Topological Insulators: Theory and Electronic Transport Calculations Vadim V. Nemytov Center for the Physics of Materials Department of Physics McGill University Montreal, Quebec 2012 A Thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Master of Science
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Topological Insulators:
Theory and Electronic TransportCalculations
Vadim V. NemytovCenter for the Physics of Materials
Department of Physics
McGill University
Montreal, Quebec
2012
A Thesis submitted to the
Faculty of Graduate Studies and Research
in partial fulfillment of the requirements for the degree of
Master of Science
Contents
Abstract v
Resume vi
Statement of Originality viii
Acknowledgments ix
1 Introduction 1
2 Theory of Topological Insulators 42.1 Topological Insulators in the context of Condensed Matter Theory 42.2 Topological Insulators – Preliminary discussion . . . . . . . . . 92.3 Quantum Spin Hall effect . . . . . . . . . . . . . . . . . . . . . 102.4 Integer Quantum Hall Effect in Graphene . . . . . . . . . . . . . 162.5 Quantum Spin Hall Effect in Perfect Graphene . . . . . . . . . . 27
3 Berry’s phase and the Topological Invariants 343.1 Berry’s Phase and Related Observables . . . . . . . . . . . . . . 353.2 Topological Insulators and the Z2 Topological Invariant . . . . . 413.3 Z2 Invariants and the Spin-resolved Berry’s phase . . . . . . . . 433.4 Summary of the Theory of Topological Insulators . . . . . . . . 45
4 Quantum transport – atomistic point of view 484.1 Tight-Binding Method . . . . . . . . . . . . . . . . . . . . . . . 484.2 Self-Consistency and the Tight-Binding method . . . . . . . . . 514.3 Tight-Binding method and the Density Functional Theory . . . 524.4 Quantum transport . . . . . . . . . . . . . . . . . . . . . . . . . 59
5 Quantum transport in Bi2Se3 nanostructures 625.1 Tight-Binding Model for Bi2Se3 . . . . . . . . . . . . . . . . . . 635.2 Transport in Bi2Se3 film with a trench . . . . . . . . . . . . . . 69
6 Discussion and Conclusions 786.1 Cd3As2 - candidate for a new Topological Insulator . . . . . . . 796.2 Berry’s phase and chirality in photonics . . . . . . . . . . . . . . 856.3 Outlook for TB-based numerical study of Bi2Se3 . . . . . . . . . 87
References 89
ii
List of Figures
2.1 SOI-induced spin-momentum locking in 2-D electron gas . . . . 152.2 Unit cell and the Brillouin Zone of Graphene . . . . . . . . . . . 162.3 Energy bands of graphene with Dirac cones at K and K ′ . . . . 172.4 Graphene in Haldane’s model. Two systems with different pa-
rameters are separated by a domain wall in the form of an edge 222.5 Haldane’s model and the equivalence of physics at different edges
and physics on the same edge but with different magnetic fields. 232.6 Edge states of Graphene in Haldane’s model. . . . . . . . . . . . 242.7 Energy bands of Graphene in Haldane’s model in a strip geometry
3.1 Time-reversal invariant momenta is identified for graphene in thebulk and on the “zig-zag” edge. . . . . . . . . . . . . . . . . . . 41
4.1 Diagram of the system in which the central region of interest isconnected to two external leads . . . . . . . . . . . . . . . . . . 59
5.1 Crystal structure of a 6 quintuple layer Bi2Se3 . . . . . . . . . . 625.2 Coordination of neighbouring atoms in Bi2Se3 . . . . . . . . . . 635.3 Energy bands of the Tight-Binding 6 quituple layer Bi2Se3 with
and without the spin-orbit interaction . . . . . . . . . . . . . . . 665.4 Two nearly degenerate Dirac cones in the energy dispersion of
the 6 quintuple layer Bi2Se3 . . . . . . . . . . . . . . . . . . . . 665.5 Helical states associated with different Dirac cones are confined
to opposite surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 675.6 Momentum-Spin Locking in the helical states of Bi2Se3 . . . . . 685.7 Conductance at different energies in 6 quituple layer Bi2Se3, in-
dicating the presence of helical states . . . . . . . . . . . . . . . 695.8 Energy bands of a 9QL slab of Bi2Se3 . . . . . . . . . . . . . . . 715.9 Different set-ups studied are shown schematically and notation
used is explained . . . . . . . . . . . . . . . . . . . . . . . . . . 715.10 Conductance vs. Energy in 9/6/9 quintuple layer type Bi2Se3
systems along primitve vectors ~a1 and ~a2 . . . . . . . . . . . . . 725.11 Conductance vs. Energy in the 9/6/9/6/9 type Bi2Se3 system . 735.12 Conductance vs Energy in 6/9/6 quintuple layer type Bi2Se3 sys-
tems at different extents of the central region . . . . . . . . . . . 745.13 Schematic view of the helical states going around the trenches . 755.14 Energy band diagram of Bi2Se3 in a slab geometry with 2, 3 and
Topological Insulators (TIs) are materials with the following characteristic prop-
erties [18],[3]. Firstly, they have a band gap in the interior of the system (also
called bulk) with the Fermi energy inside the gap. As such they cannot conduct
electric current through the bulk at low voltage bias. On the boundaries of
the system – edges and/or surfaces – TIs have states with energy dispersion
continuously crossing the bulk band gap. Topological Insulator can be a 2-
dimensional system or a 3-dimensional system and the boundaries correspond
to 1-dimensional edges or 2 dimensional surfaces respectively[19]. Herefords we
shall just call the surface states as “edge” states. These edge states exponen-
tially decay into the bulk of the system, but along the edge/surface they are not
localized. Furthermore, the edge states are helical. This means firstly that the
spin expectation value of a given edge state is 90° to its momentum expectation
value[20],[18],[19]. Therefore on a given edge/surface two states having opposite
momentum also have opposite spin. Secondly, two states with the same mo-
mentum but residing on opposite edges/surfaces have opposite spins. Here and
throughout the thesis when we speak of momentum or the spin of the state, we
mean expectation value or exact value if momentum and/or spin happen to be
good quantum numbers. The consequence of having such helical states is that
TIs exhibit Quantum Spin Hall Effect (QSHE), which means that in response to
an electric field, there is a quantized spin current in a transverse direction to the
field. Furthermore, Topological Insulators can only occur in systems possessing
time-reversal (T ) symmetry. Lastly, TIs are topologically distinct in a sense
that it is impossible to undergo a phase transition from a TI to an ordinary
insulator without closing energy gap in the bulk of the system[21],[22],[23].
The above paragraph describes the properties of a TI. However, it does not
illuminate what the TI “really is”. More precisely the following questions must
be clarified. What kind of physics make it possible to have a material with
the above described properties? What kind of materials may potentially be
Topological Insulators, i.e. how do we look for new TIs? From the mathematical
physics point of view what are the necessary and sufficient conditions that a
2: Theory of Topological Insulators 10
system must posses to be classified as a TI on a firm footing; that is, how do
we classify a TI without listing all of its properties? All of these questions are
addressed in chapters 2 and 3.
Firstly we will talk in general terms about physics such as Hall Effect and
Spin-Orbit coupling which are relevant for understanding TIs. On one hand this
will introduce the concepts necessary for further discussion. On the other hand,
it will build an intuition in terms of simple concepts as to why it is reasonable
to anticipate a material with TI properties at all.
Then in sections 2.4 and 2.5 we study thoroughly two related models which
introduce the simplest possible TI – a two dimensional Tight-Binding Hamil-
tonian with the z-component of spin conserved. The system is described by
relatively simple mathematics which is easy to understand. This allows to un-
derstand thoroughly how the non-trivial topological phase can arise. Other
more complicated TIs can then be understood in terms of the concepts learned
from this section.
While chapter 2 is meant to build an intuitive physical understanding of
TIs, the more mathematical aspects of the theory are left for chapter 3. In
particular it introduces topological invariants as a way to clasify TIs. At the
end of chapter 3, a summary of the entire theory discussion from chapters 2
and 3 is presented, concluding the first part of this thesis.
2.3 Quantum Spin Hall effect
Topological Insulators can be 2D or 3D systems. Both cases exhibit Quantum
Spin Hall Effect (QSHE) but 2D TIs are similar to some previously known 2D
Hall effect systems, while 3D TIs do not have a close relative in the “Hall effect
family”. As such it is best to build understanding about 2D TIs in terms of
well known Hall effect systems first. Having understood the 2D case, one can
then understand a 3D TI by extending the theory behind the 2D TI.
In order to understand QSHE one needs to understand Hall Effect in gen-
eral and then focus on more relevant Hall effect systems. QSHE has two “rel-
atives” in the Hall effect family; they are the Integer Quantum Hall Effect
(IQHE)[24],[18] and intrinsic Spin Hall Effect (iSHE)[20],[18],[19],[25]. The lat-
ter is close to QSHE, first of all, phenomenologically since both produce trans-
verse spin current in response to an electric field. Secondly, both exhibit the
2: Theory of Topological Insulators 11
spin Hall effect intrinsicly – i.e. without external agents such as externally ap-
plied fields or doping by magnetic impurities. The similarity between IQHE
and QSHE is a more fundamental one. They both exhibit Hall effect due to
the presence of topologically induced surface states. As such we first present a
preliminary discussion of the Hall effect in general in section 2.3. It is followed
in section 2.3 by a discussion of physics which make intrinsic Spin Hall effects
possible. Integer Quantum Hall system is left for an in-depth analysis in section
2.4. Finally, QSHE in a perfect graphene sheet will be shown to be essentially
due to identical mechanism as that of IQHE.
Thus we proceed with a brief discussion of the Hall Effect.
Hall Effect - Classically and Quantum mechanically
To understand the concept of the Hall Effect it’s worth keeping in mind the
following piece of physics. In classical Electrodynamics (of continuous media,
see for instance ref. [26]) when examining relation of the steady current due to
a constant electric field it is demonstrated that as a consequence of Maxwell’s
laws the most general expression, that of an anisotropic body, is:
ji =σikEk (2.7)
where ~j is the current density, σik – the conductance tensor and ~E – the external
electric field (eqn. 21.8 in ref. [26]). Furthermore, it is shown that while
in the absence of the external magnetic field the conductance is necessarily a
symmetric tensor, once the B-field is turned on – the conductance must acquire
a non-symmetric part as well. Thus in the presence of a B-field:
σik =σIik + σIIik (2.8)
where σIik is the symmetric part and σIIik – the antisymmetric part (eqn. 22.2 in
ref. [26]). The total current can then be written as:
ji =σIikEk + [ ~E × σII ]i (2.9)
The second term is identified as the Hall effect. This formula has an interesting
development in quantum mechanics. P. Streda had showed in 1981 [27] that for
a 2D system together with the external magnetic field (plus a few reasonable
assumptions) the conductance tensor can be written just like in equation (2.8),
with an interesting origin of each term. Equation (12) in ref [27] shows that
2: Theory of Topological Insulators 12
the first term in (2.8) is somewhat familiar; it is proportional to the trace of
the Green’s function at Fermi energy multiplied by the density of states at
Fermi energy. This roughly means that conductance is equal to the number of
charge carriers available at a given energy times the probability of getting from
point A to point B. And so in particular if Fermi-energy is in the band gap
then there will be no conductance due to this term. This is a common way to
see/anticipate conductance when looking at E − ~k graph of a given material
within the band theory. The second term of equation 2.8 is more interesting.
It is proportional to the rate of change of charge carriers’ density with respect
to the external magnetic field, evaluated at Fermi energy. This term is said to
have no classical analogies by P. Streda because classically charge carrier density
is independent of the B-field. However, another physicist A. Widom in his
(extremely short) paper [28] related this conductance to the second term in (2.8)
derived in classical Electrodynamics. This illuminates the fact that in Quantum
Mechanics you can have different mechanisms to the antisymmetric tensor in
the classical equation (2.8). The study of different Hall Effects in condensed
matter, such as QSHE, IQHE, etc is the study of different ways you can get
this antisymmetric conductance from first principles. Another important aspect
pointed out by Streda, is that while the first term depends on Green’s function
and thus all the possible impurities, symmetries of the system, etc – the second
term is quite universal, independent of system parameters and thus robust. An
example directly relevant for us is in reference [29], where by treating a periodic
Hamiltonian semi-classically it was shown that σII is proportional to Berry’s
curvature which shall be discussed in greater detail in later chapter.
Intrinsic Spin-Orbital Interaction and the intrinsic Spin Hall Effect
In some materials magnetic impurities may couple to the spin degree of freedom
of the charge carriers and consequently lead to the Spin Hall Effect (SHE) [30].
For the physics of TIs only the intrinsic mechanisms of spin-coupling are relevant
so we proceed with the discussion of the intrinsic SHE (iSHE).
Due to the relativistic effects the Hamiltonian has the term with the Spin
operator S coupled to the electric field. This can either be obtained by using
the Dirac equation and then taking the non-relativistic limit or one can use
classical Electromagnetic theory together with the Special Theory of Relativity
to derive the exact same terms and then second-quantize them in the end. The
2: Theory of Topological Insulators 13
correct term for SOI is well known from the rigorous treatment using the former
method (for example see ref. [31] p51 onwards). The correct equation is:
SOI =−ieh8m2
ec2σ · ∇ × ~E − eh
4m2ec
2σ · ( ~E × ~p) (2.10)
where σ is the vector of Pauli matrices, i.e. S = (h/2)σ. Typically electric field
can be written as the gradient of the potential and then the first term vanishes.
For our purposes ~E can be written as the gradient of a potential V (~r) so (2.10)
becomes:
SOI =− h
4m2ec
2σ · (∇V × ~p) (2.11)
We can thus use the the latter method and double check that the final quantized
term is indeed correct. This is an exercise in relativistic EM theory and can be
done; in fact there is a good pedagogical derivation in ref. [32], and we do get
the correct term. What is valuable about the second method is that it makes the
origin of (2.11) more transparent. We can now understand/anticipate 2D QSHE
at least partially using heuristic semi-classical (relativistic) arguments combined
with the results from Quantum mechanics. For details one is referenced to [32]
for an excellent discussion on this subject. In brief, a relativistically-fast moving
spin in an electric field feels a magnetic field in the frame of reference of the
spin and hence feels the force perpendicular to its direction of motion. Opposite
spins feel this force in opposite directions. This force can be expressed in terms
of the original electric field (i.e. electric field in the “lab” frame of reference)
and thus one can get the extra term in the Hamiltonian, namely as in (2.11).
The result of the relativistic EM treatment is that the SOI has the following
form[32]:
SOI =eh
4mec2σ · (~v × ~E) (2.12)
where electron spin is set to be S = (h/2)σ, ~v is the velocity and ~E is the
electric field. To put it into the form appropriate for quantization we write it
with ~v = ~p/me and ~E as a gradient of V (~r) divided by charge and change the
order for cross product, obtaining:
SOI =− h
4m2ec
2σ · (∇V × ~p) (2.13)
2: Theory of Topological Insulators 14
which is equivalent to (2.11) which we know to be correct. Now the origin of
the SOI is clear. From this classical term we can learn that moving opposite
spins feel force in opposite directions. We see that the spin is coupled to the
potential gradient which is always present in the condensed matter systems
and so with some additional constraints one can anticipate a sort of intrinsic
Spin Hall effect. That it is linear in momentum will later also prove important.
What is more, (2.13) can be rearranged in a suggestive form, using vector and
del identities (since we are still in a classical regime):
SOI =h
8mec2(σ · ~p×∇V − σ · ∇V × ~p)
=h
8mec2[σ · (~p×∇V ) +∇V · (σ × ~p)]
(2.14)
Now we see some more features of the SOI. For V (~r) being spherically sym-
metric we obtain (working with the form of the first term) the familiar SOI
applicable to an atomic Hamiltonian and in some condensed matter systems.
However, under certain conditions on V (~r), namely V (~r) giving a 2D system
in an antisymmetric well potential, working with the second term in equation
(2.14) and (dV (r)/dz)z, it turns into a Rashba SOI[32]. Thus we also see in a
simple way the heuristic argument for the Rashba-type SOI. What’s more, SOI
interaction in (2.14) appears as a sum of two terms. Indeed, for certain materi-
als such as graphene on a substrate the SOI is a sum of an atomistic term and
a Rashba term. The atomistic SOI and the Rashba SOI considered above are
perhaps the most commonly used in theoretical models. Atomistic SOI comes
from the assumption that the equation (2.14) is dominated by the spherically
symmetric potential in the vicinity of each atom. Rashba SOI is used when
the potential profile of a system is dominated by the global confining potential
which is anti-symmetric. Equation (2.14) is suggestive of these two scenarios
although it is more general. For instance, QSHE has also been realized in a
system where the SOI cames from a globally spherical potential induced by
strain gradient[33]. It is important to understand that the SOI term in (2.14)
is very general. It can sometimes appear in a system intrinsically (atomistic
SOI), sometimes as a side-effect of the experimental set-up (Rashba SOI) and
sometimes induced externally on purpose (Rashba SOI, strain gradient).
For further insight it is inevitable to proceed with more complex quantum
mechanics and unavoidable math that comes with it. SHE prior to the discovery
2: Theory of Topological Insulators 15
of the TIs was predicted in several qualitatively different systems. Focusing on
the intrinsic SHE, we look at a simple system producing SHE – a 2D electron
gas together with the SOI [21], as in equation (2.13).
SOI =~p2
2me
− a
hσ · (~z × ~p) (2.15)
This is the system more relevant to our discussion. Overall physics of the model
in ref. [20] are quite different from those responsible for the 2D QSH phase (i.e.
2D TI) but the main importance of that model with respect to ours is two-fold.
First of all, it was shown that a SHE can exist in the absence of a magnetic field
intrinsically due to SOI alone, and secondly they had demonstrated the so-called
spin-momentum “locking”. Spin-momentum locking is when the spin is at 90°
in the E-~p space, as you can see in Figure 2.1. That is, it was demonstrated
that if one only takes free electrons (kinetic terms) and SOI (strong Rashba-
type in their case) some very elegant physics come out leading to the intrinsic
SHE. In addition, in 2D and 3D low energy condensed matter systems one gets
Fermi gas (quasi-particles with ~k as a good quantum number) which in many
ways resembles free electrons and has similar Hamiltonian to the one in (2.15).
Therefore the model in ref. [20] has relevance for QSHE which occurs for a
condensed matter system with a Hamiltonian more complex than that in (2.15).
As the authors of ref. [20] themselves state in the introduction: “In this Letter
we explain a new effect that might suggest a new direction for semiconductor
spintronics research.” We omit the math from the model, because the details
are quite different from the model for the 2D TI. It is, however, worth looking
at the properties that the electrons have in the model of ref. [20] in E-~p space.
Figure 2.1: 2D electron gas with Rashba SOI exhibits “momentum-spin locking”
– electron’s spin at given ~k is perpendicular to it. Figure courtecy of Ref.[20]
2: Theory of Topological Insulators 16
We now turn our focus to graphene in the low temperature limit (T ∼ 0).
In references [18] and [24] graphene was shown to exhibit a QSHE and IQHE,
respectively, subject to some constraints. We shall study it in great detail
to understand both the IQHE and QSHE. From two models we shall get a
general insight into a) topological bulk-boundary correspondence which causes
conducting boundary states and b) the effect SOI has on a).
2.4 Integer Quantum Hall Effect in Graphene
The physics of graphene is dominated by nearest neighbour π-type interactions
of the atomic pz-orbitals (~z is normal to the plane of graphene) and graphene’s
honeycomb geometry. The lattice of the graphene is shown in figure 2.2.
Figure 2.2: Graphene unit cell in a) and reciprocal unit cell in b). v1 and v2 are
primitive crystal vectors while r1 and r2 and primitive reciprocal lattice vectors
The main properties of graphene can be studied within the Band theory
together with the Tight-binding approximation which includes arbitrary number
of nearest neighbours. In order to study low energy physics near the Fermi level
it is sufficient to use 1 orbital per atom in the unit cell, i.e. two orbitals per
unit cell, one from atom A and one from inequivalent atom B. A and B are
inequivalent because they are not connected by the primitive lattice vector.
Indeed this is what is done in references [18] and [24]. Constructing thus a
Bloch 2x2 Hamiltonian and only considering nearest neighbours one obtains
(with respect to EF=0):
H(~k)AA =H(~k)BB = 0 (2.16)
2: Theory of Topological Insulators 17
H(~k)AB =t1(ei~k·~a′1 + ei
~k·~a′2 + ei~k·~a′3) = t1
∑~a′i
cos(~k · ~a′i) + isin(~k · ~a′i) (2.17)
H(~k)BA =t1∑~a′′i
cos(~k · ~a′′i ) + isin(~k · ~a′′i ) (2.18)
where {~a′i} and {~a′′i } are vectors from atom A to its three nearest neighbour
atoms B and from B to A respectively. They are defined counter-clockwise in a
sense that ~z · ~a1 × ~a2 is positive. To stick to the convention used in ref. [24] we
let {~ai} = {~a′′i }. Then, by symmetry (−~a′2 = ~a′′3,−~a′3 = ~a′′1,−~a′1 = ~a′′2) one gets
(as expected from HBA = H ′AB):
H(~k) =t1∑~ai
cos(~k · ~ai)σx + isin(~k · ~ai)σy (2.19)
where σi are the Pauli matrices and here they act on the orbital space {|A〉, |B〉}.The energy bands are as in figure 2.3.
Figure 2.3: Graphene’s energy bands with vanishing gap at K and K ′ points
giving rise to the so-called Dirac Cones. Figure courtecy of Ref.[34]
The conductance and valence bands touch (gap closes) at E = EF = 0 at
points K and K ′. These points have the property that K = −K ′ which will be
used later on. If we are now to add the six next-nearest neighbours and write
the sum of exponents as cosines and sines, we find that sines cancel out while
cosines double.
H(~k)AA →2t2∑~b′i
cos(~k ·~b′i) (2.20)
H(~k)BB →2t2∑~b′′i
cos(~k ·~b′′i ) (2.21)
2: Theory of Topological Insulators 18
where again we define {±~bi} = {±~b′′i } and {±~b′′i } = ± { (~a′′2−~a′′1), (~a′′1−~a′′3), (~a′′3−~a′′2)}. The 2x2 matrix is now:
H(~k) =t1∑~ai
cos(~k · ~ai)σx + isin(~k · ~ai)σy + 2t2∑~bi
cos(~k ·~bi) (2.22)
where I is a 2x2 identity. This is the next-nearest neighbour TB Bloch Hamil-
tonian of graphene in the two orbital approximation. By the symmetry of the
lattice we again get the zero gap at K and K ′. This is the standard way to
study main properties of graphene and can be found in many textbooks (e.g.
ref. [35]). Written in this form the Hamiltonian can be recast into an effective
2-dimensional Dirac equation if expanded about the K and K ′ points using the
~k · ~p approximation (e.g. ref. [36]). It is possible because the Pauli matrices
acting on the orbital space respect the same commutation relations as the Pauli
matrices representing spin (in the original Dirac equation); also because near
K and K ′ one can do linear expansion in ~k (and some other assumptions). For
our purposes, namely to establish the physics behind 2D and 3D Topological
Insulators, this is not desirable. We find the underlying principle not in the
Dirac equation. Instead we shall use another convenient aspect of having our
H(~k) written in term of Pauli matrices as in (2.22). The symmetries of the
system become transparent. This will be especially important in derivations of
ref. [18], discussed in section 2.5.
In ref. [24] Haldane adds to the equation (2.22) certain extra terms and
studies the resulting effect to demonstrate the possibility of the integer quan-
tized Hall effect without the Landau levels, i.e. no magnetic flux. In ref. [18]
a spin-orbit term as in (2.13) is added and the the spin Hall conductance is
derived. We first focus on the effects of the terms added in ref. [24] to then
better understand ref. [18].
In ref. [24] two terms are added to equation (2.22). The first is added by
hand representing antisymmetric local potential at sites A and B of magnitude
|M | and is written simply as Mσz. The second term arises from adding a
magnetic field which respects full periodicity of the system (with Mσz added)
and so results no net flux through the unit cell. Such B-field can be represented
as a curl of the periodic vector potential ~A(~r) having the effect on the original
system of modifying t1 → t1 and t2 → t2ei(e/h)
∫~A·d~r = t2e
±iφ where φ is a
constant phase and the sign in front depends on the relative orientation of the
2: Theory of Topological Insulators 19
two next nearest neighbours (details are in [24]). When one includes the effect
of this periodic ~A(~r) and carefully computes H(~k) again in the {|A〉, |B〉} basis,
one finds that the terms due to next nearest neighbours proportional to sine do
not cancel out anymore. The result is:
H(~k) =t1∑~ai
cos(~k · ~ai)σx + isin(~k · ~ai)σy + 2t2∑~bi
cos(~k ·~bi)
+(M − 2t2sin(φ)
∑~bi
sin(~k ·~bi))σz
(2.23)
Now the first two terms still vanish due to the symmetry of the lattice at points
K and K ′. The term in the second line of equation (2.23) contains the constant
M which clearly doesn’t vanish everywhere unless set to zero; it also contains
the sum over sines – this also doesn’t vanish at K and K ′ unlike the sum of
cosines. In general, the Hamiltonian in (2.23) doesn’t have a vanishing gap
anywhere (provided |t2/t1| < 1/3); the gap-closing can only occur at K or K ′
similar to before provided that the third term has cancellation at K or K ′[24].
This can only occur if M = ±3√
3t2sin(φ) where the sign depends on whether
it is at K or K ′.
We have arrived at the most important feature of the Haldane’s model. The
energy spectrum of the system has, in general, a band gap. This gap varies in
E − ~k space and has local minima at K and K ′ points; in fact, at K or K ′ it
can even vanish provided the above mentioned condition on the third term is
satisfied. We then focus at our system at K and K ′ points.
H( ~K ′) =(M − 2t2sin(φ)∑~bi
sin( ~K ′ · ~bi))σz
H( ~K) =(M − 2t2sin(φ)∑~bi
sin( ~K · ~bi))σz(2.24)
we now defineK ′ as the point where the sum over sines evaluates to (−1/2)(3√
3)
and so at K = −K ′ we have (+1/2)(3√
3). Thus, equations (2.24) evaluate to
H( ~K ′) =(M − 3√
3t2sin(φ))σz
H( ~K) =(M + 3√
3t2sin(φ))σz(2.25)
Note that in both cases we get constant(KorK ′) · σz so that the constant
determines the eigenvalues, but the eigenstates never change; they are [1, 0]T
and [0, 1]T . Label these eigen-states as |1〉 and |2〉 respectively. Also note
2: Theory of Topological Insulators 20
that if |1〉 has eigen-value λ1 = constant(KorK ′) then λ2 = −λ1 = −const,because σz = [1 0; 0 − 1]. In terms of the notation of ref.[24], the constant
at K ′ and K is proportional to m− and m+ respectively. Now, note that e-
values depend on three parameters −λ2 = λ1 = λ1(M, t2, φ) and so there are
separate cases to consider depending on the values of the parameters. We define
∆E(K) = E|1〉 −E|2〉 for the energy gap at K and we similarly define ∆E(K ′).
We shall examine several cases depending on the parameters (M, t2, φ) and pay
special attention to ∆E(K) and ∆E(K ′) – these values will prove to be of
significance.
Case 1: M = 3√
3t2sinφ, or else M = −3√
3t2sinφ. We now get a single
vanishing gap in the reduced Brillouin Zone (BZ) (or 3 in the entire BZ). They
occur at, respectively, K or else K ′. At the same time at the opposite point,
i.e. K ′ and K respectively, we get a gap equal to 6√
3t2sinφ. Note, these cases
are equivalent to m− or else m+ being zero in ref. [24].
Notice that M = |M | > 0 and M = −|M | < 0 assign an identical eigenvalue
but of opposite sign to the eigenstate |1〉 (and so |2〉) in ~k-space where there is
a gap, i.e. ∆E( ~K ′) and ∆E( ~K) respectively. The sign of ∆E( ~K ′) and ∆E( ~K)
is opposite but it doesn’t matter since this relative sign difference occurs not
in the same system. We shall see later that the sign of this eigenvalue matters
and has consequences in case 3.
Case 2: M = ∆ε such that |∆ε| > 3√
3t2sinφ. Then no matter whether
∆ε < 0 or ∆ε > 0 you get an energy gap at both ~K ′ and ~K. Moreover the
e-state |1〉 has the same sign of the corresponding e-value at both points ~K ′ and
~K. Therefore ∆E( ~K ′) and ∆E( ~K) have the same sign. This case is equivalent
to m− and m+ having the same sign in ref. [24]. Note that the absolute sign of
∆E is not important since we can always redefine it to get a positive gap. The
important thing is the relative sign of ∆E( ~K) and ∆E( ~K ′), as we shall show.
Case 3: M = ∆δ such that |∆δ| < 3√
3t2sinφ. The sign doesn’t matter so
without loss of generality let ∆δ > 0. We now get a band gap at both ~K ′ and ~K
but ∆E( ~K ′) = 2(∆δ + 3√
3t2sinφ) > 0 while ∆E( ~K) = 2(∆δ − 3√
3t2sinφ) =
−|2(∆δ−3√
3t2sinφ)| < 0. This band structure can be called inverted, although
it is not a convention. What is important to point out is that one cannot go
from case 3 to any other case continuously without closing the gap. This signals
that case 3 is a topologically distinct phase from the rest. Also note that when
2: Theory of Topological Insulators 21
the Hamiltonian in equation (2.25) is recast in an effective Dirac equation, like
in ref. [24], case 3 corresponds to quasiparticles associated with wave-vectors
~K ′ and ~K as having opposite mass.
Case 3 puts the system into an Integer Quantum Hall Phase. As we shall
now argue, Case 3 exhibits states confined to the boundary (i.e. edge) which
continuously cross the bulk-gap and carry current with the quantized transverse
Hall conductance σxy = ±e2/h. The sign depends on the sign of ∆δ and is
equivalent to just defining where is “up” in our 2D system; whether current will
flow clockwise or counterclockwise. The author in ref. [24] establishes the case
of quantized Hall conductance for the case 3 by applying small magnetic field,
examining how the resultant Landau levels are filled and then taking the limit of
vanishing field. That is a perfectly valid way to demonstrate the result, however
with the hindsight of the theory of Topological Insulators, we shall argue for
the conductance in terms of topological bulk-boundary correspondence[37].
Let H(~k)i be the Hamiltonian from equation (2.23) with the parameters as
in Case i, where i = 1, 2 or 3. We take case 3, i.e. H(k)3, set ∆δ > 0, define
∆E( ~K) = E|1〉−E|2〉 at ~K and ∆E( ~K ′) = E|1〉−E|2〉 at ~K ′, as before. We thus
have a system which in the bulk has a band gap and an additional property that
in the vicinity of the points ~K and ~K ′ we have ∆E( ~K) < 0 and ∆E( ~K ′) > 0.
Relative sign of ∆E( ~K) and ∆E( ~K ′) matters, and we have an inverted band
structure. We now perform the following thought experiment to demonstrate
the existence of edge states.
Take the system defined by H(k)3 and put it on a 2D half-infinite manifold.
We define x’-y’-z’ axis with z’-axis same as in ref. [24] (normal to the plane,
pointing up, i.e. z’=z), x’-axis is parallel to v2 from figure 2.2 and positive
y’ is at 90° normal to x’ such that x′ × y′ = z′. We shall make our manifold
infinite in negative y’.The positive y’ direction we shall call the front (and -y’
– back). The edge in this way is the “zig-zag” edge. In x’-axis it is infinite
(periodic) so we only focus on the front edge effects. On manifold with edges
one can simply get a Bloch Hamiltonian with periodicity only in one dimension
and obtain bands. The purpose of this thought experiment, however, is to learn
to anticipate the results prior to actually computing such 1D-periodic Bloch
Hamiltonian. Let’s define the system of H(~k)3 together with the half-infinite
manifold as H(~k)3 −M f3 . M stands for manifold and the superscript f means
2: Theory of Topological Insulators 22
that it is semi-infinite with the edge at the front.
Now, at the front boundary of H(~k)3 −M f3 add system H(~k)2 on another
half-infinite manifold with the edge at the negative y’ i.e. at the back. This
added system we call H(~k)2−M b2 and the total system that results from fusing
the two we call H(~k)3 −M f3 /H(~k)2 −M b
2 . The total system is schematically
shown in figure 2.4. H(~k)2 has bulk band gap with ∆E( ~K) and ∆E( ~K ′) both
positive, while H(~k)3 has an inverted band structure with alternating signs of
∆E( ~K) and ∆E( ~K ′).
Figure 2.4: The system H(~k)3 −M f3 /H(~k)2 −M b
2 . It is infinite in all extents,
but has a “zig-zag” boundary betweed the systems H(~k)2 (yellow and light blue
atoms) and H(~k)3 (orange and dark blue atoms).
If we separately look at 2D-periodic H(~k)2 and H(~k)3 bulk-bands we re-
alize that one set of bands cannot be continuously transformed into the other
set without closing the energy gap. Formally one can say that they cannot be
transformed one into the other adiabatically without closing the gap due to
different topology of the bands (like a coffee cup cannot be transformed into a
sphere without closing the hole). At the same time, the energy of the above de-
fined “H(k)3−M f3 /H(k)2−M b
2” system must have energy continuously defined
throughout the entire manifold. We conclude that far away beyond and before
2: Theory of Topological Insulators 23
the boundary the energy must be well described by the 2D-periodic bulk-bands
of H(~k)2 or H(~k)3 respectively. Thus in the neighbourhood of the boundary
we must have energy states which continuously transform the bands of H(~k)3
into H(~k)2 and therefore cross the gap due to the above-discussed topology
considerations.
We thus have concluded that in the energy region corresponding to bulk-
gap of H(~k)2 and H(~k)3 there exists a continuous energy band. We do not
know the exact shape of it on E − ~k diagram, but we know it either “starts”
on the valence band and “ends” at the conduction band (as we read E − ~kgraph from left to right) or the other way around. We also know that the states
corresponding to these energies are localized along the edge. These edge state
exist due to topological distinction of H(~k)2 and H(~k)3 bulk-bands.
Note it is necessary to have an energy gap in the bulk in order to get edge
states. This is because otherwise bulk states with the same energy and the same
edge-parallel momentum component will couple to our “edge-states-hopefuls”
and result in states that are delocalized throughout the entire system – contrary
to the definition of a boundary state[38].
Figure 2.5: The system H(~k)3 −M b3/H(~k)2 −M f
2 with magnetic field B = B0
on the left is equivalent to the system H(~k)3−M f3 /H(~k)2−M b
2 with magnetic
field B = −B0 on the right.
2: Theory of Topological Insulators 24
We now repeat the entire argument but for the edge being at the back of
the half-infinite manifold on which H(~k)3 is defined. Using the above notation
we now have “H(~k)3 − M b3/H(~k)2 − M f
2 ” system. Recall that H(~k)i has a
chirality due to the magnetic field; this will now manifest itself. The situation
is visualized in figure 2.5, studying physics at opposite edges is like studying
physics at the same edge with the opposite magnetic fields. Now the back of
the system corresponds to -y’. More precisely it corresponds to -y’/z’/x’ which
is equivalent to +y’/-z’/x’ which in turn is equivalent to switching B(z′) to
B′(z′) = B(−z′) = −B(z′). Transforming B → −B is equivalent to A → −A,
which in turn is equivalent to φ→ −φ. This interchanges the sign of the gap of
the system at ~K and ~K ′, i.e. the values of ∆E( ~K) and ∆E( ~K ′) get interchanged.
Therefore if we have a positive slope of the edge-band in the above discussed
example, then here we get the negative slope. The BZ of graphene with a single
edge is shown as a projection of the bulk BZ in figure 2.6. In the figure the
energy dispersions of the edge states at the front edge and at the back edge
are shown separately, in green and burgundy respectively. The bulk energy
dispersion is in blue.
a) b)
Figure 2.6: BZ of the system H(~k)3 possessing only a) the front or b) the back
edge. Shown as a projection of the Bulk BZ. Bulk bands are in blue; green and
burgandy are the edge states. a = |v2|
2: Theory of Topological Insulators 25
The energy dispersion of graphene in a strip geometry with both the front
and the back edges is shown in figure 2.7.
Figure 2.7: BZ of H(~k)3 which has both the front and the back edges. Green
and burgundy bands correspond to states localized along the front and back
edge respectively. Bulk bands are in blue. a = |v2|
We have found that at each front and back edges of H(~k)3 we have localized
boundary states which are chiral – i.e. positive kx′ states are on one edge and
negative kx′ states are on the opposite edge. This phenomenon is called chirality.
Chirality in our system has as its ultimate origin the fact that the third term
in eq. 2.23 which is responsible for producing an inverted bulk band structure
distinguishes between different directions in real-space. The system is inherently
chiral as can be seen in figure 1 in ref. [24].
An important feature of the chiral edge states follows. Suppose we now add
e-e interactions and try to treat boundary states with an effective 1-dimensional
Hamiltonian. Then we may wish to proceed with the approximations of the Lut-
tinger model. However, in this case we will be forced to set the amplitude for
the back-scattering event to zero. The probability of one electron scattering
another one backward is essentially zero since the forward moving and back-
ward moving electrons are separated by a macroscopic distance. Thus, the
Luttinger liquid (with back-scattering=zero) reduces to the model of the free
quasi-particle rather than charge and spin waves. In other words Luttinger
model is not a suitable approximation for the effective 1-D Hamiltonian of our
edge subsystem. Instead, our edge-states are best described as quasi-particles
with crystal momentum (even in 1-D). Such a 1-D system can only exist as
a part of a bigger 2-D system, otherwise we do not get the spatial separation
of forward-movers and backward-movers. An important feature of such a 1-D
2: Theory of Topological Insulators 26
subsystem is that it’s robust against impurities or electron-electron interac-
tions because back-scattering is forbidden. This is contrary to the typical 1-D
fermionic system which is very sensitive to impurities resulting in Anderson
localization[39].
From the fact that we have chiral states it is easy to see how the transverse
Hall conductance arises. Near Fermi energy which is in the gap, the conduction
of current can only take place at the edges. If we establish a small voltage in
x’-direction – a current will flow from, say, left to right. This is equivalent to
right-moving states (+kx′) being more populated than the left-moving states
(−kx′) on our band diagram in figure 2.7. However, since our ±kx′ states reside
on front/back edges this means we have an imbalance of net charge accumulated
on each edge. The front edge with forward-moving electrons has more electrons
than the back edge. This establishes a voltage between the back and front edges
which is nothing but the Hall voltage. This gives a transverse conductance
(charge transfers from one edge to the other). In our earlier general discussion
of the Hall effect in section 2.3, we have seen that the Hall effect is precisely this.
This Hall conductance is quantized because it is proportional to the density of
states at the Fermi energy which gives 1 per edge. It is protected against back-
scattering due to impurities or e-e interaction due to chirality. Thus you get
integer quantized/quantum Hall effect with conductance ±1 e2
h.
This Hall conductance can be shown to be equivalent to the so called topo-
logical Chern number (see ref. [40],[41] for instance); it is a topological invariant
defined for energy bands. Chern number is induced by the so called Berry’s
curvature in momentum space. For now, these concepts over-complicate the
heuristic discussion of the topological boundary states but later on we shall
return to the concept of Berry’s curvature and its consequence for the TIs.
A word of caution is necessary. From the above discussion it follows that the
origin of the boundary states is in the fact that topology of bulk-bands of our
system of interest, H(~k)3, is distinct from that of the system on the other side
of the boundary, H(~k)2. It is then also intuitive that small perturbations can
change the shape of the bands but not the topology and thus the conducting
boundary states persist. It is intuitive but not conclusive and in fact it is a
subject of recent and ongoing research. A careful analysis into the origin and
stability of surface states of IQHE and QSHE had shown[42] that the existence
2: Theory of Topological Insulators 27
of gapless (and so conducting) boundary states is not just dependant on the
topology of the bulk but also on the conservation of a certain observable related
to the physics of the boundary states. For IQHE it’s the conservation of charge
and for QSHE it is the conservation of spin (or pseudo-spin in general). We
shall return to the question of the stability of these topological edge states later.
We finish this section by stressing the importance of topology of the bands.
It has been demonstrated above that the gap must close and reopen in order
to go from H(~k)2 to H(~k)3. Thus the edge states can be seen as the system
undergoing a topological phase transition across a boundary. In going from a
trivial band insulator like H(~k)2 to vacuum across a boundary the gap does
not need to close and reopen and so trivial insulators do not have gap-crossing
edge states. One can say that trivial insulators are insulators which can be
adiabatically taken to the atomistic limit without closing the gap. Vacuum and
trivial insulators are topologically equivalent in this sense. It follows that at
the edge between H(~k)3 and vacuum we shall also have the chiral edge states
described above.
2.5 Quantum Spin Hall Effect in Perfect Graphene
We now proceed to the model in ref. [18] which was one of the first to predict
and define the topological QSH phase in 2D. Initially they identified it as a
new type of iSHE system, because the spin current was due to the edge states.
Shortly after the same authors have published a paper where they defined a
Z2 topological invariant for a general class of 2D T -invariant bulk insulators to
uniquely define such a condensed matter state [3]. It deals with the 2D graphene
as defined above in (2.22) with the focus on the effect of adding a SOI term.
In ref. [18] they work on a model of a 2D graphene sheet with a SOI added
to study the effect of SOI at T=0 limit. An important feature of graphene
is that its bands form Dirac cones with Fermi energy falling precisely at the
Dirac point. Given that the gap is closed only at the Dirac points, adding SOI
(a relatively small effect normally) dominates the existence of the energy gap.
This is, in general, the requirement for potentially having a TI.
As has been already discussed in the previous section, the energy eigen-
states near the Fermi energy are near the nonequivalent points ~K and ~K ′. As
such, to study the low energy physics, in ref. [18] they focus on H( ~K) and
2: Theory of Topological Insulators 28
H( ~K ′) and model the effective Hamiltonian in terms of the Bloch functions at
~K and ~K ′. In terms of the Tight-Binding Bloch Hamiltonian that we already
introduced in equation (2.22) and thoroughly discussed, what is done in ref.
[18] can be explained as follows. The ~k-space can be represented as an infinite
enumerable set {~ki}, and then we can write H(~k) in a matrix form 〈~k|H|~k′〉 as
an infinite dimensional diagonal matrix:
〈~k|H|~k′〉 =
. . . 0 0
0 H(~ki) 0
0 0. . .
. (2.26)
If we now ask which eigen-vectors correspond to eigenvalue equal to Fermi
energy we shall find that it is a linear combination of those vectors with non-
zero entries only at positions i corresponding to ~K or ~K ′. Therefore we can
write an approximation as in (2.27) leading to the form studied in ref. [18] in
equations (13)-(15).
〈~k|H|~k′〉 =
. . . 0 0
0 H(~ki) 0
0 0. . .
≈(H( ~K) 0
0 H( ~K ′)
). (2.27)
Where each element of the final matrix is a 2× 2 block matrix in {A,B} orbital
space (or equivalently A,B sublattice space). Finally, the authors in ref. [18]
use the result of the k · p expansion [36] around ~K and ~K ′ to get an effective
Hamiltonian in equation (2.29) in terms of slowly varying envelope functions
The effective Hamiltonian in equation (2.29) is written in a matrix form with
respect to the basis { |A, ~K〉,|A, ~K ′〉, |B, ~K〉, |B, ~K ′〉 }.
H0 = hvF ψ†(~r)
(
0 0
0 0
) (kx + iky 0
0 −kx + iky
)(kx − iky 0
0 −kx − iky
) (0 0
0 0
) ψ(~r).
(2.29)
2: Theory of Topological Insulators 29
Notice in particular that points ~K and ~K ′ are decoupled. This Hamiltonian
gives gapless energy spectrum E(~q) = ±vF |~q|, where ~q is the wave-vector with
respect to ~K or ~K ′. We now rewrite it in a notation originally used in ref. [18]:
H0 = hvF ψ†(~r)(σxτz ~kx − σyI ~ky)ψ(~r) (2.30)
where I is the 2× 2 identity on {| ~K〉,| ~K ′〉}, σj and τi are Pauli matrices acting
on {|A〉,|B〉} and {| ~K〉,| ~K ′〉} subspaces respectively. In this representation it’s
easier to see which additional term can induce an energy gap and at the same
time respect the existing symmetries.
The Hamiltonian in (2.30) has inversion symmetry with the centre of inver-
sion being the point between (any) atom A and B, it also has time-reversal (T )
symmetry. The time-reversal symmetry is easy to see since the Hamiltonian of
pure graphene has no magnetic terms. The inversion symmetry of the Hamilto-
nian in equation (2.30) can be demonstrated in a few steps. If a Hamiltonian H
has an inversion symmetry, it means that it commutes with the parity operator
π. Therefore, it suffices to show that Hπ|Ψ〉 = πH|Ψ〉 for any state |Ψ〉. We
show this is true for the basis ket |A, ~K〉; it can be shown for the rest of the
basis kets similarly.
Hπ|A, ~K〉 =H|B,− ~K〉
=H|B, ~K ′〉
=(−kx + iky)|A, ~K ′〉
(2.31a)
πH|A, ~K〉 =π(kx − iky)|B, ~K〉
=(−kx + iky)π|B, ~K〉
=(−kx + iky)|A,− ~K〉
=(−kx + iky)|A, ~K ′〉
(2.31b)
The last line of equation (2.31a) equals the last line of equation (2.31b) and so
the inversion symmetry of H in equation (2.30) follows.
We now rewrite equation (2.25) of Haldane model to make easy compar-
isons:
H( ~K ′) =(M − 3√
3t2sin(φ))σz
H( ~K) =(M + 3√
3t2sin(φ))σz(2.32)
2: Theory of Topological Insulators 30
Recall, that here σz acts on the { |A〉,|B〉 } subspace. For further convenience,
we rewrite it in { |A, ~K〉,|A, ~K ′〉, |B, ~K〉, |B, ~K ′〉 }, and add a subscript “Hal-
dane”:
HHaldane = MσzI − 3√
3t2sin(φ)σzτz (2.33)
As before, here σz acts on {|A〉,|B〉} subspace, while I and τz act on {| ~K〉,| ~K ′〉}.We can now easily see that the term MσzI which was added in Haldane’s
model opens the gap but also breaks the inversion symmetry:
MσzIπ|A, ~K〉 =MσzI|B,− ~K〉
=MσzI|B, ~K ′〉
=Mσz|B, ~K ′〉
=M |A, ~K ′〉
(2.34a)
πMσzI|A, ~K〉 =πMσz|A, ~K〉
=πM |A, ~K〉
=M |B,− ~K〉
=M |B, ~K ′〉
(2.34b)
The last lines of equations (2.34a) and (2.34b) are not equal. The term MσzI
is extrinsic and naturally it is not present in a perfect graphene. We can also
see that the other term, the extrinsic term due to periodic magnetic field,
3√
3t2sinφ · σzτz, opens the gap but breaks another symmetry of (2.30) – the
T -symetry. The T operator is T = exp(−iπσy/h)K, where K is the complex
conjugation. We check the commutation [σzτz, T ]; note that T | ~K〉 = |− ~K〉 in
~k-space:
σzτzT |A, ~K〉 =σzτz|A,− ~K〉
=σzτz|A, ~K ′〉
=σz|A,− ~K ′〉
=σz|A, ~K〉
=|A, ~K〉
(2.35a)
2: Theory of Topological Insulators 31
T σzτz|A, ~K〉 =T σz|A, ~K〉
=T |A, ~K〉
=|A,− ~K〉
=|A, ~K ′〉
(2.35b)
One can see that [σzτz, T ] 6= 0 since the last lines in equations (2.35a) and
(2.35b) are not equal. T -symmetry is broken by the term 3√
3t2sinφ ·σzτz, and
so this term is not present in a naturally occurring graphene either.
Now, as was discussed in the section 2.3 the SOI term is always present in
a condensed matter system. What was realized in ref. [18] is that if we add the
spin degree of freedom in (2.30) and consider the naturally occurring SOI then
it will be of the form ∆SOσzτzsz, where sz is a Pauli matrix acting on spin. An
additional term of this form gives a system of two decoupled Hamiltonians –
one for spin up and one for spin down. Each of these subsystems corresponds to
Haldane’s model in equation (2.33) with M = 0, ∆SO = 3√
3t2sinφ and gives a
non-trivial topology, i.e. a system we reffered to as H(~k)3 in section 2.4. The
spin degree of freedom made it possible to have an effective magnetic field for
each spin without breaking the T -symmetry. To see that the SOI is indeed
of the form ∆SOσzτzsz consider the following three arguments. Firstly, rewrite
SOI and observe that SOI( ~K) = −SOI( ~K ′).
SOI(~k) = − h2
4m2ec
2~σ · (~∇V (~r)× ~k) (2.36)
Now, using the fact that ~K ′ = − ~K, plugging K and K ′ for k in (2.36) we see
that SOI( ~K) = −SOI( ~K ′). Therefore SOI is proportional to τz in {| ~K〉,| ~K ′〉}subspace. Secondly, ~σ can be re-written sz because the z-component of spin is
conserved. Lastly to see that it is proportional to σz use the fact that |A〉 = π|B〉where π is the parity operator and where we define ~r = 0 to be the inversion
centre.
〈A|SOI(~k)|A〉 =(〈B|π)SOI(~k)(π|B〉)
=〈B|(πSOI(~k)π)|B〉
=〈A|SOI( ~−k)|A〉
(2.37)
where we used π = π†, and that parity effect on the operator is to take ~k to
−~k. Now it follows from SOI(−~k) = −SOI(~k) that SOI is also proportional
2: Theory of Topological Insulators 32
to σz in {|A〉,|B〉 } subspace. Therefore the full Hamiltonian of graphene is