Download Topological properties and surface states in a Weyl semimetal lattice

University of Trento
Department of Physics
Master Degree in Physics
Final Thesis
Topological properties and surface states
in a Weyl semimetal lattice model
Supervisor:
Prof. Iacopo CARUSOTTO
Graduant
Co-Supervisor:
LORENZO CRIPPA
Prof. Giuseppe SANTORO
ACADEMIC YEAR 2014-2015
Introduction
Both in electronic and optical systems, the study of topological effects has been
one of the hottest research topics in recent years. Depending on the mapping
between k-space and Hilbert space, known systems such as electronic insulators,
waveguides or, as in the case we will study, optical lattices exhibit unique surface
properties.
Due to their great customizabiliy, such systems can even be used to study from a
condensed matter point of view particles that have never been observed in relativistic quantum field theory, as Weyl fermions. In particular, this thesis is focused
on 3D materials known as Weyl semimetals, that under suitable conditions present
a vanishing bulk gap in a finite number of points and peculiar topological surface
states at zero energy known as Fermi arcs.
Weyl systems exhibit a variety of interesting features, like the equivalent of the
field theory Adler-Bell-Jackiw anomaly (here known as Chiral anomaly) and many
related phenomena, as negative magnetoresistence, anomalous Hall effect and nonlocal transport [12].
It has to be stressed though that, while many theoretical models for Weyl semimetals have been studied, experiments have still a long way to go: the first materials
in which Weyl behaviour was predicted are in the pyrochlore idrates family; more
recently, systems of the TX family (where T= Ta or Nb and X=As or P) have
been proposed and in 2015 confirmed through angle-resolved photoemission spectroscopy [18]. Again in 2015, Weyl points were also observed by Soljacic et al. [17]
in gyroid photonic crystals, validating predictions done in 2013. However, surface
states detection through light scattering still remains a challenging task, mainly
because of the difficulty to isolate them from the bulk signal.
This work is based on a different approach to the subject, using not a particular
material but an optical lattice trapping a BEC. The proposed Hamiltonian is an
1
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IS-breaking stack of 2D Harper lattices with phase per plaquette π, realized via
laser-assisted tunneling. The study was initially focused on the propagation of
wavepackets on the surface of such a lattice, under the hypothesis that the shape
of the zero energy surface states smoothly depended on the direction of the surface
plane. However, this idea proved not to be entirely true, as for specific cuts we
stumbled upon a peculiar situation of flat surface bands, with no group velocity
for the wavepacket. Moving the Weyl point projections in the Brillouin Zone also
produced an interesting behaviour of surface bands, especially when those projections coincide.
The following work is divided into four parts: in the first, we sketch the framework we will be working in, giving the basic notions of topology in physics, Weyl
Hamiltonian and semimetallic phase. The second part describes the proposed optical lattice we have studied, and illustrates the surface state behaviour as the cut
direction of the slab is changed. The third part provides a quick analytical justification of some of the obtained results, while in the fourth part some simulations
of wavepacket propagation on an optical Weyl semimetal lattice chunk are shown.
Contents
1 Theory
1.1 Topology in physics, a brief introduction . . . . . . . . . . . . .
1.1.1 The adiabatic theorem . . . . . . . . . . . . . . . . . . .
1.1.2 An interesting parallel . . . . . . . . . . . . . . . . . . .
1.1.3 Chern number and topological effects: the IQHE . . . . .
1.2 Some interesting hamiltonians . . . . . . . . . . . . . . . . . . .
1.2.1 The Weyl equation . . . . . . . . . . . . . . . . . . . . .
1.2.2 Harper-Hofstadter hamiltonian . . . . . . . . . . . . . .
1.3 Weyl semimetals . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Introduction to Weyl nodes . . . . . . . . . . . . . . . .
1.3.2 Stability of Weyl nodes . . . . . . . . . . . . . . . . . . .
1.3.3 Parallel with monopoles . . . . . . . . . . . . . . . . . .
1.3.4 Surface states . . . . . . . . . . . . . . . . . . . . . . . .
1.3.5 The Nielsen-Ninomiya theorem . . . . . . . . . . . . . .
1.3.6 More intuitive arguments for Nielsen-Ninomiya theorem .
2 3D
2.1
2.2
2.3
lattice Weyl Hamiltonian
Introduction . . . . . . . . . . . . . . . . .
Extension to the 3D case . . . . . . . . . .
Cutting the lattice . . . . . . . . . . . . .
2.3.1 Cut along the x − y = 0 plane . . .
2.3.2 Cut along the x − ay = 0 plane . .
2.3.3 Cut parallel to one cartesian axis .
2.3.4 Cut along the x − y + az = 0 plane
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3 Analytical approach
3.1 Analytical derivation of 3d Weyl surface states
3.1.1 Our system: cut along x − y = 0 . . .
3.1.2 Arc direction with energy tilt . . . . .
3.1.3 Numerical verification . . . . . . . . .
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4
CONTENTS
4 Wavepacket propagation
65
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Cut along the plane x − y = 0 . . . . . . . . . . . . . . . . . . . . . 66
4.3 Cut along the plane x − y + z = 0 . . . . . . . . . . . . . . . . . . . 67
Conclusions
73
Appendices
74
A Useful material
A.1 Adiabatic theorem . . . . . . . . . . .
A.2 Peierls’ substitution . . . . . . . . . . .
A.3 A 2D toy model for border states . . .
A.4 The SSH Hamiltonian . . . . . . . . .
A.4.1 Exact calculation of edge states
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84
B MATLAB code
B.1 Band diagrams . . . . . . . . . . . . .
B.2 Numerical way to find Weyl points . .
B.3 State localization . . . . . . . . . . . .
B.3.1 Numerical verification of Witten
B.4 Direct space propagation . . . . . . . .
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state expression
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References
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100
Chapter 1
Theory
1.1
Topology in physics, a brief introduction
Broadly speaking, topological properties are those which are robust under smooth
deformations. Since the discovery of Quantum Hall effect in the ’80s, a new class of
materials has been under growing investigation in condensed matter and quantum
optics physics. These materials, which are especially interesting for the properties
they show at an interface with other materials or the vacuum, are said to be
topological.
In quantum mechanics, wave functions are maps defined in a parameter space
taking values in the Hilbert space. In crystals, where the wave vector k becomes a
good quantum number, the wave function maps the k-space into a manifold in the
Hilbert space. Depending on the topology of this manifold, wavefunctions will have
specific non-trivial characteristics. The trivial or non-trivial topology is usually
associated to an integer quantity (called topological invariant), with 0 representing
the trivial system. The topological invariant is a characteristic of gapped energy
states, so it can’t change as long as the gap remains open. As the invariant changes
at the boundary between a topological insulator and a trivial material, the energy
gap must vanish at the interface, i.e. we expect gapless surface states localized at
5
6
CHAPTER 1. THEORY
the interface between a topological insulator and a trivial material (or the vacuum).
As a starter, we have to introduce the key quantities that enter in the definition
of topological invariants, the Berry connection and the Berry curvature. This will
incidentally drive us to establish a close connection between topological insulators
and magnetic monopoles.
It all starts from a well known theorem of quantum mechanics
1.1.1
The adiabatic theorem
Let’s consider a generic time-dependent Schrödinger problem, with the Hamiltonian and the state depending on a vector in a parameter space R(t), in which the
t dependence is sufficiently smooth (given a path in the parameter space from R0
to R1 it takes a long time T for the parameters to evolve). The time-dependent
Schrödinger equation is
i~
d
|Ψ(t)i = Ĥ(R(t)) |Ψ(t)i
dt
(1.1)
where H has the eigensystem
Ĥ(R) |Φn (R)i = En (R) |Φn (R)i
(1.2)
for a given value of R. We furtherly assume that, chosen the initial condition to be
an instantaneous eigenstate that we label with 0, at each time the corresponding
energy eigenvalue remains non-degenerate, that is it exists a finite energy gap
between E0 (R) and the other Ek (R). Then the Adiabatic Theorem states
Theorem 1. For long enough evolution time T, the state remains proportional
to the initial eigenstate, up to a phase that consists in a trivial dynamical phase
factor and a non-trivial geometrical phase factor.
A rigorous proof of this theorem is given, for example, in the book by Messiah
[19]. In Appendix A a less rigorous but easily understandable proof is given, which
7
CHAPTER 1. THEORY
will be enough for our purpose.
In short, we rewrite the state |Ψi as
|Ψ̃(s)i =
X
iT
Cn (s)e− ~
Rs
0
ds0 En (Rs0 )
|Φn (Rs )i
(1.3)
n
where s is time normalized to 1, with the initial C being nonzero only if n=0.
Then it’s a matter of writing down the time evolution of the C coefficients, which
is found to be
Ċk (s) = iγ̇k (s)Ck (s) +
X
...
(1.4)
m6=k
where γk is a real quantity called Berry phase defined as
Z
γk (s) = i
s
ds0 Ṙs0 hΦk (Rs0 |∇R Φk (Rs0 )i
(1.5)
0
Then it can be shown that, if the evolution is indeed adiabatic and the gap between
eigenvalues is preserved, one can write the expression for the state of the timedependent problem as
i
|Ψ(t)i ≈ eiγ0 (t) e− ~
R t0
0
E0 (R(t0 ))
|Φ0 (R(t))i
(1.6)
where γ0 (t) has the form we stated before. This is what we called the geometrical
phase factor, as it is independent from the parametrization of the curve described
by R(t). Let’s define the Berry connection
A(R) = i hΦ0 (R)|∇R Φ0 (R)i
(1.7)
that depends on the phase we choose for |Φ0 (R)i in the same way the magnetic
potential depends on the gauge choice:
|Φ00 (R)i = e−iΛ(R) |Φ0 (R)i
→
A0 = A + ∇R ΛR
(1.8)
8
CHAPTER 1. THEORY
which clearly is not relevant if we move on a closed loop. Therefore, on a closed
path the Berry connection and the Berry phase are gauge-invariant.
As we said, the Berry connection behaves just like a magnetic potential; let’s
therefore introduce the analogous of the magnetic field, here called Berry curvature,
which in 3D is defined, as we would expect, as the curl of A.
In a general dimension, we must be a little more cautious: the vector field A is
said to be a 1-form, and it’s related via Stokes theorem to a so-called 2-form, that
is the Berry curvature, defined as
Fαβ (R) = −2Im h∂α Φ0 (R)|∂β Φ0 (R)i
= i h∂α Φ0 (R)|∂β Φ0 (R)i − h∂β Φ0 (R)|∂α Φ0 (R)i
(1.9)
which is manifestly antisymmetric. We can also show that this quantity is gaugeinvariant: first we note that
Im h∂α Φ0 (R)|Φ0 (R)i hΦ0 (R)|∂β Φ0 (R)i = 0
(1.10)
because hΦ0 (R)|∂β Φ0 (R)i is purely imaginary. This means we can insert the projector Q(R) = I − |Φ0 (R)i hΦ0 (R)| in the expression for Fαβ retaining its meaning:
Fαβ (R) = −2Im h∂α Φ0 (R)|Q(R)|∂β Φ0 (R)i
(1.11)
Now, if we apply a gauge transformation
0
|Φ0 (R)i = e−iΛ(R) |Φ0 (R)i
(1.12)
|∂β Φ0 (R)i = −i∂β Λe−iΛ(R) |Φ0 (Ri + e−iΛ(R) |∂β Φ0 (R)i
(1.13)
we have that
0
and so the extra term is proportional to ∂β Λ and to |Φ0 (R)i, and therefore is
9
CHAPTER 1. THEORY
erased by the projector Q. Hence the gauge-invariance of the Berry curvature.
1.1.2
An interesting parallel
As we said, the Berry connection and curvature closely resemble magnetic potential
and field. Let’s now treat a familiar Hamiltonian, that of a spin 1/2 electron in a
magnetic field, and try to exploit these similarities.
Let’s start from the Hamiltonian
(1.14)
Ĥ(R(t)) = gµB Ŝ · B(t) := R(t) · ~σ
where R absorbs the coefficients. Now we express R using spherical coordinates
and write down the eigenstates of the spin in direction R as

|Φ+1/2 (R)i = 
cos θ/2
iφ
e sin θ/2



|Φ−1/2 (R)i = 
−iφ
e
sin θ/2
− cos θ/2

.
(1.15)
These expressions are fine around the North pole, where θ = 0, but singular
at θ = π because there the exponential in φ is undetermined. It is possible to
reparametrize the states regularizing them around the south pole by multiplying
by e∓iφ , with the drawback of making them singular at the north pole. Now we
calculate the Berry connection in both cases, expressing it in spherical coordinates
as
(1.16)
A = AR R̂ + Aθ θ̂ + Aφ φ̂
From its definition we get
AN
± = 0, 0, ∓
1 − cos θ 2R sin θ
AS± = 0, 0, ±
1 + cos θ 2R sin θ
(1.17)
10
CHAPTER 1. THEORY
which have a vortex respectively at the south and north pole. As we are in 3D,
we readily calculate the Berry curvature as a curl, obtaining
F± = ∓
1 R̂
2 R2
(1.18)
which has a singularity at the origin and is regular everywhere else.
Now, in our parallel with the magnetic monopole it’s interesting to calculate the
flux of the magnetic field exiting from such an object (that, if existing, would satisfy
∇·B = 4πqM δ(R) with qM being the "magnetic charge" of the monopole) through
a spherical surface. Taking into account that a regular-everywhere parametrization
of the sphere is not possible in polar coordinates (for the same reason involving θ
and φ), we split the integral in two at the equator, and use Stokes theorem with
two different parametrizations for the two semi-shells ΣN and ΣS obtaining
Z
Z
AN · dR
B · n da =
P
C
N
Z
Z
B · n da = −
P
(1.19)
S
A · dR
C
S
with C being the equator and the minus in the second formula being due to the
direction on which the path is oriented. Now, we easily see that we can use the
expressions AN and AS we derived before as magnetic potential, provided we
substitute −1/2 → qM . Then we have
AN − AS = 2qM
1
φ̂ = ∇(2qM φ)
R sin θ
(1.20)
(gradient in spherical coordinates). Therefore the total flux is given by
Z
N
S
Z
(A − A ) · dR =
C
2qM ∇φ · dR = 4πqM
(1.21)
C
as expected. It’s important to notice, for our purposes, that the line integral
of ∇φ around the equator gives 2π times an integer called winding number. This
11
CHAPTER 1. THEORY
quantity is closely related to the Chern number, which is one very useful topological
invariant that will come across during our study [24].
1.1.3
Chern number and topological effects: the IQHE
Before proceeding, let’s study an easy example of topology at work [2]. The first
discovery of a topological quantum phenomenon dates back to 1980, when quantum Hall effect in a 2D semiconductor under high magnetic fields was discovered.
At very low temperatures, localization of electrons and Landau quantization (when
the chemical potential is located in the gap) account for a vanishing longitudinal
conductivity and a quantized transverse Hall conductivity. The topological invariant associated to this system is called TKNN invariant and is in fact the first Chern
number or winding number we introduced before. It signals the mapping between
the k-space and a non-trivial Hilbert space. The invariant is proportional to the
Berry phase of the Bloch wave function integrated around the Brillouin zone.
The system we are going to study is a 2D lattice of size L × L, on which we apply
an electric field E directed along axis ŷ. Treating the electric potential as the effect
of a perturbation V = −eEy, we could approximate the perturbed eigenstate as
|niE = |ni +
X hm| (−eEy) |ni
|mi + ...
E
n − Em
m6=n
(1.22)
and derive the expectation value for the current density along x
hjx iE = hjx i0 +
X
n
eνx |niE
L2
X hn| eνx |mi hm| (−eEy) |ni
f (En ) hn|E
1 X
f (En )
= hjx i0 + 2
L n
m6=n
+
hn| (−eEy) |mi hm| eνx |ni En − Em
En − Em
(1.23)
12
CHAPTER 1. THEORY
where νx is the velocity along x and we used the Fermi distribution function. Then,
Heisenberg equation tells us
1
1
d
y = νy = [y, H] → hm| νy |ni = (En − Em ) hm| y |ni
dt
i~
i~
(1.24)
and so the Hall conductivity is
σxy
i~e2 X
hjx iE
hn| νx |mi hm| νy |ni − hn| νy |mi hm| νx |ni
=− 2
=
f (En )
E
L n6=m
(En − Em )2
(1.25)
Now, using the Heisenberg equation for νy in k-space (if the system we consider is
in a periodic potential and therefore we can apply Bloch theorem) we can rewrite
σxy
∂
∂
∂
∂
ie2 X X
f (Enk )
hunk |
unk i −
hunk |
unk i .
=− 2
~L k n6=m
∂kx
∂ky
∂ky
∂kx
(1.26)
From the definition of Berry connection this is
σxy = ν
where
ν=
XZ
n
BZ
e2
h
X
d2 k ∂Ay ∂Ax
1 X
νn = −
γn
−
=
2π ∂kx
∂ky
2π
n
n
(1.27)
(1.28)
where γn is the Berry phase for the nth band, which we know from our analogy to
be related to the winding number as
γn = 2πm
(1.29)
with integer m.
Now, a trivial system does not show quantum hall conductance, and its TKNN
invariant ν is zero. We saw that a system for which the Berry connection is not
single-valued around the BZ (which is, in our analogy, that we are integrating over
13
CHAPTER 1. THEORY
a surface that contains a magnetic monopole) the system has non-trivial properties
that, being related to an integer quantity, we call topological.
Summing up, till now we introduced via the adiabatic theorem the concept of
geometrical phase and Berry connection, which we compared with the case of a
magnetic monopole emitting flux through a spherical surface, and showed that
systems possessing a non-zero Berry phase around the Brillouin zone exhibit nontrivial properties such as a quantized hall conductance in presence of electromagnetic fields.
In general if we write a generic 2D Hamiltonian as
H(kx , ky ) = (k) + R(k) · ~σ
(1.30)
then the ground state is a map from the BZ to a spinor space, which can be
parametrized using spherical coordinates as we saw. Then the wave function is
actually a map from a torus to a spherical shell. Now, depending on whether
the vector R describes on the sphere a surface enclosing or not the origin (the
monopole), the system will be (won’t be) topological.
It’s important to stress again the fact that the topological invariant is characteristic
for a specific insulating system, that means as long as the gap remains open it
cannot change. To make it change, the gap needs to be closed, therefore we expect
gapless states localized at the surface. This will be used in 3D to search for surface
states in our Weyl semimetal system.
1.2
Some interesting Hamiltonians
Before entering the study of the 3D Weyl semimetal, we had better sum up what
a Weyl fermion is, and sketch the broader category of which the Weyl lattice
Hamiltonian is a particular case.
14
CHAPTER 1. THEORY
1.2.1
The Weyl equation
Let’s start from the Dirac equation
(∂/ + m)Ψ = 0
(1.31)
where Ψ is a Dirac spinor living in a 4-dimensional space S which naturally splits
into two invariant subspaces
1 ± γ5
S
2
W± =
(1.32)
whose elements are called Weyl spinors. Hence we can depict the Dirac spinor in
a chiral representation:

Ψ=
φ
χ

(1.33)

γ 5 is called chirality operator, because its eigenstates are precisely the two Weyl
spinors and its eigenvalues are ±1, the associated chiralities. This is easily seen
from the form of the operator

γ5 = 
I
0
0 −I


(1.34)
where I is the 2x2 identity.
In the chiral representation the Dirac equation is expressed as a system of two
coupled equations for Weyl spinors:
i(∂0 − σ · ∂)φ = mχ
i(∂0 + σ · ∂)χ = mφ
(1.35)
15
CHAPTER 1. THEORY
In the zero mass limit the two equations decouple, and we obtain a pair of independent Weyl equations:
i(∂0 − σ · ∂)φ = 0
i(∂0 + σ · ∂)χ = 0
(1.36)
Looking at those, we recognize that they are exactly of the form we discussed for
the spin in a magnetic field. Therefore, an interesting connection between Dirac
theory and condensed matter emerges, in the fact that if we can create a system
in which the Hamiltonian is, at least locally, the Weyl Hamiltonian, topological
effect will be likely to come out.
1.2.2
Harper-Hofstadter Hamiltonian
One of the most beautiful results in condensed matter physics is the fractal band
diagram of the Harper-Hofstadter Hamiltonian. As our study will be focused on a
lattice which is a 3D extension of a subcase of this Hamiltonian, it’s worth doing
a brief excursus on its origin and definition.
We start from a two-dimensional square lattice of spacing a, on which acts a
magnetic field perpendicular to the plane. The Hamiltonian of a charged particle
moving in a magnetic field s the usual
2
e
1
p− A .
H=
2m
c
(1.37)
which, as we know, gives origin to Landau levels. We fix the Landau gauge for the
vector potential (A = B(0, x, 0)) [?].
If we are in a periodic potential for a 2d square lattice, we can put the Hamiltonian
for one single Bloch band in the following form (in the Wannier basis)
H=−
X
mn
tx a†m+1,n am,n + ty a†m,n+1 am,n + h.c.
(1.38)
16
CHAPTER 1. THEORY
where the creation operator is labeled with the indices of the site, and we have
separated the hopping in the two directions.
Now we want to include a magnetic field in the picture, with an associated vector
potential A = (0, x, 0). The usual technique is called Peierls substitution, and a
sketch proof is given in the Appendix A. In short, it amounts to substitute
m1n1
a†m1 ,n1 am2 ,n2 → eiAm2n2 a†m1 ,n1 am2 ,n2
(1.39)
where the coefficient of the exponential is the integral of the vector potential along
a path connecting the starting and ending sites:
1 n1
Am
m2 n2
m1 n1
Z
A · dr.
=
(1.40)
m2 n2
(incidentally, we are using a "discretized" version of our magnetic potential, in
which the continuous variable x is replaced by the index m). Along x direction
then periodicity changes but still persists, as the new coefficient is a phase, hence
defined modulo 2π.
The Hamiltonian is therefore changed as follows:
H=−
X
tx a†m+1,n am,n + ty ei2πφ m a†m,n+1 am,n + h.c.
(1.41)
mn
where φ is rational, and gives the flux of the magnetic field in each plaquette
of the lattice. Now we define the usual Fourier transform for the creation and
annihilation operators to pass in momentum space: from
a(†)
m,n
1
=
(2π)2
Z
π
Z
π
dkx
π
−π
dky e±i(mkx +nky ) akx ,ky
(1.42)
17
CHAPTER 1. THEORY
we get
1
H=
(2π)2
Z
π
π
Z
dky − 2tx cos(kx )a†kx ,ky akx ,ky
.π
−π
iky †
−iky †
− ty (e
akx +2πφ,ky akx ,ky + e akx −2πφ,ky akx ,ky )
dkx
(1.43)
which is diagonal in ky but mixes kx with kx ± 2πφ.
Let’s continue by remembering we assumed rationality for φ = p/q and define
kx = k̄x + 2πφn, rewriting therefore the Hamiltonian as
1
H=
(2π)2
Z
π/q
Z
π
dk̄x
−π/q
dky Hk̄x ,ky
(1.44)
−π
where
Hk̄x ,ky =
q−1 X
− 2tx cos(k̄x + 2πφn)a†k̄x+2πφn ,ky ak̄x+2πφn ,ky
n=0
−ty (e−iky a†k̄x +2πφ(n+1),ky ak̄x +2πφn,ky
iky †
+ e ak̄x+2πφ(n−1),ky ak̄x +2πφn,ky ) .
(1.45)
This Hamiltonian is diagonal in both k̄x and ky , then we can use it for a specific
couple of these momenta.
An eigenstate of such a reduced Hamiltonian satistifes
Hk̄x ,ky |ψi = Ek̄x ,ky |ψi
(1.46)
in the new magnetic Brillouin zone k̄x ∈ [−π/q, π/q], ky ∈ [−π, π]. Then, expanding it in the basis
|ψi =
q−1
X
m=0
cm a†k̄x +2πφm,ky |0i
(1.47)
18
CHAPTER 1. THEORY
we obtain for each m the following eigenvalue equation, which is the celebrated
Harper equation
−2tx cos(k̄x + 2πφm)cm − ty (e−iky cm−1 + eiky cm+1 ) = Ek̄x ,ky cm
(1.48)
The fractal diagram known as Hofstadter butterfly appears plotting the energy
bands versus the flux, the latter being continuously varying. For our purposes,
however, is is sufficient to consider the case φ = 1/2; the Hamiltonian for the 2D
optical lattice system we will introduce in the second chapter as a starting point
to obtain a Weyl semimetal will be trivially obtained by (??) in real space, and
by (??) in momentum space.
1.3
1.3.1
Weyl semimetals
Introduction to Weyl nodes
First of all, we make clear the concept of semimetals. While it is well-known that
metals are materials that have partially-filled bands and insulators are materials in which completely filled (valence) bands and empty (conduction) bands are
separated by an energy gap, the case in which the energy gap vanishes only in a
finite number of k-points gives rise to these peculiar materials. We focus on Weyl
semimetals, that are those which, around the points where the gap vanishes (called
Weyl nodes or Weyl points) are described by a Weyl-like Hamiltonian, of the type
H=
X
~vij ki σj
(1.49)
i,j=[x,y,z]
(the chirality being defined as det[vij ]). The question now is: given the discussion
we did in the previous section about the topological invariants being associated
with the presence of a gapped system, how can we expect to find topological effects
in a material whose gap by requirement closes at specific points?
19
CHAPTER 1. THEORY
First of all, let’s state the requirement that the bands touching at Weyl nodes
are non-degenerate. If they were, terms hybridizing states within a degenerate
subspace could in general gap out the spectrum. Then, the at least one among
time-reversal and inversion symmetries must be broken, because the combined
presence of those symmetries would make the bands doubly degenerate. Let’s
consider, as an example, an Hamiltonian of the form
H=−
X
[2tx (cos kx −cos k0 )+m(2−cos kx −cos ky )]σx +2ty sin ky σy +2tz sin kz σz .
k
(1.50)
For pseudospin 1/2 system, the three σi behave under TR in two possible ways:
either they are all odd, or two or them are even and one odd ([29]). As in H two
terms with sin(ki ) appear, it’s easy to see this Hamiltonian is (at least) not TRS.
It’s also easy to see it possesses nodes at k = ±k0 x̂. Interestingly, these nodes will
be proven indestructible unless they are annihilated by making them coincide.
If the bands are filled with fermions right up to the nodes, then this is a semimetal
with vanishing density of states at the Fermi energy. It can be easily seen that the
Hamiltonian, expanded around the nodes where we define p± = (±kx ∓ k0 , ky , kz ),
takes the form
±
±
H ± = vx p±
x σx + vy py σy + vz pz σz
(1.51)
which is similar to the particle physics Weyl equation
H ± = ±cp · σ
(1.52)
Now let’s take a step back to Dirac theory: we remember the time-dependent
Schrödinger equation
∂
ψ = Hψ
∂t
(1.53)
H = αi ∂i + mβ
(1.54)
i
with a Dirac-type Hamiltonian
20
CHAPTER 1. THEORY
where αi and β are suitable matrices. Differentiating again with respect to time,
if we choose as state a plane wave we get the condition
(p0 )2 = −
1 i j
α , α pi pj + i αi , β pi + m2 β 2
2
(1.55)
then, from the condition p20 = pi pi + m2 we would have to search for a β matrix
that is a root of the identity and anticommutes with all the α, that in our case
are the 2 × 2 Pauli matrices. But such a matrix does not exist, therefore it’s
not possible to add a mass term to the Weyl equation and gap out the spectrum.
The Hamiltonian around the node is then that of a massless fermion with a single
chirality, fixed by the ± coefficient at the RHS of the expression for H ± , depending
on which the velocity is along or opposite to the spin. We will later see that, due to
Nielsen-Ninomiya theorem, Weyl points always come in pairs of opposite chirality.
1.3.2
Stability of Weyl nodes
Let’s consider only a pair of energy levels that touch at a Weyl node. We express
the Hamiltonian in the proximity of the nodes in the following way

H=
δE
ψ1 + iψ2
ψ1 − iψ2
−δE

.
(1.56)
p
Then the energy splitting is in general ∆E = ± δE 2 + ψ12 + ψ22 , and to have it
disappear we have three real numbers that have to be 0 separately, which means
we have three equations in three variables (and three tunable parameters). Then,
if there is a solution, a perturbation slightly changing the Hamiltonian will only
shift the position of the Weyl points in the BZ but cannot destroy the node. This
means that the node is robust with respect to continuous deformations, i.e. it’s
topologically protected. While this equation does not tell us anything about the
energy of the node, it is convenient to stick to the previous situation E = 0, which
21
CHAPTER 1. THEORY
is also the one we will be considering in our case study.
1.3.3
Parallel with monopoles
Let’s try now to answer the question we asked before: how can a system with
vanishing bulk gap present topological properties?
First of all, we remark that Weyl point position is very important, as moving them
may lead to annihilation of pairs. So, they must have well-defined coordinate in
k-space, which implies the presence of translation symmetry.
As we saw, Weyl Hamiltonian closely resembles the "spin in a magnetic field" case,
which possesses a non-zero Berry connection and curvature, once we express the
wave function of filled bands in the usual way

ψ(θ, φ) = 

sin(θ/2)
− cos(θ/2)e
iφ
.
(1.57)
It’s easy to make a parallel with section 1.1.2, associating the chirality of the node
to the "magnetic monopole charge". The Berry flux through a sphere surrounding
the Weyl point is therefore 2πκ, κ = ±1 being the chirality which enters the
Weyl Hamiltonian. Then, by Gauss law, the flux is the same through any surface
containing the node. It is the non-zero Berry flux through this surface to guarantee
the presence of the node and topologically protect it. Incidentally, this offers an
intuitive proof (rigorous one will come later) for Nielsen-Ninomiya theorem (called
in general Fermion doubling theorem): as the overall Weyl charge of the BZ has
to be 0, Weyl points come in pairs.
1.3.4
Surface states
As we saw topology is involved in this system. Then we expect surface states at its
interface with a trivial material (or the vacuum). First of all, given the existence
of such states, what keeps them from hybridizing with bulk states? If there is
22
CHAPTER 1. THEORY
translational symmetry and the surface momentum is conserved, surface states
can be stable at any momenta where there are no bulk states at the same energy.
So, for example, if in a fermionic system Fermi energy is fixed to be at the Weyl
point energy, only at the projections of these points onto the surface there will be
bulk states; everywhere else surface states will be well defined and separated with
a gap from bulk states.
The surface states of a 3D Weyl semimetal at 0 energy have a peculiar form:
they are Fermi arcs connecting the projections of Weyl points on the surface.
The reason for this behaviour can be explained in many ways: let’s start with a
somehow intuitive view.
Dimension reduction argument for Fermi Arcs
To clarify the topological nature of Weyl points, we make a correspondence between
Weyl semimetals (3D gapless systems) and 2D topological insulators (gapped), as
it can be seen the formers inherit the topology of the latters. Let’s consider a curve
in the surface Brillouin zone encircling the projection of the Weyl point, which we
parametrize with λ ∈ [0, 2π] running counterclockwise; then the momentum states
on the curve are kλ = [kx (λ), ky (λ)].
Now let’s rewrite the Hamiltonian as
H(kλ , kz ) = H(λ, kz )
(1.58)
which we interpret as the hamiltonian of a 2D gapped system (our surface never
intercepts the node) the Brillouin zone of which is the usual torus (both λ and
kz have 2π periodicity). Now, this system has non-zero Chern number, because
it encloses a Weyl point with Berry flux 2πχ; it is therefore a 2D quantum Hall
insulator.
So, given a border (for example at z = 0), we expect a chiral gapless surface
state for this system, that will cross 0 energy at some value λ0 , that is at certain
23
CHAPTER 1. THEORY
(a)
(b)
(c)
Figure 1.1: (a)Depiction of the states as a function of (kx , ky ): the bulk states are inside
the cone; the red cylinder has a 1D circular BZ as a base.(b) The "unrolled"
cylinder gives the spectrum of the 2D system H(λ, kz ) with a boundary
in the z direction. (c) Chern number of the 2D "slices" of material cut
perpendicularly to the finite surface. Picture taken by [27]
[kx (λ0 ), ky (λ0 )]. This reasoning can be done for any surface, as long as the total
chirality of the enclosed monopoles is 6= 0. All the surface states at 0 energy
then "sum up" in an arc connecting the projections of two Weyl nodes of opposite
chirality on the cut. If the projections coincide, no arc will (in general) be shown,
so the Fermi arcs are a combined effect of both the Weyl points and the direction
of the finite surface.
Next, we show an argument which uses a stack of lattice layers, which is how the
system we will study in the next chapter is constructed.
SSH model argument for Fermi Arcs
Let’s follow [12]. This picture is interesting because it allows us to consider cases
in which, given a slab of material, the surfaces in the finite directions are composed
by elements of one or the other sublattice (remember, the hamiltonian lives in a
2-dimensional space).
Let’s start with a Weyl semimetal consisting in a stacking of L planes with two
Weyl nodes at (kx , ky , kz ) := (K1,2 , 0) and Fermi arcs connecting the projections
24
CHAPTER 1. THEORY
Figure 1.2: Layer stacking for the SSH model argument for Fermi Arcs. On the left
even-planes case, on the right odd-planes case. Picture taken from [12]
K1 and K2 along a segment S or S 0 on the z = 1 and z = L surface Brillouin
Zones, as in figure 1.2. It can be proven that the two arcs coincide if the number
of stacked layers is even, while they do not if the number is odd.
We assume for the system the Hamiltonian
Hk =
L
X
a†z,k (−1)z Ek az,k
+
L−1
X
a†z,k hz,k az+1,k + h.c.
(1.59)
z=1
z=1
Ek is a phenomenological function that vanishes along a contour C that comprises
S in the first case and is made up from the union of S and S 0 in the second (this
guarantees Fermi arcs are at 0 energy). The interplanar hopping hz,k is defined as
−tk (∆k ) for z even(odd). The relation between these two quantities is
tk =


> ∆k
k∈S

< ∆k
k ∈ C/S
(1.60)
that means tk = ∆k at the Weyl point projections.
First we consider the bulk hamiltonian: if we have an infinite system in the three
directions (or a system with PBC along z, which is a safe assumption if the size of
CHAPTER 1. THEORY
25
the chunk is big enough that translational invariance can be assumed in the bulk),
we can rearrange (1.59) considering a 2-component "spinor", that consists of sites
belonging to planes among which the hopping parameter is ∆k . Apart from the
first term that trivially becomes Eσz , there will therefore be an in-spinor hopping
term
∆k a†k,A ak,B + ∆k a†k,B ak,A = ∆k σx
(1.61)
(where A, B indicate the two planes in the "spinor" and σx is the first Pauli matrix)
and an inter-spinor hopping term (assume the inter-spinorial distance =1)
− teikz a†A aB − te−ikz a†B aA
= − t(cos(kz ) + i sin(kz ))a†A aB − t(cos(kz ) − i sin(kz ))a†B aA
(1.62)
= − t cos(kz )σx + t sin(kz )σy
where we have introduced the other two Pauli matrices.
The bulk hamiltonian will therefore be
bulk
= Ek σz + ∆k − tk cos(kz ) σx + tk sin(kz )σy .
HK,k
z
(1.63)
It’s easy to see this is gapped everywhere except at the Weyl points. Moreover, it
can be approximated near these points as
bulk
HK
≈
p
·
∇
E
σ
+
p
·
∇
∆
−
t
σ
+
t
p
σy
k
K
z
k
K
K
x
K
z
i
i
i
i
i +p,0+pz
(1.64)
= p⊥ v1 (Ki )σz + pk v2 σx + ∆0 pz σy
where the momentum components are that along ẑ and those parallel and perpendicular to C and v1 and v2 are appropriate coefficients. This Hamiltonian is
Weyl-type just like (1.49), and chirality is defined accordingly.
Now let’s consider a finite number of layers and let’s focus on a fixed in-plane
k: using the z coordinate as an index, our system can be thought of as a simple
SSH model, that is essentially a dimer chain with different intra-cell and inter-cell
26
CHAPTER 1. THEORY
TRS
k2
IS
k
k2
k
k1
k1
-k
-k
Figure 1.3: An illustrative justification of the statement on the number of couples of
Weyl points: schematic 2D Brillouin zones have been divided in "slices"
according to the sign of tk − ∆k . It’s easy to understand from the text
that in the TRS case opposite quadrants have equal colorization, while in
the IS case colorizations are inverse. The number of "slices" (and also of
slice borders and so of Weyl points) is ≡ 0 mod 4 in the TRS case and ≡ 2
mod 4 in the IS case.
hoppings. It is known that, in the complete dimerization limit with no intra-cell
hopping (and by adiabaticity even if intra-cell hopping is nonzero but smaller than
inter-cell hopping) the system presents zero energy surface states1 .
So, let’s consider a surface of our system, for example z = L in figure 1.2. For
∆k < tk , that is if k ∈ S, our SSH model is topologically non-trivial; then, surface
states at 0 energy will be present, and a Fermi arc will appear. So, if the number
of layers is even, the reasoning exactly applies for the other surface of the chunk,
and Fermi arcs overlap.
If the number of surfaces is odd the situation is a bit trickier: we have to redefine
the cell so that intra-cell hopping is tk , because our model is a chain consisting
of complete dimers, even at its end. We must also be in the topological phase for
the SSH model, that is intra-cell hopping < inter-cell hopping. This happens if
0
tk < ∆k that is if k ∈ S . Thus, finally, Fermi arcs will run on a close contour in
the 2D BZ of the slab.
Using this picture, we can also make a statement on the number of Weyl point
1
A complete study of the SSH model can be found in [3]; a short resume of the features we
are interested in is given in Appendix A.
27
CHAPTER 1. THEORY
pairs the material presents: as we said, it is essential that at least one among TRS
and IS be broken. Now, let’s assume Ek = E−k (otherwise both symmetries would
be broken); if TRS is preserved, then it must hold tk = t−k and ∆k = ∆−k , that
is tk − ∆k = t−k − ∆−k and this means the number of points at which tk = ∆k
is a multiple of 4; conversely, inversion around a layer interchanges t and ∆, so
tk = ∆−k preserves IS. But this implies tk − ∆k = −(t−k − ∆−k ) so an odd number
Weyl point couples is expected. This feature is best understandable via a graphic
example, such as the one give in figure 1.3.
1.3.5
The Nielsen-Ninomiya theorem
Earlier on we briefly mentioned the fact that Weyl points always come in pairs
of opposite chirality (and are connected by Fermi arcs). Now we sketch a more
accurate proof of this statement.
We start from a generic hamiltonian H(k) describing a Weyl semimetal, therefore
gapped everywhere in the BZ except in a finite number of isolated points. If we
attach an integer "label" to these points, we can demonstrate that these integers
have to add up to 0.
As it is widely done in condensed matter physics, we only focus on the two touching
bands, and define the usual "spin in a field" Hamiltonian
H(k) = B(k) · σ.
(1.65)
Now, away from the nodes B is not null, so we define the versor
n(k) =
B
B
(1.66)
and consider the map k → n(k), which maps a sphere parametrized by k to
another, parametrized by the versor n. As known from the theory of homotopy
28
CHAPTER 1. THEORY
groups
(1.67)
πn (S n ) = Z
this means that a continuous mapping between two spheres of the same dimension
has an integer winding number, that is how many times the first wraps around the
second. In our case, the Hamiltonian is
(1.68)
H = ±σ · k
and then B = ±k. We consider the node at k = 0 and the initial sphere to be
the unit sphere k = 1; then the map is n = ±k, which is an identity with winding
number given by the chirality.
Let’s now state the argument for the sum of all the winding numbers (i.e. all chiralities) being 0. Now, from the definition of the winding number w and remembering
the map k → n we can write
1
w=
4π
Z
1
d k n · (∂µ n × ∂ν n) =
4π
S
2
µν
Z
S
d2 k µν abc na
∂nb ∂nc
∂k µ ∂k ν
(1.69)
where and S is a sphere around a "bad" point where B(k) = 0. On the other
hand we note that
0 = ∂λ (λµν n · ∂µ n × ∂ν n) = λµν ∂λ n · ∂µ n × ∂ν n
(1.70)
because all the vectors at RHS are normal to n and all the other terms vanish
because of the antisymmetry given by .
Now, as depicted in figure ??, for each node kα in the BZ, we call Uα a small
ball around it, whose boundary is a sphere Sα . Let B 0 be the BZ minus the balls.
29
CHAPTER 1. THEORY
U1
U2
U4
B
0
U3
Figure 1.4: BZ depiction for the Nielsen-Ninomiya theorem demonstration. The dots
are Weyl points, with color matching their chirality.
Its boundary is ∂B 0 = ∪α Uα . From Stokes’ theorem
2
we get the thesis:
Z
1
3
λµν
0=
d k∂λ n · ∂µ n × ∂ν n
4π B 0
X 1 Z
d2 k µν n · (∂µ n × ∂ν n)
=
4π Sα
α
X
=
wα
(1.71)
α
1.3.6
More intuitive arguments for Nielsen-Ninomiya theorem
The following arguments are not proofs: they are simple results in favour of
Nielsen-Ninomiya theorem, and for simplicity we will consider only Weyl nodes
with unit chirality. The first argument is similar to the one we used to account
for Fermi arcs: if we consider a BZ containing Weyl nodes and start slicing it in a
direction parallel to the axis of the Weyl cones, we will get a set of 2D slices that,
if they do not contain the node, are in effectively 2D insulators. Now, if two slices
2
see for example [26], pag. 84
30
CHAPTER 1. THEORY
Figure 1.5: BZ slices according to their chirality. Because of periodicity, the total number of Weyl points is even, and the number of points of each chirality is the
same. Image taken from [13].
contain a monopole between them there is a net flux through them, and then the
Chern numbers of the two differ by one. This, as we saw, accounts for Fermi arcs
in presence of a surface. But then, as the Chern number of slices is periodic across
the Brillouin zone, there are the same number of Weyl points for each chirality.
Another argument stems from the so-called chiral anomaly, which is to say current for each chirality is not conserved under applied electric and magnetic fields
according to
∂µ jχµ
e3
= −χ 2 2 E · B
4π ~
(1.72)
This is actually a very interesting feature, that calls for many experimentallydetectable effects, such as negative magnetoresistance, chiral magnetic effects and
nonlocal transport, that are however outside the goal of our tractation. A summary
of the most important properties related to the chiral anomaly can be found in
[13]. As for the argument on Nielsen-Ninomiya theorem: suppose the system only
contains Weyl electrons of a certain chirality; then in presence of electric and
magnetic fields the electromagnetic current of these electrons would satisfy (1.72),
so charge would not be conserved. In fact, as Weyl nodes come always in pairs
and (1.72) depends on χ, the total current j+µ + j−µ is conserved.
Chapter 2
3D lattice Weyl Hamiltonian
2.1
Introduction
Our study of 3d topological effects in Weyl semimetals will be focused on a specific Hamiltonian, which is built starting from a subcase of the Harper one we
introduced earlier. A proposal for an experimental realization is found in a recent
paper by Ketterle et al. [7]; the experimental setup relays on laser-assisted tunneling [21] to engineer hoppings in a cubic optical lattice in a site-dependent fashion,
thereby breaking time or space inversion symmetry, which is the starting point to
get non-trivial Weyl behaviour.
Let’s start with a 2d square lattice described by the usual Hamiltonian in
tight-binding picture
H=−
X
Jx a†m+1,n am,n + Jy a†m,n+1 am,n + h.c.
(2.1)
hm,ni
Now, tunneling along x̂ is suppressed by adding an energy tilt ∆ proportional to
the Bloch oscillation (for example by means of a magnetic field). Then, two lasers
are used to produce 2-photon Raman excitation, with 2-photon Rabi frequency Ω,
31
CHAPTER 2. 3D LATTICE WEYL HAMILTONIAN
32
Figure 2.1: The experimental setup of the 2D optical lattice at hand. An energy tilt
∆ is added in the x̂ direction, and hopping is restored by 2-photon Raman
scattering process. Ω is the 2-photons Rabi frequency (which is far from
resonance) ki , ωi are characteristics of the lasers. Egap is the energy gap
between atomic states; the lattice is engineered so that δω ≈ ∆ Egap .
Image taken from [21].
detuning δω = ω1 − ω2 and momentum transfer δk = k1 − k2 . The two Raman
beams couple different sites, but do not change the internal state of the atoms.
If we assume ∆ larger than J and much smaller than the atom energy level gap
E, for resonant tunneling δω = ∆/h the energy tilt effectively vanishes, as in the
dressed atom picture the state at site (m, n) with j, k photons is degenerate with
the state at m + 1, n with j + 1, k − 1 photons. In this situation, it can be proven
([21]) that the resulting effective Hamiltonian is Harper-like:
H=−
X
Ke−iΦm,n a†m+1,n am,n + Jy a†m,n+1 am,n + h.c.
(2.2)
hm,ni
where the phase is spatially-varying and defined as Φm,n = δk · Rm,n = mΦx + nΦy .
If we choose [Φx , Φy ] = [π, π], then for each 4-site plaquette the accumulated phase
is π, and this realizes the Harper hamiltonian for φ = 1/2 (in Chapter 1 notation).
This setup retains both inversion 1 and time-reversal 2 symmetries and the Hamil1
as can be seen from figure 2.2, where the centres of inversion in the lattice are highlighted
as orange crosses, or in k-space, where I operator for this system is simply σx .
2
the phase π in each plaquette is equal to the phase −π, T for spinless particles is simply
complex conjugation, T iT −1 = −i. With these ingredients, checking the invariance of 2.2 is easy
(in direct space). In k-space, TRS for this system operates like (σx , σy , σz ) → (σx , −σy , σz ) and
CHAPTER 2. 3D LATTICE WEYL HAMILTONIAN
33
tonian can be written in quasimomentum representation in a "spinorial" fashion,
as the lattice can be divided into two sublattices (say A and B) according to
the position of the hopping with phase π. The hamiltonian in quasimomentum
representation will therefore be a simplified version of (??):
H2d = −2 · cos(ky a)σx + sin(kx a)σy
(2.3)
which has 2 energy bands that touch at two Dirac points at (0, ±π/2a).
2.2
Extension to the 3D case
The key idea now is to extend this lattice in 3D by stacking such layers one atop the
other, and studying the resulting system. The Hamiltonian would be something
of this sort:
HStack = −2 · cos(ky a)σx + sin(kx a)σy + cos(kz a)1
(2.4)
where 1 is the 2x2 identity matrix.
However, it is not sufficient to make a pile of 2D lattices: in fact, along ẑ both
TRS and IS are still preserved in this setup, and we need one of those to be broken
for Weyl point to emerge.
The above Hamiltonian has in fact the following eigenvalues
q
E± = −2 cos(kz a) ± 2 sin2 (kx a) + cos2 (ky a)
(2.5)
and it’s easy to see that 2D Dirac points become line nodes (with generally nonzero
energy) in the 3D BZ. In order for the Weyl hamiltonian to appear we should add
a component proportional to σz ; this is easily done imposing that, depending on
to which sublattice each point belongs (that is, whether the sum of the x and y
checking T H(k)T −1 = H(−k) is again easy.
CHAPTER 2. 3D LATTICE WEYL HAMILTONIAN
34
Figure 2.2: 3D extension of the 2D tilted optical lattice. (a) the experimental realization
with the two Raman lasers directer along x̂ + ŷ. (b) three planes of the 3D
lattice described by the effective Hamiltonian (2.6). Image taken from [7].
coordinates is even or odd), ẑ hopping acquires a phase respectively of 0 and π.
The Raman laser technique can be used again, by applying a tilt in the direction
x̂ + ẑ and two lasers accordingly orientated, with parameters tuned to match the
effects of the tilt. In 3D, the generalization of (2.2) is
H=−
X
Kx e−iΦm,n,l a†m+1,n,l am,n,l + Jy a†m,n+1,l am,n,l
(2.6)
hm,n,li
+
Kz e−iΦm,n,l a†m,n,l+1 am,n,l
+ h.c. .
We choose as a phase [Φx , Φy , Φz ] = [π, π, 2π] (there has to be momentum transfer
in the tilt direction, hence Φz 6= 0), so that Φm,n,l = (m + n)π.3
3
an explanation for this statement can be found in [20] or in [21]: in short (and in 2D), given
Raman lasers with wavevectors k1 and k2 , they create a time-dependent potential
V (r, t) = Ω cos(δk · r − ωt)
where Ω is the 2-photon Rabi frequency. Now, in rotating wave approximation we can express
the laser-assisted tunneling term as
R
K = Ω2 d2 rw∗ (r − R)w(r − R − a)e−iδk·r =
R
R
e−iδk·R Ω2 dxw∗ (x)w(x − a)e−iδkx x dyw∗ (y)w(y)e−iδky y
where we have extracted the phase we refer to in the text, and the functions in the overlap
integrals are Wannier functions in the ŷ direction and Wannier-Stark functions in the x̂ direction.
It can be seen plotting the absolute value of the integrals that the first one has oscillatory
behaviour, and is 0 for kx = 0, so there must be momentum transfer in the tilt direction for
the Raman coupling to be strong enough to produce tunneling. Luckily, as we are considering a
phase, 2π will do.
35
CHAPTER 2. 3D LATTICE WEYL HAMILTONIAN
Now, in quasimomentum representation this amounts to adding to the 2D Harper
Hamiltonian a third term: in crystal momentum space we have kz ∈ [−π/a, π/a],
and we can readily see that, separating the Hamiltonian for the two sublattices as
H = HA + HB,
Hz =
B
X
e−iΦm,n,l a†m,n,l+1 am,n,l + h.c. −
A
X
hm,n,li
hm,n,li
=
A
X
a†m,n,l+1 am,n,l + h.c. −
=
B
X
a†m,n,l+1 am,n,l + h.c.
hm,n,li
hm,n,li
A
X
A
X
eiRm,n,l (k−k’) eik·az a†k ak’
B
B
X
X
+ h.c. −
[...]
k,k’ hm,n,li
k,k’ hm,n,li
=
=
A
X
k
A
X
e
e−iΦm,n,l a†m,n,l+1 am,n,l + h.c.
ikz a
−ikz a
+e
a†k ak
−
B
X
eikz a + e−ikz a a†k ak
k
2 cos(kx a)a†k ak −
k
B
X
2 cos(kx a)a†k ak
k
(2.7)
where we have used several known formulas as trivial trigonometric identities and
P iR·K
= δ(K). Rewriting Hz in a spinorial fashion we get
Re

Hz = a†A a†B 
2 cos(kx a)
0


a
  A
−2 cos(kx a)
aB .
0
(2.8)
= 2 cos(kz a)σz
Adding this term to H in quasimomentum representation we finally obtain
HWeyl = −2 · Jy cos(ky a)σx + Kx sin(kx a)σy − Kz cos(kz a)σz .
(2.9)
This hamiltonian breaks inversion symmetry (it be can trivially understood from
the fact that, in the lattice if figure 2.2, inversion around the orange crosses sends
36
CHAPTER 2. 3D LATTICE WEYL HAMILTONIAN
kz
π
a
E
ky
π
a
π
a
kz
ky
kx
Figure 2.3: On the right: Brillouin zone of the 3D Hamiltonian: the reciprocal lattice
vectors are directed along the directions x̂ + ŷ, x̂ − ŷ and ẑ; in this way,
the system is
√a Bravais lattice of 2-component spinors and lattice spacing a
along ẑ and 2a in the other directions. The black axes represent the rotated
coordinate system kx , ky , kz , since H is defined with respect to those. On
the left, bulk bands of the kx = 0 plane. The four Weyl points projections
can be clearly seen.
sites with 0-phase hopping along ẑ to sites with π-phase hoppings along ẑ 4 ), and
is expected to present Weyl nodes. Indeed the two energy bands
q
E1,2 (k) = ±2 Kx2 sin2 (kx a) + Jy2 cos2 (ky a) + Kz2 cos2 (kz a)
(2.10)
touch at (kx , ky , kz ) = (0, ±π/2a, ±π/2a).
Around these points the linearized H is
H=
X
(2.11)
vij qi σj .
i,j=[x,y,z]
where q is the momentum with respect to the momentum of the node. The v term
is a 3x3 matrix structured in this way:

0

±2Jy a

0
4
−2Kx a
0

0
0



0
±2Kz a
or, in k-space, checking that IH(k)I −1 = H(−k) is not valid anymore
(2.12)
CHAPTER 2. 3D LATTICE WEYL HAMILTONIAN
37
with signs depending on the Weyl point. As we said in the theoretical overview
the Weyl points can be classified according to their chirality, defined here as
χ=sign(det[vij ]).
They are also robust, this meaning that the only way they can be destroyed is
by making points with opposite chirality coincide and annihilate. This can be
achieved by making them move in the BZ, for example by adding a tunable A-B
sublattice offset in the form of an on-site energy ±λ, with sign depending on the
sublattice. This adds a term λσz to the hamiltonian, and makes the Weyl points
shift along the ẑ direction until they annihilate (at kz = 0 or at the edge of the
BZ, respectively for λ = ∓2). It is also possible to make them shift along the ŷ
direction (but not along the x̂, as they all sit on the x=0 plane) by adding a term
σx , which amounts to tuning the hopping amplitude along the x̂ direction.
2.3
Cutting the lattice
Now we come to the main subject of this tractation, that is the study of the
presence and characteristics of the peculiar surface states known as Fermi arcs.
As we stated before, their appearance is expected whenever the projections of the
Weyl points on the surface of the cut do not coincide; then we will see a line
of k-points for which the band gap vanishes; furthermore, the states are highly
localized on the surface and the group velocity will have a specific direction for
each surface.
In each of the following cases, the system is a 3D lattice which is cut along a certain
direction to obtain a slab made of Np lattice planes. Now, one may consider the
lattice in the usual way as Bravais lattice plus base function, the latter consisting
of the two sites in the same "spinor". However, especially for cuts in nontrivial
directions, it may happen that the Bravais lattice unit vectors are not orthogonal,
forcing us to obtain the reciprocal vectors as
bi = 2π
a j × ak
ai · (aj × ak )
CHAPTER 2. 3D LATTICE WEYL HAMILTONIAN
38
that may not be the most convenient choice. Indeed, our study always starts from
finite samples as those of Chapter 4: given a chunk of material, we pick a face to
study and imagine the sample infinitely extended along that plane. As long as the
face is rectangular, its sides serve well as a 2D orthogonal frame, and passing to
reciprocal space is trivial. So, we will always consider the system as a 2D Bravais
lattice extending in the infinite plane to which the face belongs. The basis function
for each point of the lattice will be composed of as many 2-sites "spinors" as the
number of stacked planes in the slab. This approach is very useful especially for
cuts along planes of the form x − y + az = 0, as we will consider planes stacked
along the direction ẑ, having therefore in principle a non-orthogonal 3D reference
frame (see figure 2.10).
2.3.1
Cut along the x − y = 0 plane
This is the original cut presented in the article [7]. It’s very convenient for many
reasons, like orthogonality. However, the main feature of this cut is that, because
of the alternance of phase-π and phase-0 hoppings in the xy plane, all the sites
along the cut belong to the same sublattice; then, the spinor is easily defined with
2 neighbouring sites alonx x̂. The slab will be considered as a 2D Bravais lattice
plus a basis; the unit vectors of the Bravais lattice of the slab are a1 = a(x̂ + ŷ)
and a2 = aẑ; so the BZ will also be 2D, with reciprocal lattice vectors b1 =
π/a(x̂ + ŷ) and b2 = 2π/aẑ. A generic k-point in the BZ will be addressed as
√
k = kk (x̂ + ŷ)/ 2 + kz ẑ. It’s also simple to see that the Weyl points projections
√
on the slab surface are at (kk , kz ) = (±π/2 2a, ±π/2a). Regarding the basis
function, to each point in the BZ there will be 2Np associated sites, that is the
number of planes times 2 (the components of the spinor). So, if we plot E(kk , kz ),
we will get 2Np eigenstates.The general procedure to obtain the band diagram is
presented in Appendix B.
From now on, we will assume lattice spacing 1 for simplicity. The presence
of Fermi arcs can be clearly seen from figure 2.4b. As predicted, they connect
CHAPTER 2. 3D LATTICE WEYL HAMILTONIAN
39
x̂ + ŷ
Finite
ẑ
(a) Lattice depiction for the original cut; the green rectangles highlight the "spinors".
(
)
(b) Band diagram for alternated slices of 50 A-sublattice and 50 B-sublattice sites in the finite
direction
(
)
(c) Band diagram for alternated slices of 50 A-sublattice sites and 49 B-sublattice sites in the
finite direction
Figure 2.4: The lattice with the cut proposed by [7]. The k values are in units of π.
CHAPTER 2. 3D LATTICE WEYL HAMILTONIAN
40
the projections of Weyl points of opposite chirality (why the couples are formed
in this way is still not clear, see [12]). As we see, the Fermi arcs are well defined
√
√
for kk ∈ (−π/2 2; π/2 2), while outside this region the bands join with the bulk
ones and there is no surface state. Inside this region, there are two surface bands
which, close enough to the Fermi arc, present linear dispersion with opposite group
velocity. We will show that they are localized on opposite surfaces of the slab.
But first let’s recall the Hosur SSH model argument for the appearance of the
arcs: the spinorial fashion of our Hamiltonian has forced us to consider a case in
which the type of sites on one surface differ from the type of sites on the other (2n
sites in x̂ or ŷ direction). It’s however legitimate to think of a slab of material cut
along the same direction, but with sites belonging to the same sublattice on both
surfaces. Intuitively, we would expect zero energy surface states to form a closed
contour in the diagram, and indeed the result is shown in figure 2.4c. The bulk
is unaffected by boundary conditions: as it can be clearly seen, it always presents
the semimetallic structure with four touching points between the bands. We have
one band left, the surface band. For kz = kzWeyl there are zero energy states at
π
π
; √
] the surface state is analogous to the even
any kk : in the region kk ∈ [− 2√
2 2 2
case and lives on the same surface of the even case; outside that region there is an
exact replica of the band, living however on the other surface.
Localization of the states
As stated before, the states bands that present Fermi arcs are generally localized
on the surface. Figure 2.5 shown the localization of the states for the "even lattice
planes" case: on the x̂ axis we have the finite direction coordinate, while on the ŷ
the square modulus of the wave function (obtained directly from the eigenvectors
of the hamiltonian corresponding to the bands). The two states live respectively
on sites of type A and B, as expected from surface made up respectively of each site
type only. As we see, the states are strongly localized at the border on the Fermi
arc, and they spread in the bulk getting closer to the Weyl point. At the Weyl
CHAPTER 2. 3D LATTICE WEYL HAMILTONIAN
41
point, they cease to be fittable with a decaying exponential in the finite direction.
In the "odd lattice planes" case the situation does not vary drastically: there
is obviously only one surface state that lives on sites of only one sublattice, and
p
√
is localized on one surface for kk ∈ (−π/2 (2); π/2 2) and on the other outside
this interval.
As stated in the theoretical introduction, while Weyl points are a bulk topological
characteristic of the semimetal, Fermi arcs strongly depend on the surface direction. It is therefore interesting to see what happens for different choices of cuts,
starting with the most trivial ones
2.3.2
Cut along the x − ay = 0 plane
This is merely an extension of the case presented before: the cut is now directed
along a different line, such as for example x − ay = 0. It is important that a be an
odd integer, otherwise the chunk would not be translationally invariant using the
spinor we have chosen, as different sublattice sites would align along the cut planes.
The band diagram computation is very similar to the case before; we only have to
redefine the Bravais lattice vector directed along the cut. The reciprocal lattice
√
vector in the direction parallel to the cut will now be in modulus 2π/ a2 + 1. We
don’t expect anything strange from the band diagram, which indeed presents Fermi
arcs connecting the projections of the same Weyl points. Interestingly enough, if
we set up the inverse situation and cut along ax − y = 0 we have to redefine the
spinor as in figure 2.6a to preserve translational invariance in the infinite directions,
√
√
and the result is that Fermi arcs now run along kk ∈
/ (−π/2 a2 + 1; π/2 a2 + 1).
2.3.3
Cut parallel to one Cartesian axis
One might ask why the most intuitive cuts were not used in the first place: as we
shall see, none of them is suitable for the appearance of Fermi arcs.
42
CHAPTER 2. 3D LATTICE WEYL HAMILTONIAN
1
0.9
0.8
0.7
|psi|
2
0.6
0.5
0.4
0.3
0.2
0.1
0
0
20
40
60
80
100
x-y
(a) Away from the Weyl point: (kk ,kz )= (0,π/2)
0.15
|psi|
2
0.1
0.05
0
0
20
40
60
80
100
x-y
√
(b) Near the Weyl point: (kk ,kz )= 0.95·(−π/2 2,π/2)
0.02
0.018
0.016
0.014
|psi|
2
0.012
0.01
0.008
0.006
0.004
0.002
0
0
20
40
60
80
100
x-y
√
(c) At the Weyl point: (kk ,kz )= (−π/2 2,π/2)
Figure 2.5: Localization of the border states in three different BZ points.
CHAPTER 2. 3D LATTICE WEYL HAMILTONIAN
43
ŷ
x̂
(a) Lattice depiction
(b) Band diagram
Figure 2.6: (a) Lattice depiction for a chunk of material cut along the line 3x + y = 0
(depicted in green). The lattice is shown here in the xy plane to highlight
the new definition for the spinor; however, the Bravais lattice will be the one
with unit vectors directed along ẑ and along the cut. (b) Band diagram of
the system (with normalized axes).
Finite along x̂
We know that Weyl points in our model sit on the k̂x = 0 plane. So in this case
the projections do not coincide, and we should expect to see Fermi arcs. As we
can see from the band diagram they are instead missing.
In fact, as the "spinors" in the infinite plane pile up exactly one atop the other,
we have to use the BZ ky ∈ [−π/2, π/2], kz ∈ [−π, π]. Along the finite x̂ direction,
spinors are not aligned: however, this is not actually a problem, as it amounts to
considering a base function consisting of Np spinors shifted by a lattice site along
ŷ. With our new BZ, as we recall from the position of the original Weyl point
projections, points of opposite chirality are at the boundary, hence coinciding tue
to PBC of the BZ. So, as it can be seen from figure 2.7 no Fermi arcs are shown.
CHAPTER 2. 3D LATTICE WEYL HAMILTONIAN
44
ẑ
ŷ
finite
(a) Lattice depiction
(b) Band diagram
Figure 2.7: Lattice depiction and band diagram for a chunk of material finite in x̂ direction. The k values are in units of π.
Finite along ŷ
The situation varies slightly from the previous one. Now, it is obvious that with
this cut the Weyl points are pairwise projected onto (0, ±π/2), hence chiralities
sum up to 0 and no arcs are shown (as seen in figure 2.8).
Finite along ẑ
Here we have simply Np copies of the original 2D lattice stacked one atop the
√
√
√
√
other. The BZ is k1 ∈ [−π/ 2, π/ 2], k2 ∈ [−π/ 2, π/ 2], with k1 and k2 along
the directions x̂ + ŷ and x̂ − ŷ. As expected there are two band-touching points at
√
√
(±π/2 2, ∓π/2 2), that means (kx , ky ) = (0, ±π/2) in the (kx , ky , kz ) frame that
enters the definition of H. Now, these are the coinciding projections of two pairs
of opposite-chirality Weyl points, hence no Fermi arcs are expected and none are
shown (cfr figure 2.9).
CHAPTER 2. 3D LATTICE WEYL HAMILTONIAN
45
ẑ
x̂
finite
(a) Lattice depiction
(b) Band diagram
Figure 2.8: Lattice depiction and band diagram for a chunk of material finite in ŷ direction. The k values are in units of π.
x̂ + ŷ
x̂ − ŷ
finite
(a) Lattice depiction
(b) Band diagram
Figure 2.9: lattice depiction and band diagram for a chunk of material finite in ẑ direction. The k values are in units of π.
46
CHAPTER 2. 3D LATTICE WEYL HAMILTONIAN
2.3.4
Cut along the x − y + az = 0 plane
In the following, in plane equation a is of the form 1/n. The reason for trying
this non-trivial cut direction follows from the study presented in Chapter 4; given
a cubic chunk of material, the idea was to initialize a wavepacket on a face and
watch it pass over the edge of the cube into the neighbour face and so on. To this
end, a cube cut along x − y = 0 is no good: in fact, the top and bottom surface
are precisely those studied in the previous section, and as clearly shown in figure
2.9b no Fermi arcs are present.
So a new cut was proposed, along a plane not parallel to any Cartesian axis.
Intuitively, with this setup Weyl point projections should be distinct and Fermi arcs
should be present. However, as shown in Chapter 4, no wavepacket propagation
is seen.
An unexpected feature
This scenario is actually trickier than imagined. First, the cut direction implies
that the points on the surface will no longer all belong to the same sublattice,
and there will be no allowed hopping within the same plane in general and on the
surface in particular (this was not the case of the previous cuts, for which we had
allowed hopping along ẑ on the surfaces).
Most importantly, the pseudo-spinor has to be redefined to preserve translational
invariance in the two infinite directions. The lattice then looks like the one sketched
in figure 2.10
For a = 1, the unit vectors of the Bravais lattice are r̂1 = −x̂ − ŷ and
r̂2 = x̂ − ŷ + ẑ; they are orthogonal, so the unit vectors of the reciprocal space
will be accordingly directed. The basis function of the Bravais lattice consists of
the 2 sites of the spinor times the Np stacked planes. The reciprocal lattice vec2π √
, 2π6 ). All allowed hoppings are between neighbour
tor lengths are (k1 , k2 ) = ( √
2
planes.
For a = 1/2 the situation is similar: what varies is the length of k2 =
2π
√
18
and the
CHAPTER 2. 3D LATTICE WEYL HAMILTONIAN
47
r̂2
Finite
y
z
x
r̂1
Figure 2.10: Lattice cut along the x − y + z = 0 plane. The rotated and original axes
are presented. The black dotted lines represent the planes stacked along ẑ.
fact that hopping along x or y directions now changes plane index by 2.
For a = 1 we find the band diagram shown in figure 2.11, which has two peculiarities: first, it shows only two Weyl point projections. Naively, if we take the
original Weyl point coordinates (ky , kz ) and rotate them to the (k1 , k2 ) coordinate
system we obtain noncoinciding values; however, it can be seen that two points are
π
π
π
√ )
rotated respectively to ±( 2√
, √
), while the other two are rotated to (− 2√
, 3π
2 2 6
2 2 6
π
√ ), outside the Brillouin Zone. Their Bloch equivalents inside the
and ( 2√
, − 23π
2
6
BZ boundary coincide with the other two projections.
The second peculiarity of the diagram is the presence of two flat and degenerate
surface bands, that account for the zero group velocity of the surface state. While
the reason for their flatness is still not completely clear, their behaviour under a
conveniently tuned on-site energy will be exposed in the following.
If we put a = 1/2, that is increase the steepness of the cut along ẑ, we get figure
2.12: we see that the two surface bands are still present, although no more at zero
CHAPTER 2. 3D LATTICE WEYL HAMILTONIAN
48
Figure 2.11: Band diagram for a lattice cut along the x − y + z = 0 plane.
energy; the only points at zero energy are the Weyl point projections, which again
coincide for the same reason as in the previous case.
Moving the Weyl points
We would like to see what happens when the point projections do not coincide:
then we can exploit one of the features previously introduced, that is moving about
the points in the BZ by applying an extra energy term to the hamiltonian. We start
with an on-site energy, that amounts to adding to the bulk hamiltonian a term
λσz , as the spinor still consists obviously of sites belonging to different sublattices.
Indeed, Fermi arcs at 0 energy do appear, and the bands split up elsewhere. Let’s
study the case for cut along the plane x − y + z/2 = 0, as seen in figure 2.13.
Fermi arcs and the Weyl points projections have also been added; the latter
have been obtained from the original Hamiltonian by taking the lowest energy
value for the upper surface bands (which ought to coincide to the highest of the
lower surface band), rotating them and projecting them on the cut plane. The
projections move in the BZ and come to coincide in five cases: two of them are
CHAPTER 2. 3D LATTICE WEYL HAMILTONIAN
49
Figure 2.12: Band diagram for a lattice cut along the x − y + z/2 = 0 plane. Due to
resolution, Weyl cones do not touch. Increasing the number of planes in the
finite direction they get closed and eventually exhibit the usual behaviour
(this has been proven by plotting only the surface bands at high resolution,
as it would be computationally too heavy to plot the entire diagram).
the original ones, for λ = ±2, at the boundary of the semimetal phase. The third
is simply the case without on-site energy. The last two cases are more interesting;
a complete resume is found in the table.
√
The nontrivial values of ± 2 for which Weyl point projections join can be calculated analytically as follows: let’s start from our bulk hamiltonian with on-site
energy tilt
H = −2 cos(ky )σx + sin(kx )σy − cos(kz )σz + λσz .
(2.13)
then we define the tilt matrix


2/3
−2/3
1/3


√
√

M =
2
−1/
2
0
−1/


√
√
√
−1/ 18 1/ 18 4/ 18
(2.14)
that change coordinate basis from the Cartesian to the tilted one, and its inverse M −1 . Let’s take, for example, the point which is projected onto (k1 , k2 ) =
50
CHAPTER 2. 3D LATTICE WEYL HAMILTONIAN
1
0.5
E 0
-0.5
-1
1
0.5
0
1
0.5
E 0
-0.5
-1
0
1
0.5
0
k2
0.5
0
-0.5
1
1.5
k1
2-1
2-1
(a) λ = ±2
E
1
0.5
0
-0.5
-1
(b) λ = ±1.65
1
0.5
0
1
0.5
E 0
-0.5
-1
0
1
0.5
0
k2
0.5
0
1.5
k1
2-1
2-1
√
(c) λ = ± 2
1
0.5
E 0
-0.5
-1
(d) λ = ±1
1
0.5
0
0
k2
0.5
k1
1
0.5
E 0
-0.5
-1
1
0.5
0
0
k2
0.5
-0.5
1
1.5
2-1
(e) λ = ±0.6
-0.5
1
1.5
k1
k2
0.5
-0.5
1
-0.5
1
1.5
k1
k2
0.5
-0.5
1
k1
1.5
2-1
(f) λ = 0
Figure 2.13: Surface bands for the cut along x − y + z/2 = 0
CHAPTER 2. 3D LATTICE WEYL HAMILTONIAN
51
π
3π
(− 2√
, √
). Let’s choose a "dummy" third coordinate k3 = − π4 so that kx = 0
2 2 18
in the (kx ky kz ) basis, as it should. The coordinates of the point in this latter basis
are given by k̃ = M −1 · k. We then put those in expression (2.13) and diagonalize
the resulting matrix. Its eigenvalues are
E1/2
q
√
= ∓ λ2 + 2 2λ + 2
(2.15)
√
which are obviously degenerate and 0 for λ = − 2. A similar procedure exhausts
the other case.
λ
k1
k2
K Description
-2
0.5 -0.5
0.5 -0.5
-0.5 0.5
-0.5 0.5
0
0
0
0
Lowest λ for which the bands touch; Weyl points are
coinciding and total chirality in each band touching point
is 0. No Fermi arcs are shown. Cfr figure 2.13a.
√
− 2
0.5 0.5
0.5 -1.5
-0.5 1.5
-0.5 -0.5
0
1
1
0
Two points are projected outside the BZ. Fermi arc
direction is outwards, so arcs are defined everywhere. Cfr
figure 2.13c
0
0.5 1.5
0.5 -2.5
-0.5 2.5
-0.5 -1.5
1
1
1
1
No on-site energy: although none of the points is
projected in the 1st BZ, they can all be reconducted inside
it by a reciprocal lattice vector of the same modulus. Cfr
figure 2.13f.
√
+ 2
0.5 2.5
0.5 -3.5
-0.5 3.5
-0.5 -2.5
1
2
2
1
Case specular to the second; two poins are outside the BZ
by a double-length reciprocal lattice vector.
2
0.5 3.5
0.5 -4.5
-0.5 4.5
-0.5 -3.5
2
2
2
2
Case specular to the first: all points can be reconducted
to the first BZ by a reciprocal lattice vector of the same
length.
CHAPTER 2. 3D LATTICE WEYL HAMILTONIAN
52
√
√
In the previous table, k1 is in units of π/ 2, k2 is in units of π/ 18 and we
also list the modulus of the reciprocal lattice vector K (directed along r̂2 ) that
√
joins the k-points with their equivalents in the 1st BZ (1 corresponds to 2π/ 18
and so on).
The system presents another interesting feature: for two values of the tuning parameter the bands become almost flat and degenerate. For the less steep cut it
happens for λ = 0, for the steeper cut it happens for two values of lambda (with
the same modulus, due to symmetry). Varying λ with a step δλ = 1 · 10−5 gives
band degeneracy for |λ| = 1.07457, and the corresponding band diagram is shown
in figure 2.13d. The value has been calculated by taking the band energy difference
of the two surface bands on border of the BZ, where the gap is maximum.
As we remember, if we cut along the plane x − y + z = 0 surface band flatness
occurs when Weyl point projections coincide (at λ = 0); this turns out to be a
coincidence due to the specific cut, as in general the two phenomena are unrelated.
Finally, an interesting remark: let’s take the bulk Hamiltonian (2.13). If instead
of adding an on-site energy λσz we tune the intra-spinor hopping adding a σx
term, we merely shift the positions of the Weyl points along the kk direction (ky
being associated to σx in the hamiltonian and the points being bound to the kx = 0
plane). However, we do not obtain the same effect in the last two cases we studied,
because we have redefined the spinor that now consists of sites among which there
is no hopping.
Let’s try to add such a term anyway: the result for the surface bands is shown
in figure 2.14. Weyl points and Fermi arcs are still present; the arcs, however,
have been curved. In the "dummy" coordinate system (kx , ky , kz ) the new term
53
CHAPTER 2. 3D LATTICE WEYL HAMILTONIAN
corresponds to
ei(kx −kz ) a†A aB + e−i(kx −kz ) a†B aA
=(cos(kx − kz ) + i sin(kx − kz ))a†A aB + (cos(kx − kz ) − i sin((kx − kz ))a†B aA
= cos(kx − kz )σx − sin(kx − kz )σy
(2.16)
where A and B represent the sublattice. The position of the Weyl points are
readily found: from (2.9) and trivial trigonometric identities we have
(kx , ky , kz ) =
2 /4
π
arctan , ± arccos p
,±
,
2
1 + 2 /4 2
(2.17)
the Hamiltonian has to be linearized around these points to find the Fermi arc
directions exiting the nodes. The reason for the curved shape of the arcs is at
present not fully understood.
Localization of the states
To prove that what we are studying are not bulk bands, we use the same procedure
as before, that is take the eigenvectors corresponding to the two band eigenvalues
(for various values of k). The spatial coordinate represent the plane index. We
can still see that the maximum of the states is on the two surfaces when we are
away from the Weyl point, while the states are delocalized at the node, as it can
be seen from figure 2.15. However, we don’t see such a good localization as we
did for the cut along x − y = 0. This depends on the shape of the band diagram
2.12: as we can see, the gap between the surface and bulk bands is smaller than
that of diagram 2.4b, and the Weyl cones are extended almost throughout the
entire direction k̂2 . The finite size of our sample (100 planes) also accounts for
imprecision in the localization diagram.
54
CHAPTER 2. 3D LATTICE WEYL HAMILTONIAN
-1
E
0.5
-0.5
0
0
-0.5
2
1.8
0.5
1.6
1.4
1.2
k2
1
0.8
0.6
k1
0.4
1
0.2
0
Figure 2.14: Surface bands for = 0.5 in the x − y + z = 0 cut case. Weyl point
projections and Fermi arcs have been highlighted.
1
0.09
0.9
0.08
0.8
0.07
0.7
0.06
0.6
2
|psi|
|psi|
2
0.1
0.05
0.5
0.04
0.4
0.03
0.3
0.02
0.2
0.01
0.1
0
0
0
20
40
60
80
plane
(a) Localization at (normalized) coordinates
(k1 , k2 ) = (1/2, −1/2)
100
0
20
40
60
80
plane
(b) Localization at (normalized) coordinates
(k1 , k2 ) = (1/2, 0)
Figure 2.15: State localization for λ = 0 in the x − y + z/2 = 0 cut case.
100
CHAPTER 2. 3D LATTICE WEYL HAMILTONIAN
55
Summing up: final remarks on the surface bands
• First of all, the bands we are considering are indeed surface bands, as proven
in the previous subsection. As such, they are respectively localized on either
surface in the finite direction of the slab. Varying the on-site energy, their
relative difference in energy varies accordingly, and (almost) vanishes for
certain values of λ depending on the cut. Checking the surface localization
of the state before and after the "flat bands" situation, it can be seen that
the surface on which each band is localized swaps (a check has been made
for points at the border of the BZ, where the gap is maximum). At present,
we have found no analytical reason for the flatness of the bands to occur.
• Differently from the original cut case, here the length of the Fermi arcs is
affected by the on-site energy: they shrink until they disappear and then
reappear. They can also come to span the whole Brillouin zone.
• If we classify our material phase as a function of the on-site energy λ, we see
that it is a Weyl semimetal (i.e. with vanishing gap in a finite number of
points) in the region λ ∈ [−2, 2]. Inside this region, for 5 values of λ the point
projections in the "cut along x − y + z/2 = 0 case coincide. If we simply take
the Weyl point positions and rotate them in the new reference frame, they
happen to be projected outside the first BZ. In three cases (λ = 0, ±2) all
projection points differ from their equivalents in the BZ by reciprocal lattice
vectors of the same modulus. This intuitively corresponds to a vanishing
√
Fermi arcs. For λ = ± 2, projections of Weyl points joined by an arc differ
from their Bloch corrispectives in the BZ by reciprocal lattice vectors of
different modulus. In this case, the Fermi arcs starting from both projections
get to the border of the Brillouin zone (as their end is at a points outside it).
Effectively, we have coinciding projections of the points, and an arc spanning
the whole BZ in direction k2 . In [23], [25] and [6] similar situations are
considered. If it were possible to make the bulk bands detach, once achieved
CHAPTER 2. 3D LATTICE WEYL HAMILTONIAN
56
a close Fermi contour, we would have a system that is a 3D topological
insulator with gapless surface bands. In our case, however, when the arcs
span the whole BZ we are still in the region λ ∈ [−2, 2], so the system is
bound to remain gapless at 4 points in the BZ, i.e. a semimetal.
• The flatness of the bands and the position of Weyl points, although both
depending on the on-site energy, do not seem to be correlated: for the cut
along x − y + z = 0 the two features occur for the same value λ = 0, while for
the cut along x − y + z/2 = 0 the situation is different: band flatness occurs
at |λ| = 1.07457 while Weyl point projection coincidence (in a non-trivial
√
fashion) occurs at λ = ± 2. While the first result is purely numerical,
the second can be easily calculated as we saw in the previous subsection.
Fermi arcs only depend on the position of the Weyl point projections. In the
x − y + z = 0 case, their existence for λ = 0 is masked by the flatness of the
bands.
Chapter 3
Analytical approach
Up to now, we didn’t ever write an analytical form for the surface states on the
arcs. This can indeed be done, as we’ll prove in the following. A 2D derivation of
gapless surface states is presented as a toy model in Appendix A.
3.1
Analytical derivation of 3d Weyl surface states
Let’s imagine to have a Weyl semimetal which is semi-infinite in one direction
(that is, it has an interface on which we expect the surface states we discussed
earlier on). Let’s start with a non-chiral massless Dirac fermion ψ (it is irrelevant
whether this system is actually feasible in condensed matter physics). Now, what
would be a suitable boundary condition for our system? We should impose angular momentum conservation, that is the component of angular momentum on the
direction perpendicular to the surface should be conserved. Since along this axis
the direction of motion is reversed, also helicity, which is the direction of the spin
projected onto the momentum direction, has to be reversed. This is possible since
a Dirac spinor has both helicities.
What happens then when we have fermions which obey Weyl equation, i.e. with
only one possible helicity? The boundary condition will not be able to reverse it,
since the other helicity doesn’t exist. This will reflect in the presence of a prefer57
CHAPTER 3. ANALYTICAL APPROACH
58
ential direction in the boundary plane.
Let’s ask ourselves what is a good boundary condition for the real-space Hamiltonian
H = −iσ · ∇
(3.1)
Let’s say we impose, at x1 =0 the condition
M ψx1 =0 = ψx1 =0
(3.2)
M = σ2 cos α + σ3 sin α
(3.3)
where
This condition makes the Hamiltonian hermitian, and is therefore good: we want
to prove that
hψ|Hψi = hHψ|ψi
(3.4)
If we integrate by parts LHS, the only difference consists in a boundary term that,
since the boundary is x1 = 0 reduces to
Z
∂B
hψ|σ · ∇ψi · (1, 0, 0) = hψ|σ1 ψi |x1 =0
(3.5)
but now at other boundary
1
hψ|σ1 ψi =
hψ|σ1 ψi + hψ|σ1 ψi
2
1
=
hψM |σ1 ψi + hψ|σ1 M ψi
2
1
=
hψ, {σ1 , M }ψi
2
(3.6)
=0
where we used the fact that Pauli matrices are hermitian and they anticommute.
Now, the choice of angle is not crucial, as it can be absorbed in a rotation of
CHAPTER 3. ANALYTICAL APPROACH
59
the x2 x3 plane. Let’s furtherly assume that σ2 ψ = ψ everywhere, not only on
the boundary, and also that ∂ψ/∂x2 = 0. By assuming these conditions, we are
actually throwing away some possible solutions. However, in this way calculations
are heavily simplified, and moreover these conditions are still consistent with our
case study: assuming σ2 ψ = ψ is still justified for states we want exponentially
decreasing in the bulk, while ∂ψ/∂x2 = 0 means that Fermi arcs live at a specific
constant value of k2 , which again is what happens for our system. We will therefore
keep these assumptions (though we are not completely satisfied with those) and
simplify our study.
The Hamiltonian becomes
Hψ = σ1 ∂1 + σ2 ∂2 + σ3 ∂3 ψ = σ1 ∂1 − i∂3 ψ,
(3.7)
where we used the Pauli matrix property σ3 = −iσ1 σ2 and the condition σ2 ψ = ψ.
Solving Hψ = 0 gives
ψ = eikx3 −kx1 ψ0
(3.8)
where ψ0 is a constant spinorial term obeying σ2 ψ0 = ψ0 . This expression is normalizable if k > 0, and in this case it is exponentially decaying away from the
border. In the plane wave picture, k is actually the momentum component p3 , and
the spectrum of allowed border states is a line in the positive p3 direction. Near
p3 = 0, i.e. near a Weyl point, the state ceases to be localized at the boundary.
Therefore the only way a Fermi arc exiting from a Weyl point can end is to intercept another Weyl point, which has opposite chirality for the argument given in
Chapter 1.
3.1.1
Our system: cut along x − y = 0
Let’s apply now the procedure we have outlined to our optical lattice system:
obviously the Hamiltonian is not everywhere Weyl-like, so we will have to linearize
CHAPTER 3. ANALYTICAL APPROACH
60
it around the points.
Let’s consider the arc that connects the projection of (0, π/2, π/2) to the projection
of (0, −π/2, π/2) along the cut considered in [7], and let’s study the region near
the surface of the slab consisting of sites belonging to sublattice A.
First, we have to linearize the bulk Hamiltonian.
Linearize around (0, π/2, π/2)
• −2 sin(kx ) ≈ −2 sin(0) − cos(0)(k0 ) = −2kx
• −2 cos(ky ) ≈ −2 cos(π/2) − sin(π/2)(ky − π/2) = +2(ky − π/2)
• +2 cos(kz ) ≈ +2 cos(π/2) − sin(π/2)(ky − π/2) = −2(kz − π/2)
so that the Hamiltonian becomes
Happrox (k) = 2(ky − π/2)σx − 2kx σy − 2(kz − π/2)σz
(3.9)
Looking at the coefficients and remembering (2.11) we see this Weyl point has
χ = −1. It is convenient to rotate by the reference frame according to

 √
√
1/ 2 −1/ 2 0

 √
√

M =
1/
2
1/
2
0


0
0
1
to get

√


k
=
(k
+
k
)/
2

x
y
k


√
k⊥ = (kx − ky )/ 2




kz
(3.10)
(3.11)
61
CHAPTER 3. ANALYTICAL APPROACH
Now, (3.9) becomes
kk + k⊥
kk − k⊥
π
√
√
σx − 2
σy − 2 kz −
σz
Happrox (k) = 2
2
2
2
σ2 + σ1
√
= 2kk σ1 − 2k⊥ σ2 − 2kz σz − π
+ πσz
2
(3.12)
where we have defined
σ1 =
σx − σy
√
,
2
σ2 =
σx + σy
√
,
2
σz .
(3.13)
Proceeding grouping the terms according to the σs we get
π
π
π
σz .
Happrox (k) = 2 kk − √ σ1 − 2 k⊥ + √ σ2 − 2 kz −
2
2 2
2 2
(3.14)
Now, let’s rewrite the Hamiltonian as
HA = qA · ~σ
(3.15)
√
√
where qA = (2(kk −π/2 2), −2(k⊥ +π/2 2), −2(kz −π/2)) and ~σ = (σ1 , σ2 , σz ).
and repeat the same procedure around the other end of the Fermi arc, which
we know is at (0, −π/2, π/2).
Linearize around (0, −π/2, π/2)
• −2 sin(kx ) ≈ −2 sin(0) − cos(0)(k0 ) = −2kx
• −2 cos(ky ) ≈ −2 cos(−π/2) − sin(−π/2)(ky + π/2) = −2(ky + π/2)
• +2 cos(kz ) ≈ +2 cos(π/2) − sin(π/2)(ky − π/2) = −2(kz − π/2)
Looking at the coefficients we see this Weyl point has positive χ. Let’s again
write the Hamiltonian in the form (3.15): it amounts to redefine qB = (2(k⊥ −
CHAPTER 3. ANALYTICAL APPROACH
62
√
√
π/2 2), −2(kk + π/2 2), −2(kz − π/2)).
As the system has a border along x⊥ , we pass back to direct space in its direction
in both cases, by substituting q⊥ → ±i∂x⊥ , considering the negative (positive)
dependence of q⊥ from x⊥ in each case. Now we have to choose a boundary
condition for the function along the cut. Following Witten we choose
M ψ|x⊥ =0 = ψ|x⊥ =0
(3.16)
M being a linear combination of the other two Pauli matrices. In particular,
we choose M = σz and extend the condition everywhere, furthermore assuming
∂z ψ = 0, which means qz = 0, which is consistent with position of the Fermi
arc at hand in the band diagram. In both cases, the linearized Hamiltonians are
simplified and respectively become


HA = σ1 (+i∂⊥ − iqk )
(3.17)

HB = σ2 (−i∂⊥ + iqk )
where we used σz = iσ1 σ2 and {σ1 , σ2 } = 0.
Now we can solve HA/B ψ = 0, which has solutions of the type
 
1
ψ = eiqk xk +qk x⊥  
0
(3.18)
where the spinor easily satisfies the boundary condition.
This means that, around (0, π/2, π/2), our border state is normalizable for
√
√
qk = 2(kk − π/2 2) < 0, that is kk < π/2 2, while around (0, −π/2, π/2) the
√
√
border state is normalizable for qk = −2(kk + π/2 2) < 0, that is kk > −π/2 2.
This is exactly the region where we saw the Fermi arc of the very first band diagram we considered.
CHAPTER 3. ANALYTICAL APPROACH
3.1.2
63
Arc direction with energy tilt
If we add a term σx to the original Hamiltonian, the Weyl points shift in the
kk direction. Then, repeating the procedure above, we trivially get the following
condition for kk :
sin(kW )(kk − kW ) < 0
(3.19)
√
where kW = arccos(−/2)/ 2. Obviously the condition has to be independently
√
√
stated in the regions U1 = (−π/ 2, 0) and U2 = (0, π/ 2). So, if = 0 and we
√
√
are in U1 we find again kk < π/2 2, if we are in U2 we have kk > −π/2 2, that
are the two conditions we found in the previous section. As we can see, as long as
Weyl point projections do not coincide (that happens at = ±2 for points at the
edge of the BZ or at kk = 0 respectively, that is at the border of U1 and U2 ) the
Weyl point position varies smoothly with , but crucially the Fermi arc direction
depends only on sin(kW ), i.e. on the region in which the projection of the point is.
A similar reasoning can be put up for the "cut along x − y + z/2 = 0" situation.
Now, the perturbation is λσz , the arcs run along k2 and the projections of the
√
points (once reconducted in the first BZ) coincide for certain λ at k2 = ±π/2 18.
So, the arc direction will depend on whether the point projection is inside or
√
√
outside the region (−π/2 18, π/2 18). An analytical calculation would follow
the same procedure as before; however, it’s completely analogous to the case we
have studied and quite more cumbersome, and the numerical calculation already
shows that points inside the region give rise to arcs in the decreasing k2 direction,
while points outside the region give rise to arcs in the increasing k2 direction (cfr.
figure 2.13). This accounts for the two different situations seen in Chapter 2: let’s
√
consider the arc at k1 = π/2 18: if we consider the limit λ → 0, bringing the point
√
projection closer and closer to k2 = −π/2 18 ± 0, the arcs will get shorter and
√
shorter and finally disappear; if instead we consider the limit λ → 2 the points
√
will approach k2 = +π/2 18 ± 0 and the arcs will eventually span the whole BZ.
64
CHAPTER 3. ANALYTICAL APPROACH
angular coefficient=1.001229
0.11
0.11
0.1
0.1
0.09
0.09
0.08
0.07
0.08
0.07
0.06
0.06
0.05
0.05
0.04
0.04
0.05
0.06
0.07
0.08
0.09
angular coefficient=-1.001229
0.12
decay coefficient
decay coefficient
0.12
0.1
0.11
0.12
k//-kw//
0.04
-0.12
-0.11
-0.1
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
k//-kw//
√
√
(a) Weyl point (−π/2 2, −π/2), positive δkk (b) Weyl point (π/2 2, −π/2), negative δkk
Figure 3.1: Decay coefficient behaviour at the extremities of a Fermi arc. In the directions in which the arc is present the behaviour is strongly linear.
3.1.3
Numerical verification
The formula we derived for states on the Fermi arcs around the Weyl points can
be put to the test by numerical means. Let’s start near a Weyl point, then proceed
along the Fermi arc getting far away from the point. If the analytical hypothesis
is correct, the momentum kk , which is at the exponential of the decaying part of
ψ, should increase linearly. If we fit the squared amplitude of the surface state
in each case with an exponential e−ax⊥ , then plotting a against k should give us
a linear dependence. The diagrams of figure 3.1 illustrate the results obtained
near the two Weyl points (0, −π/2, −π/2) and (0, π/2, −π/2). They confirm our
theoretical hypothesis pretty closely. A more extensive tractation of the numerical
procedure used is given in Appendix B.
Chapter 4
Wavepacket propagation
4.1
Introduction
The topological properties of the Fermi arcs have interesting consequences in real
space. To observe wavepacket propagation in real space and convince ourselves of
the existence of a preferred propagation direction based on the initial momentum
of the packet, we consider a chunk of material (obviously this is finite in all the directions, but we suppose it to be sufficiently big to consider approximately fulfilled
translational invariance in all three directions, while at the same time retaining
the Fermi arc behaviour because of the presence of borders) cut along different
directions and shapes and study the behaviour of the system when a wavepacket is
initialized at a certain position with a specific phase. To all good extents, we can
treat the real-space sites basis as a discretized space coordinates triplet (i, j, k) ≈ r;
then the elementary quantum mechanics of the gaussian wavepacket should apply.
The general procedure for a chunk is the following:
• Choose the system diagonal size N: in the case of a cubic chunk of material,
we know that the site species on the surface, which in turn depends on the
number of lattice planes, gives rise to two different band diagrams. However,
since the bulk is unaffected by the boundary conditions, it’s actually fine to
65
66
CHAPTER 4. WAVEPACKET PROPAGATION
use an even or odd N, provided we give the wavepacket an appropriate phase
depending on to which surface we want it to remain localized.
• Write the equation for the boundary planes
• For all the points contained in a cube of dimension N, check if they are inside
the boundaries; assign to every good point (x, y, z) an index l = z+N y+N 2 x
• generate the Hamiltonian matrix, which will be l × l with hopping elements
obtained by the original definition of the lattice, and on-site energy varies in
sign depending on to which sublattice the site belongs
• initialize a gaussian wavepacket on a surface of the cube:
−
Ψ=e
x − xc 2 y − yc 2 z − zc 2 +
+
2σx2
2σy2
2σz2
· ei·k0 R δ(b(x, y, z))
(4.1)
where we are using the original site coordinates, R is the displacement vector
from the center in real space, k0 is the initial phase, conveniently chosen from
the band diagram, and δ fixes the condition for the initial state to be surfacelocalized, b(x, y, z) being the equation for the plane to which the face belongs
(for example, for the sample in figure 4.1, b(x, y, z) = x − y −
N +1
).
2
• Diagonalize the Hamiltonian and get the eigenvectors φn , the eigenvalues En
and the basis expansion coefficients cn = hφn |Ψi.
• Evolve the packet as Ψ(t) =
4.2
P
n cn e
−iEn t
φn
Cut along the plane x − y = 0
This is the original cut: when we initialize the packet with a phase (in the basis
√
(kk , kz )) of (π/2 2, −π/2), that is at a Weyl point, we see that a considerable
part of the state spreads out in the bulk (figure 4.2). If we start at (0, −π/2),
CHAPTER 4. WAVEPACKET PROPAGATION
67
just half-way along the Fermi arc (figure 4.1), the state is clearly bound to be
localized on the surface; the propagation direction is determined by the group
velocity vg = ∂ω/∂k, the dispersion being, near the Fermi arc, a sine function
(hence approximately linear) along kz , and flat along kk . Now, when the state
comes to an edge of the cube there are no other possibilities for it than going
back: the upper surface does not support surface states anywhere (it is the case
of cut perpendicular to ẑ), so the edge behaves like an infinite potential barrier;
the wavepacket is reflected, wavevector is reversed and the state is now localized
on the other Fermi arc.
4.3
Cut along the plane x − y + z = 0
This cut was originally chosen to allow the surface state to pass over the border into
the top surface of the chunk. As in general Weyl point projections did not seem to
coincide (we then discovered they actually were coinciding), the behaviour of 4.3
was totally unexpected. Once plotted the band diagram the reason became clear,
and so became the solution to get surface states moving on this surface. Adding
an on-site energy λσz , Fermi arcs appear in the band diagram. So, a conveniently
initialized wavepacket has a group velocity (though small) and moves accordingly,
as it can be seen in 4.4.
68
CHAPTER 4. WAVEPACKET PROPAGATION
(a) t=0
(b) t=2
(c) t=5
(d) t=9
(e) Band diagram
Figure 4.1: Propagation of a gaussian wavwpacket on a chunk of topological material,
with initial phase set to (kk , kz ) = (0, −π/2). The surface is made of sites
belonging to sublattice A. The initial phase is that of point A. s we can see
from (e), the initial group velocity is positive in the ẑ direction. Once at the
edge of the face, it gets reflected to opposite kz , that is point B. The group
velocity is also reversed (as represented from the orange arrow).
69
CHAPTER 4. WAVEPACKET PROPAGATION
(a) t=0
(b) t=2
(c) t=4
(d) t=9
(e) Band diagram
Figure 4.2: Propagation of a gaussian wavepacket on
√ a chunk of topological material,
with initial phase set to (kk , kz ) = (−π/2 2, −π/2), at a Weyl point. There
is still a surface component that retains the behaviour of figure 4.1, but now
the bulk component is non-negligible; however, as we saw from diagrams
in Chapter 2, the maximum is still at the surface. Unfortunately the bulk
component of the state is not clearly visible due to plotting issues (empty
sites are actually white dots).
70
CHAPTER 4. WAVEPACKET PROPAGATION
(b) t=20
(a) t=0
(c) Band diagram
Figure 4.3: Propagation of a gaussian wavwpacket on a chunk of topological material
cut along x − y + z = 0, with no on-site energy. As long as the packet is not
centered at a Weyl point. the initial phase is irrelevant. Here the example
is for k0 = 0.
71
CHAPTER 4. WAVEPACKET PROPAGATION
(a) t=0
(b) t=10
(c) t=20
(d) t=30
(e) Band diagram
Figure 4.4: Propagation of a gaussian wavepacket on a chunk of topological material cut
along x − y + z = 0, with on-site energy = 1. (g) shown the two surface
bands, with the Weyl point projections denoted as black dots; the initial
phase is that of point A, the band is that localized on the upper surface. We
see that the wavepacket has group velocity, which although
small
√
√ is positive
along k1 (remembering that k1 is directed along (−1/ 2, −1/ 2, 0)).
Conclusions
In this work, we have analyzed some of the properties of a specific Weyl semimetallic optical lattice. Disregarding the experimental realization complexity of such a
system, we can say the bulk structure is relatively simple to imagine. However,
simply inserting an interface in the picture produces a surprisingly big variety of
topological effects.
At present, some issues are not yet resolved: while the nature of the surface bands
and the presence of the gap everywhere but at the Weyl point projections is theoretically accounted for, it is not clear enough the behaviour such surface bands
present as a function of the on-site energy: in particular, the reason why they
become flat (and indeed if they are in each case really flat or if their gap -though
extremely small- has physical meaning and is not a numerical effect) for some values of λ has still to be completely clarified.
Another interesting study to be done concerns the topological phase transition of
the system: depending on the way the Weyl points merge, Fermi arcs disappear
or form a closed contour, that should remain at zero energy even if bulk bands
detach, giving rise to a 3D topological insulator. This feature is even more interesting if combined with the idea that, adding more hopping terms and/or tuning
the existing ones, the shape of Fermi arcs can be modified. As an example, it
would be interesting to "curve" the Fermi arcs in the original cut, and then make
the Weyl point projections coincide and the bulk bands detach: we should then
be in the 3D topologically non-trivial insulating phase. To this end, however, the
exact relation between added terms and Fermi arc shape must first be established.
72
CHAPTER 4. WAVEPACKET PROPAGATION
73
Finally, in terms of propagation of wavepacket, it would be interesting to check if
it is possible, for a specific chunk shape and initial phase, to create a sample in
which a wavepacket can overcome the edge and propagate in two adjacent faces.
This, that was the initial goal of our study, should be obtained comparing the
surface dispersion of such faces, and conveniently tuning the parameters (on-site
energy, inter-spinor hopping, added hopping terms) to make propagation possible.
It is interesting, as a closing remark, to note that all these features are specific to
the Hamiltonian we studied, which is only one of the many possible realizations
of the Weyl Hamiltonian in condensed matter physics; this shows how rich and
promising the field of topological phases of matter is.
Appendices
74
Appendix A
Useful material
A.1
Adiabatic theorem
As stated in Chapter 1, we start from the Hamiltonian Ĥ(R), depending on an
external parameter set R living in an n-dimensional space and slowly varying along
the path Rs where s ∈ [0, 1].
The eigenvalue equation for the system is
Ĥ(R)|Φm (Ri = Em (R)|Φm (R)i
(A.1)
and the time-dependent equation as a function of the reduced time s = t/T is
i~
d
|Ψ̃(s)i = T Ĥ(Rs )|Ψ̃(s)i
ds
(A.2)
where |Ψ̃(s)i = |Ψ(sT )i.
We incidentally recall that the projector on the first eigenvalue and its complementar operator (used in the text) are:
P(R) = |Φ0 (R)ihΦ0 (R)|
75
(A.3)
76
APPENDIX A. USEFUL MATERIAL
Q(R) = I − P(R) =
X
(A.4)
|Φm (R)ihΦm (R)|
m6=0
Now, what we are going to prove is that
Theorem 2. if E0 (Rs ) is non-degenerate and separated by a finite energy gap
~∆k0 (s) from all the other eigenvalues, for a sufficiently large T the eigenstate
remains proportional to |Φ0 (Rs i) up to a phase factor.
Let’s start by the following hypothesis on the state at reduced time s:
|Ψ̃(s)i =
X
iT
Cm (s)e−i ~
Rs
0
ds0 Em (Rs0 )
(A.5)
|Φm (Rs )i
m
If we plug it into (A.2) and solve the equation we get that the phase factor cancels
against the term T Ĥ(Rs ) |Ψ̃(s)i, and we are left with an eqation for the coefficients
Cm (s)
X
e
− iT
~
Rs
0
ds0 Em (Rs0 )
Ċm (s) |Φm (Rs )i + Cm (s)Ṙs · |∇R Φm (Rs )i = 0
(A.6)
m
where it’s understood that |Φm (R)i are differentiable with respect to R.
Now we take the scalar product of the expression above with a given eigenstate
|Φk (Rs )i and get
Ċk (s) = −
X
iT
Cm (s)e− ~
Rs
0
ds0 (Em −Ek )
Ṙs · hΦk (Rs )|∇R Φm (Rs )i
(A.7)
m
which can be rewritten as
Ċk (s) = −Ck (s)Ṙs · hΦk (Rs )|∇R Φk (Rs )i + ... := iγ̇k (s)Ck (s) + ...
(A.8)
where we isolated the diagonal term. Note that Ṙs · hΦk (Rs )|∇R Φk (Rs )i is purely
imaginary, because its real part is
hΦk |∇R Φk i + h∇R Φk |Φk i = ∇R hΦk |Φk i = 0
77
APPENDIX A. USEFUL MATERIAL
and therefore can be expressed in terms of a real quantity
Z
s
γk (s) := i
(A.9)
ds0 Ṙs0 · hΦk (Rs0 )|∇R Φk (Rs0 )i
0
Now we define the new quantities ck (s) = e−iγk (s) Ck (s) and rewrite (A.7) as
ċk (s) =
X
iT
e~
Rs
0
ds0 (Ek −Em )
Fkm (s)cm (s)
(A.10)
m6=k
where
Fkm (s) = −e[γm (s)−γk (s)] Ṙs · hΦk (Rs )|∇R Φm (Rs )i
(A.11)
If we integrate these equations for s ∈ [0, s̄] we have a system of integral equations
of the form
ck (s̄) = δk,0 +
XZ
m6=k
s̄
iT
ds e ~
Rs
0
ds0 (Ek −Em )
Fkm (s)cm (s)
(A.12)
0
where the δs are given from the trivial initial condition Ck,0 = δk,0 . Now, if k 6= 0
we can assume (not in fact so safely) that the most important contribute comes
from the m = 0 term, again because of the initial values of the cm . So we have
Z
s̄
iT
ds e ~
ck (s̄) =
Rs
0
ds0 (Ek −E0 )
Fk0 (s)c0 (s) + [...]
(A.13)
0
where [...] stands for all the other terms. Recalling the definition of Fk0 (s) we have
Fk0 (s) = −ei[γ0 (s)−γk (s)] Ṙs hΦk (Rs )|∇R Φ0 (Rs )i
(A.14)
Now the initial hypothesis on the non-degeneracy of the ground state ~∆k0 (s) =
Ek (s) − E0 (s) 6= 0 ∀ s ∈ [0, s̄] becomes useful: if we take the gradient of the original
78
APPENDIX A. USEFUL MATERIAL
time-independent Schrödinger equation we get
E0 ∇R |Φ0 (Rs )i = ∇R Ĥ(Rs ) |Φ0 (Rs )i
= ∇R Ĥ(Rs ) |Φ0 i + Ĥ(Rs ) |∇R Φ0 (Rs )i
(A.15)
Taking the scalar product with hΦk | we get
hΨk (Rs )|∇R Φ0 (Rs )i = −
hΦk (Rs )|∇R Ĥ|Φ0 (Rs )i
Ek (Rs ) − E0 (Rs )
(A.16)
that in turn gives the following expression for Fk0 :
Fk0 (s) = ei[γ0 (s)−γk (s)]
Ṙs · hΦk (Rs )|∇R Ĥ|Φ0 (Rs )i
.
Ek (Rs ) − E0 (Rs )
(A.17)
If the energy gap gets larger, Fk0 gets smaller.
Now, integrating (A.13) by parts we get
R s̄ 0
s̄
eiT 0 ds ∆k0 (s)
Fk0 (s)c0 (s)
ck (s̄) =
iT ∆k0 (s)
0
R
Z s̄
iT 0s̄ ds0 ∆k0 (s)
e
d
(Fk0 (s)c0 (s)) + [...].
ds
−
iT ∆k0 (s) ds
0
(A.18)
The important bit is the T at the denominator; this means that, if adiabaticity is
guaranteed, that is
Fk0 (s) T ∆k0 (s) 1,
(A.19)
their contribution can be neglected, so the state remains proportional to |Φ0 (R)i,
acquiring a phase, namely
i
|Ψ(t)i ≈ eiγ0 (t) e− ~
Rt
0
dt0 E0 (R(t0 ))
|Φ0 (R(t))i
(A.20)
where the trivial phase factor is explicitly written, while the non-trivial geometrical
79
APPENDIX A. USEFUL MATERIAL
phase is
Z
γ0 (t) := i
t
dt0 Ṙ(t0 ) · hΦ0 (R(t0 ))|∇R Φ0 (R(t0 ))i .
(A.21)
0
It is this phase we have to study to find topological properties.
A.2
Peierls’ substitution
It all starts from the necessity of including a magnetic field in a tight binding
picture. The result of this tractation will be to state that it is not sufficient to
substitute p̂ → p̂ + ec A in the Hamiltonian, it’s also necessary to vary the hopping
term as
e
Kr,r’ → Kr,r’ e−i ~
R r’
r
A·dl
(A.22)
where the integral is calculated along a straight line from r to r’.
We follow the proof given by Feynman in [8]. Let’s start from the known quantum
mechanics fact that, in presence of a magnetic potential, the amplitude that a
particle goes from one place to another following a determined path acquires a
phase
iq
e~
Rb
a
A·s
.
(A.23)
Then let’s recall the time-dependent Schrödinger equation of a charged particle in
an electric potential
~ ∂ψ
1
~
~
−
=
∇ ·
∇ ψ + qV ψ
i ∂t
2m~ i
i
(A.24)
The statement that has to be proven is that adding the phase (A.23) amounts to
modifying the above Hamiltonian as
1
~
~
~ ∂ψ
−
=
∇ − qA ·
∇ − qA ψ + qV ψ
i ∂t
2m~ i
i
(A.25)
80
APPENDIX A. USEFUL MATERIAL
that is Schrödinger equation for a charged particle moving in an electromagnetic
field V,A.
Let’s consider our tight binding case, a discrete chain of atoms along x̂ with spacing
b, and a hopping amplitude −K in absence of any field. If a potential Ax (x, t)
iq
is added, the amplitude wil be altered by a factor e ~ Ax b , where we substitute
the integral as the vector potential is directed along x̂. From now on, we will
call(q/~)Ax = f (x).
So, if the amplitude of probability to find the electron at site i is initially C(x) = Ci ,
in presence of the potential it changes according to the following:
−
~∂
C(x) = E0 C(x) − Ke−ibf (x+b/2) C(x + b) − Ke+ibf (x+b/2) C(x − b). (A.26)
i ∂t
In the previous equation, the first term on RHS is the probability amplitude of the
electron being in C by the energy of that case, the second and third term are the
probabilities for the electron to have hopped, adjusted adding phase (A.23), where
the value of f (x) is approximated at the midpoint of the spacing, which is not too
rough an approximation if Ax does not change heavily in one atomic spacing.
If C(x) is smooth enough, and the atoms get closer, this discrete equation will
approach the behaviour of an electron in free space. So, if we expand the RHS of
the previous equation in powers of the small quantity b up to order 2 we get
−
~ ∂C(x)
=E0 C(x) − 2KC(x)
i ∂t
2
”
0
0
(A.27)
2
− b K C (x) − 2if (x)C (x) − if (x)C(x) − f (x)C(x)
where differentiation is done with respect to x.
The expression above can be trivially seen to be equal to
~ ∂C(x)
∂
2 ∂
−
= E0 − 2K C(x) − Kb
− if (x)
− if (x) C(x).
i ∂t
∂x
∂x
(A.28)
APPENDIX A. USEFUL MATERIAL
81
Now we set E0 = −2K and restore the definition f (x) = (q/~)Ax , remembering
also that in a lattice without potential for small k the energy is E = 2K cos(kb) ≈
Kk 2 b2 and the relations v =
~2
2Kb2
that yelds Kb2 =
1 dE
~ dk
and E = 12 mef f v 2 hold, so that we have mef f =
~2
.
2mef f
Now (A.28) is exactly analogous to (A.25), that means that changing the hopping
amplitudes as in (A.22) in a tight-binding picture amounts exactly to replacing in
the usual way the momentum as p̂ → p̂ + ec A in the continuum.
A.3
A 2D toy model for border states
The system can be a lattice as the one we sketched in Chapter 2, or an electronic
system with spin, as long as it can be described by 2-component spinors.
Let’s start by the Hamiltonian
H=
X
n
σ3
a†n
X
− iσ1
†n σ3 − iσ2
a†n σ3 an
an+x̂ + a
an+ŷ + h.c. + m
2
2
n
(A.29)
which, diagonalized in k space as usual, gives
H = sin(kx )σ1 + sin(ky )σ2 + m + cos(kx ) + cos(ky ))σ3 .
(A.30)
It can be proven that gapless states localized at the boundary exist if |m| < 2,
otherwise the system is trivial. In fact, if for example m = −2, it’s easily proven
that the energy gap between the bulk bands
q
2
E(kx , ky ) = ± sin2 (kx ) + sin2 (ky ) + m + cos(kx ) + cos(ky )
(A.31)
is closed for k = (0, 0), and we know that closing the gap is necessary at the
transition between a trivial state and a topological state. So, let’s study the
system for the point k = (0, 0), cutting it perpendicular to the x̂ direction.
First, kx is no more a well-defined quantity because translational invariance along x̂
82
APPENDIX A. USEFUL MATERIAL
is broken. So we have to get back to the real space in this direction, by substituting
kx → −i∂x . Also, we approximate the hamiltonian near (0,0) as
H ≈ kx σ1 + ky σ2 + m + 2 − kx2 − ky2 )σ3
(A.32)
that when the cut is made becomes
H = −i∂x σ1 + ky σ2 + m + 2 − ky2 + ∂x2 σ3
(A.33)
To simplify calculations, let’s assume ky = 0 and rewrite (A.33) in matrix form

H|ky =0 = 
m+2+
−i∂x
∂x2
−i∂x
−(m + 2 + ∂x2 )


(A.34)
so, if we expect an exponentially decaying state away from the border φ ∝ e−λx ,
we get
H|ky =0


m + 2 + λ2
−iλ

=
−iλ
−(m + 2 + λ2 )
(A.35)
and, as we are looking for zero energy states, we have to put
det H|ky =0 = 0 → m + 2 + λ2 = ±λ.
(A.36)
This gives us 4 solutions, but we need those that make φ exponentially decaying,
that is those with Re(λ) < 0. In the end, the good λs and eigenstates turn out to
be
1
λ1,2 = − ±
2
r
1
− (m + 2)
4
 
1
φ1 =  
i
(A.37)
83
APPENDIX A. USEFUL MATERIAL
Figure A.1: Dimer chain described by the SSH Hamiltonian; the dimer is enclosed in
the dotted region, inter-cell hoppings are denoted as w, intra-cell hoppings
as v. The coloured regions highlight the borders. Image from [3]
The final zero-energy bound state is
φ(x) = αφ1 eλ1 x + βφ1 eλ2 x = αeλ1 x + βe
λ2 x
 
1
 
i
(A.38)
with suitable boundary conditions, for example
 
0
φ(0) =   → α + β = 0.
0
A.4
(A.39)
The SSH Hamiltonian
The Su-Schrieffer-Heeger Hamiltonian describes a long dimer chain with staggered
hopping amplitudes. The chain consists of N unit cells, each of them made up
from two sites belonging to different sublattices. Interaction between the electrons
and the spin of the electrons are neglected, so the Hamiltonian is written like
H=v
N
X
(|m, Bi hm, A| + h.c.) + w
m=1
N
−1
X
(|m + 1, Ai hm, B| + h.c.)
(A.40)
m=1
An equivalent form, that highlights the spinorial fashion of the Hamiltonian, is
H=v
N
X
m=1
|mi hm| σx + w
N
−1
X
m=1
|m + 1i hm|
σx + iσy
+ h.c.
2
(A.41)
84
APPENDIX A. USEFUL MATERIAL
We are interested in the edge states of the model. Let’s consider two cases: in
one case, the intra-cell hopping v is set to 1 and the inter-cell hopping w to zero,
the other case is the opposite. In this limit, that is called fully dimerized, we can
choose a set of energy eigenstates restricted to one dimer. These consist of an even
(odd) superposition of the two sites forming the dimer:
H(|m, Ai ± |m, Bi) = ±(|m, Ai ± |m, B)i
(A.42)
where the energy is E = ±1.
In the opposite case, each dimer is split in two adjacent unit cells, so we have
H(|m, Bi ± |m + 1, Ai) = ±(|m, Bi ± |m + 1, A)i
(A.43)
Now, in the trivial case all energy eigenvalues of the SSH chain are those presented
in (A.42). In the topological case, instead (A.43) is missing the two eigenstates
living on the ends of the chain, with 0 energy as there is no on-site potential. We
have
H |1, Ai = H |N, Bi = 0
(A.44)
What happens if we move away from the dimerized limit, that is increase v? The
zero of energy is in the bulk band gap, so states of almost zero energy have to
remain confined ad the edges. This is true as long as v is less then w, then the
system enters the trivial phase.
A.4.1
Exact calculation of edge states
Let’s take our SSH Hamiltonian
H=v
N
X
(|m, Bi hm, A| + h.c.) + w
m=1
N
−1
X
(|m + 1, Ai hm, B| + h.c.)
m=1
(A.45)
85
APPENDIX A. USEFUL MATERIAL
for a finite chain, and look for a zero-energy eigenstate, that is
H
N
X
(am |m, Ai + bm |m, Bi) = 0.
(A.46)
m=1
This condition gives a system of equations for the coefficients:
vm am + wm am+1 = 0
(A.47a)
wm bm + vm+1 bm+1 = 0
(A.47b)
vn aN = 0
(A.47c)
v1 b1 = 0
(A.47d)
that are solved by
am =
m−1
Y
j=1
−vj
a1
wj
N −1
−vN Y −vj
bm =
bN
wN j=m+1 wj
m>1
m<N
b 1 = aN = 0
(A.48a)
(A.48b)
(A.48c)
that in the generic case give no solution, as all together they imply am = bm = 0.
However, in the thermodynamical limit N → ∞ when v < w if we define
N −1
1 X
log |v| =
log |vm |
N − 1 m=1
N −1
1 X
log |w| =
log |wm |
N − 1 m=1
(A.49)
then (A.48a) and (A.48b) can be rewritten as
|aN | = |a1 |e−(N −1)/ξ
|b1 | = |bN |e−(N −1)/ξ
(A.50)
86
APPENDIX A. USEFUL MATERIAL
where we conveniently defined the localization length
ξ=
1
(A.51)
log |w| − log |v|
and N → ∞ satisfies (A.48c).
In the thermodynamic limit, the two values at the denominator of the previous
formula can be seen as "bulk average values". If the localization length is positive,
then, we have two approximate zero-energy states
|Li =
N
X
am |m, Ai
m=1
|Ri =
N
X
bm |m, Bi
(A.52)
m=1
where the coefficients a1 and bN fix the norm and all the others are calculated
following (A.48a) and (A.48b).
Appendix B
MATLAB code
In the following appendix are included significant sections of the MATLAB scripts
used to obtain the results listed in the thesis, with an explanation of the used
procedure.
B.1
Band diagrams
The following code was used to produce the band diagram in the Weyl spinor
picture. First of all, the matrices have to be initialized. Disregarding the reference
frame chosen, we will always express the Hamiltonian as
H = Hx + Hy + Hz + V
(B.1)
where Hi contains the hopping elements in the i direction, and V takes into account
the external tuning used to move about the Fermi arcs.
1
Hx=cell(Np,Np);
2
[Hx{:}] = deal([0,0;0,0]);
3
Hy=cell(Np,Np);
4
[Hy{:}] = deal([0,0;0,0]);
87
APPENDIX B. MATLAB CODE
5
Hz=cell(Np,Np);
6
[Hz{:}] = deal([0,0;0,0]);
7
P=cell(Np,Np);
8
[P{:}] = deal([0,0;0,0]);
9
Q=cell(Np,Np);
10
88
[Q{:}] = deal([0,0;0,0]);
All the variables before are of cell type, that is in this case N × N matrices of
2 × 2 matrices. For the original cut along x̂ + ŷ the initialization of V is simple:
1
for n=1:Np
P{n,n}=epsilon*[0,1;1,0];
2
3
end
4
for n=1:Np
Q{n,n}=lambda*[1,0;0,-1];
5
6
end
7
V=cell2mat(P)+cell2mat(Q);
where the last line flattens the matrix so that it can be diagonalized afterwards
and we have taken the general case V = σx + λσz . By the way, the part tunes
the intra-spinor hopping, and this naive implementation of the code gives rise to
the curvilinear arcs we saw in Chapter 2.
The next step is to fill the matrices using a nested loop spanning the Brillouin zone
√
√
and the band number (from 1 to 2N ). For kp ∈ [−π/ 2, π/ 2] and kz ∈ [−π, π],
For Hx
1
right=exp(-i*kp*sqrt(2));
2
left=exp(i*kp*sqrt(2));
3
for n=1:Np
4
Hx{n,n}=[0,-1;-1,0];
5
if n+1 < Np+1
APPENDIX B. MATLAB CODE
Hx{n,n+1}=[0,0;right,0];
6
7
end
8
if n-1 >0
Hx{n,n-1}=[0,left;0,0];
9
end
10
11
89
end
For Hy
1
right=exp(-i*kp*sqrt(2));
2
left=exp(i*kp*sqrt(2));
3
for n=1:Np
4
Hx{n,n}=[0,-1;-1,0];
5
if n+1 < Np+1
Hx{n,n+1}=[0,0;right,0];
6
7
end
8
if n-1 >0
Hx{n,n-1}=[0,left;0,0];
9
end
10
11
end
For Hz
1
for n=1:Np
Hz{n,n}=[2,0;0,-2]*cos(kz);
2
3
end
It is then merely a matter of flattening via cell2mat, summing up and diagonalizing. the plotting is done via the standard scatter MATLAB function.
This is the standard code to obtain all the band diagrams presented in the main
text. All that changes is the definition of the ks and the matrix elements.
APPENDIX B. MATLAB CODE
B.2
90
Numerical way to find Weyl points
To superimpose the position of the Weyl points to the band diagrams, they can
either be calculated as the top or bottom values of the bands immediately preceding
or following the surface ones (that present Weyl cones). It is though better to
calculate them from the bulk Hamiltonian: they are indeed a bulk property and
this procedure serves as a cross-check.
Let’s follow the procedure for the cut along x̂ − ŷ + ẑ case: first we have to write
the rotation matrix
1
R=[1/sqrt(3),-1/sqrt(3),1/sqrt(3);
2
-1/sqrt(2),-1/sqrt(2),0;
3
-1/sqrt(6),1/sqrt(6),2/sqrt(6)];
Then we span the Brillouin Zone creating the bulk Hamiltonian (note how in this
case the tilt term in adds new intra-spinor hoppings, accounting for the curviness
of the arcs)
1
newhop=[0,exp(-i*dot([1,0,-1],[kx,ky,kz]));
2
exp(-i*dot([-1,0,1],[kx,ky,kz])),
3
0];
4
V=epsilon*+lambda*sz;
5
H=-2*(sin(kx)*sy+cos(ky)*sx-cos(kz)*sz)+V;
6
energy=eig(H);
7
value=abs(energy(1));
8
points=[points;kx/pi,ky/pi,kz/pi,value];
APPENDIX B. MATLAB CODE
91
Now we sort the points vector, keeping only the lowest energy points: the
steepness of the cones makes it unlikely to get points belonging to the same cone
and differing by a small k.
1
points=sortrows(points,4);
2
tmp=points(1:4,4);
3
points=points(1:4,1:3);
The points must now be rotated to the new coordinate system, their coordinate in
the finite direction can be forgotten and replaced by the energy value (which was
stored in tmp).
1
points=points.';
2
coord=R*points;
3
coord(2,:)=sqrt(2)*coord(2,:);
4
coord(3,:)=sqrt(6)*coord(3,:);
5
coord=coord.';
6
coord(:,1)=[];
7
coord(:,3)=tmp;
Finally, the point projections must be reconducted back into the first BZ:
1
2
for l=1:4
while coord(l,1)>1
coord(l,1)=coord(l,1)-2;
3
4
end
5
while coord(l,1)<-1
coord(l,1)=coord(l,1)+2;
6
7
end
8
while coord(l,2)>1
coord(l,2)=coord(l,2)-2;
9
10
end
APPENDIX B. MATLAB CODE
while coord(l,2)<-1
11
coord(l,2)=coord(l,2)+2;
12
end
13
14
92
end
B.3
State localization
Here we present the code used to obtain the state localization diagrams: the
procedure is actually the same as in the previous section; however, we are no more
interested in the eigenvalues: what we need are the N -th and N +1-th eigenvectors,
that are 2N -component vectors where the dimension is the length of the slab in
the finite side. For the original cut in the even-planes case, we expect each state
to be localized only on sites belonging to one sublattice. So, fixed kp and kz we
can plot both bands in a single diagram in this way:
1
[eigenvectors,eigenvalues]=eig(H);
2
for m=1:Np
if mod(m,2)== 0
3
out1(m/2,:)=[m/2,abs(eigenvectors(m,Np/2))^2];
4
else
5
out2((m+1)/2,:)=[(m+1)/2,abs(eigenvectors(m,(Np/2)+1))^2];
6
end
7
8
end
B.3.1
Numerical verification of Witten state expression
The previous code can be easily modified to obtain a numerical proof of the correctness of the states analytically derived in Chapter 3. For each step counted by
a counter, to which correspond a distance k_dist from the Weyl point, we obtain the desired eigenstate as the corresponding column of the eigenvectors matrix.
93
APPENDIX B. MATLAB CODE
Then we fit the output with an exponential function, and plot k_dist versus the
negative coefficient at the exponent, as the following code shows
1
[eigenvectors,eigenvalues]=eig(H);
2
for m=1:Np
if mod(m,2)== 0
3
eigenstate(m/2,:)=[m*sqrt(2),abs(eigenvectors(m,Np/2))^2];
4
end
5
6
end
7
f = fit(eigenstate(:,1),eigenstate(:,2),'exp1');
8
toplot(counter,1)=k_dist;
9
toplot(counter,2)=f.b;
Note that the x value in eigenstate is mutiplied by
√
2, to account for the lattice
plane spacing in the finite direction (as in our construction the distance along the
Cartesian directions is between sites, not spinors).
The same procedure is then repeated for the new data, only using a 1st order
polynomial and not an exponential:
1
g = fit(toplot(:,1),toplot(:,2),'poly1');
2
plot(g,toplot(:,1),toplot(:,2))
3
angularcoefficient=g.p1;
The results of the code are shown in Chapter 3.
B.4
Direct space propagation
Here is an example of the code used to generate the wavepacket propagation animations of Chapter 4. This example refers to the cut along x̂ − ŷ + ẑ/2 case, with
on-site energy λ and hopping parameter J.
The first step is to generate a vector of good points, and to do so we must map
94
APPENDIX B. MATLAB CODE
a triplet of numbers (x, y, z) into an index l and vice-versa. This is done via the
function indices(q,Np) defined as
1
function y=indices(q,Np)
2
y = [fix(q/(Np*Np)),
3
fix((q-Np*Np*fix(q/(Np*Np)))/Np),
4
rem((q-Np*Np*fix(q/(Np*Np))),Np)];
Now we start considering all points in a big parallelepiped Np × Np × 4Np , and put
the indices of the "good ones", i.e. those enclosed in the planes parametrizing the
surfaces of our chunk of material in a vector sitelab
1
for x= 0:Np
for y=0:Np
2
for z=0:4*Np
3
if (x+y≥Np/2) && (x+y≤3*((Np/2))) &&
4
5
(x-y≤(Np/2)) && (x-y≥(-Np/2)) &&
6
(z
≥
2*((Np/2)-(x-y))) && (z
≤
2*((3*Np/2)-(x-y)))
sitelab=[sitelab;z+(4*Np+1)*y+(4*Np+1)*(4*Np+1)*x];
7
end
8
end
9
end
10
11
end
12
L=length(sitelab);
The real space Hamiltonian will be L × L, and it’s constructed in the following
way: first we determine from the indices of the site if it belongs to sublattice A or
B, and initialize the on-site energy sign accordingly
1
for m=1:L
2
label=sitelab(m);
3
triplet=indices(label,4*Np+1);
95
APPENDIX B. MATLAB CODE
if mod(triplet(1)+triplet(2),2)==0
4
sublattice=-1;
5
else
6
sublattice=1;
7
8
end
9
H(m,m)=sublattice*lambda;
Then we write the hopping elements for adjacent sites (that is, the non-zero hopping element will be the Hmn element of the Hamiltonian, where m and n are the
0
0
0
site labels of the two lattice sites (x, y, z) and (x , y , z ). So
For hopping along x̂
1
x=triplet(1)+1;
2
y=triplet(2);
3
z=triplet(3);
4
if (x+y≥Np/2) && (x+y≤3*((Np/2))) &&
5
(x-y≤(Np/2)) && (x-y≥(-Np/2)) &&
6
(z
≥
2*((Np/2)-(x-y))) && (z
≤
2*((3*Np/2)-(x-y)))
7
[¬,n]=ismember(z+(4*Np+1)*y+(4*Np+1)*(4*Np+1)*x,sitelab);
8
H(m,n)=-J*sublattice;
9
end
10
x=triplet(1)-1;
11
y=triplet(2);
12
z=triplet(3);
13
if (x+y≥Np/2) && (x+y≤3*((Np/2))) &&
14
(x-y≤(Np/2)) && (x-y≥(-Np/2)) &&
15
(z
≥
2*((Np/2)-(x-y))) && (z
≤
2*((3*Np/2)-(x-y)))
16
[¬,n]=ismember(z+(4*Np+1)*y+(4*Np+1)*(4*Np+1)*x,sitelab);
17
H(m,n)=J*sublattice;
18
end
96
APPENDIX B. MATLAB CODE
For hopping along ŷ
1
y=triplet(2)+1;
2
z=triplet(3);
3
if (x+y≥Np/2) && (x+y≤3*((Np/2))) &&
4
(x-y≤(Np/2)) && (x-y≥(-Np/2)) &&
5
(z
≥
2*((Np/2)-(x-y))) && (z
≤
2*((3*Np/2)-(x-y)))
6
[¬,n]=ismember(z+(4*Np+1)*y+(4*Np+1)*(4*Np+1)*x,sitelab);
7
H(m,n)=-J;
8
end
9
10
x=triplet(1);
11
y=triplet(2)-1;
12
z=triplet(3);
13
if (x+y≥Np/2) && (x+y≤3*((Np/2))) &&
14
(x-y≤(Np/2)) && (x-y≥(-Np/2)) &&
15
(z
≥
2*((Np/2)-(x-y))) && (z
≤
2*((3*Np/2)-(x-y)))
16
[¬,n]=ismember(z+(4*Np+1)*y+(4*Np+1)*(4*Np+1)*x,sitelab);
17
H(m,n)=-J;
18
end
For hopping along ẑ
1
y=triplet(2);
2
z=triplet(3)+1;
3
if z
≤
2*((3*Np/2)-(x-y))
4
[¬,n]=ismember(z+(4*Np+1)*y+(4*Np+1)*(4*Np+1)*x,sitelab);
5
H(m,n)=J*sublattice;
6
end
7
8
x=triplet(1);
9
y=triplet(2);
APPENDIX B. MATLAB CODE
10
z=triplet(3)-1;
11
if z
2*((Np/2)-(x-y))
≥
12
[¬,n]=ismember(z+(4*Np+1)*y+(4*Np+1)*(4*Np+1)*x,sitelab);
13
H(m,n)=J*sublattice;
14
97
end
Now the gaussian wavepacket has to be initialized: we assume it is centered in
the middle of the top surface, with width in the two plane direction adjusted to
match the length of the sides. The two-dimensional wavepacket is discretized on
the surface, so the initial wave we will evolve is a L-element vector, of which only
some elements, corresponding to sites on the top surface, are initially 0. The
phase, that positionizes the wavepacket in the band diagram, is chosen before the
elements are initialized.
1
gausscenter=[Np/2;Np/2;3*Np];
2
sigma1=2;
3
sigma2=6;
4
wave=[];
5
tilt=[[1/sqrt(2),1/sqrt(2),0];
6
[-1/sqrt(18),1/sqrt(18),4/sqrt(18)];
7
[2/3,-2/3,1/3]
8
];
9
10
for m=1:L
11
label=sitelab(m);
12
triplet=indices(label,4*Np+1).';
13
if triplet(1)-triplet(2)+triplet(3)/2==3*Np/2
14
dist=tilt*(triplet-gausscenter);
15
phase=exp(i*kpar*dist(1)+i*kz*dist(2));
16
wave=[wave;exp(-(dist(1)/(sqrt(2)*sigma1))^2
-(dist(2)/(sqrt(2)*sigma2))^2)*phase];
17
18
19
else
wave=[wave;0];
98
APPENDIX B. MATLAB CODE
end
20
Then, the wave has to be normalized and the eigenvalues and the coefficients of
the eigensystem of the Hamiltonian, which enter the time evolution equation for
the state
Ψ(t) =
X
cn e−iEn t φn
(B.2)
n
have to be determined by
1
normalization=norm(wave);
2
wave=wave./normalization;
3
4
[evecs,evals]=eig(H);
5
evals=diag(evals);
6
7
for m=1:L
c(m)=dot(evecs(:,m),wave);
8
9
end
and finally the wavepacket is evolved in the usual way
1
wavetime=zeros(L,1);
2
for m=1:L
wavetime=wavetime+(c(m)*exp(-i*evals(m)*t)*evecs(:,m));
3
4
end
5
wavetime=wavetime./norm(wavetime);
6
modsquare=abs(wavetime).^2;
This concludes the physics of the process. It is now a matter of plotting the states
in a convenient way; for each time step, we create a normalized vector greyscale,
each element of which is the squared modulus of the amplitude of the wave at the
corresponding site normalized to 1. Although this is not a perfect convention, as it
favours very broadened wavepackets over narrow ones, which tend to be depicted
APPENDIX B. MATLAB CODE
99
as very few dots cutting the tails, it still is adequate, as it depicts well the center
of the packet, that to all extents evolves like a classical wave with group velocity
obtained by the band diagram, and is the point that matters in following the
evolution of the wave. So, what the program will print as output file will be given
by
1
greyscale = modsquare./max(modsquare);
2
for m=1:L
3
label=sitelab(m);
4
triplet=indices(label,4*Np+1);
5
output(m,:)=[triplet,greyscale(m)];
6
end
for each frame. To plot the data files in this case the standard tool splot of the
gnuplot program with greyscale color palette has been used.
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