equate to
zero at that region.
We note that ux changes of integral multiples of 0 = hce leave the system
gauge invariant. Moreover we observe that paths can encircle the ux several
times, which pertains to some winding number. This corresponds to the topology of the space, indicating a deep relation between the Aharonov-Bohm eect
and topology.
Appendix H
Atiyah-Singer theorem
In this appendix we comment on the Atiyah-Singer index theorem[12, 17, 77, 79],
which is the most general theorem that relates topology and analysis. Specically, it states that the analytical index of an elliptic dierential operator on a
compact manifold equals the topological index.
Let E and F be complex vector bundles over a m-dimensional manifold M
and D a dierential operator, i.e a linear map D : (M; E ) ! (M; F ). Roughly
speaking a dierential operator of order n in k variables is characterized by a
symbol, which is given by the number of variables xni and derivatives @in of
degree n [12]. This symbol allows for the denition of an elliptic operator,
being a dierential operator with a non-zero symbol. Furthermore, one can
then dene the analytical index of an elliptic operator as
ind(D) = dim(kerD) dim(kerDy ):
Moreover one can consider the topological index of the map D, which is essentially dened in terms of the Chern character of the top dimensional cohomology
class of the global homology class of the manifold M [12]. The Atiyah-Singer
index theorem then states
Atiyah-singer theorem. The analytical index of D equals the topological
The Atiyah-Singer index theorem is extremely useful as it directly reveals the
number of zero-energy states of a Dirac Hamiltonian, with chiral symmetry. We
shall illustrate this in what follows.
Consider such a Dirac operator that, due to the chiral symmetry, takes the
Atiyah-Singer theorem
0 Dy
D 0
in terms of the operator D, which maps a subspace M+ onto a subspace M .
Dene the number of zero-energy solutions of D by + and the number of
zero-energy states of Dy by . The chirality operator, referred to as 5 , has
eigenvalues depending on the subspaces m and is thus represented by
5 =
In order to calculate the number of zero energy states we then consider
Dy D 0
0 DDy
We notice that DDy and Dy D have the same number of non-zero eigenvalues.
Indeed, for 6= 0 it follows that
DDy = ! Dy D(Dy ) = Dy ;
which means that the operator Dy D has the same eigenvalue , corresponding
with the eigenstate Dy . Subsequently we compute the trace
Tr(5 e tK ) = Tr(e tD D ) Tr(e tDD ) =
e t+
e t ;
where indicate the eigenvalues of the operators Dy D and DDy , respectively.
Since every eigenvalue of Dy D is in one-to-one correspondence with an eigenvalue of DDy , all pairs of non-zero eigenvalues cancel out. Hence, we obtain
Tr(5 e tK ) = +
Now we should analyze the index of K . This can be done practically, using
an alternative method known as the heat expansion. It states that for general
^ and D^ on a two-dimensional manifold we can expand
1 X 2l
Tr( e tD ) =
t b ( ; D );
4t l0 l
with bl ( ; D) being expansion coecients. Specically, for = 5 and D^ = K 2 ,
we have to obtain an expression independent of t, in view of Eq. (H.7). This
implies that the coecients bl vanish for all l, except for l = 2, allowing for
an explicit evaluation of b2 even in the presence of curvature. Explicitly, let us
take D = ie (r ieA ), where are the Pauli matrices and the Einstein
summation convection for ; = 0; 1; 2 is used. In addition, e are the usual
zweibeins associated with the curved metric g , which dene a local at frame
= g e e . Furthermore, A denotes a gauge eld and r is the standard
covariant derivative.
Explicit evaluations then shows that
b2 = 2
B dS;
i.e the ux of the magnetic eld, associated with the gauge eld, resulting from
integration over the complete surface. As a result, we obtain that that the
index of K is equal to 21 B dS, which can also be derived using the usual
topological derivation[12]. In conclusion we thus nd
B dS:
This is obviously nothing but a specic case of the Atiyah-Singer index theorem.
Nonetheless, it tells in advance how many zero-energy solution there are and
therefore has may application in theoretical physics[80, 81, 82].
+ = ind(D) = dimkerD dimkerDy =
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It has been a great pleasure completing my master thesis, which is greatly due
to the people I have been working with. I would like to thank Jan Zaanen
for his inspiring comments. Additionally, I want to thank Andrej Mesaros and
Vladimir Juricic. I came to work with Andrej and especially Vladimir on an
almost every-day basis, which shows how much time they invested in our project
and ultimately my thesis. We had a lot of interesting discussions and I think it
is safe to say that we became to appreciate each other on a professional as well
as a personal level. Furthermore I thank my roommates Thomas, Bart and Bob
for their companionship. We really had an awesome time. Last, but certainly
not least, I want to thank my family for their love and support. I am truly
indebted to them for all they have given me.

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