Bulk and edge band structure of the Haldane model

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Topology in condensed matter physics

The Berry Phase

Consider some time-varying Hamiltonian H(R) where is R ≡ R(t) is a vector of parameters that depends on time. Now, we want to investigate the evolution of the system when moving adiabatically, i.e. slowly in comparison to other energy scales [11] along some path through parameter space. To this end, we diagonalize the Hamiltonian H(R) at each point in the parameter space and orthonormal eigenbasis |n(R)〉. In fact, the eigenbasis |n(R)〉 can be determined up to a phase factor. In order to avoid arbitrariness (which can be interpreted as choice of gauge) of this phase factor, we require the phase to evolve smoothly when moving along a path in parameter space spanned by the components of the parameter vector R.
Now assume that we start moving along a path in parameter space, where we start with the inital eigenstate |n(R(0))〉. According to the adiabatic theorem [28], the system described by the Hamiltonian H(R) and starting in the inital, in-stantaneous eigenstate |n(R(0))〉 will along an adiabatic drive through parameter space remain in its eigenstate. Now, we assume that the phase factor mentioned above is indeed the only degree of freedom that remains to be computed [11]. Let us define the phase factor θ (t) as [12, 11] |Ψ〉 = exp(−iθ (t))|n(R(t))〉. (2.1)
Therefore, the energy evolution of the system is described by the equation [11] H(R(t))|Ψ〉 = ih¯ d |Ψdt
The phase θ (t) can in fact not be zero, since it needs to contain at least capture the energetic evolution of the eigenstate through parameter space, the so-called dynamic phase [28]. To our surprise, solving the differential equation associated with Eq. 2.2 [11] yields more than that, i.e. θ (t) = 0 En (R(t′))dt′ − i 〈n(R(t′))| d |n(R(t′)〉dt′. (2.3) h¯ 0 dt
The first term is the dynamic phase related to an energy integral. The second part is an – a priori – unexpected term which is called the Berry phase which we denote by γBerry , i.e. γBerry = i 0 〈n(R(t′))| ddt |n(R(t′)〉dt′. (2.4)
The Berry phase arises from the fact that the states at t and t + dt are not iden-tical [11], or in other words, it originates from the geometrical properties of the parameter space of the Hamiltonian [12]. From now on, let us consider only closed paths C in paramter space. First, let us write Eq. 2.4 without the direct time dependance as [11] γBerry = i 〈n(R)|∇R |n(R)〉dR. (2.5)
Then, we can in analogy to transport on manifolds define the Berry connection as A(R) = i〈n(R)| ∂ |n(R〉, (2.6) ∂R and then again with this definition the Berry phase is γBerry =dR • A(R). (2.7) C
Now, we remind ourselves that the eigenstates |n(R)〉 are determined up to a phase, i.e. |n(R)〉 → exp(iθ )|n(R)〉, where θ is here a smooth function in pa-rameter space θ ≡ θ (R). From a physical point of view, it would be appropriate to call the Berry connection A(R) in fact the Berry vector potential. As such, it is gauge dependent according to the the choice of θ, i.e. A(R) → A(R) − ∂θ(R) ∂R . (2.8)
Hence, the Berry phase as line integral of the Berry vector potential, Eq. 2.7 changes by [11] γBerry → γBerry + θ (R0 ) − θ (R1 ) (2.9)
where R0 and R1 are the start and end point of the path C, respectively. Since C is a closed path, we must have R0 = R1, and because the eigenstate basis is here chosen single valued [11], we also have |n(R0 )〉 = |n(R1 )〉 since we moved along C adiabatically. Therefore, in the case under consideration here, the only possible solution for the closed path is [11] θ(R0) − θ(R1) = 2πω (2.10)
where ω must be an integer. The number ω can be interpreted as a winding number where the sign of ω determines the orientation with which we move around the path C
Furthermore, note that in the case of graphene, which we will have a closer look at in chapter 2.2.2, we consider a lattice Hamiltonian with chiral symmetry. We will point out that encircling the so-called Dirac points of the hexagonal lattice will yield Berry phases of ±π such that we can identify these points as topological defects.

Berry curvature, first Chern number and Hall conductivity

For a closed path, we can make use of Stokes theorem [7] so that we can Eq. 2.7 express as γBerry = dR • A(R) = dS • (∇R × A(R)) . (2.11)
Here, we transformaed a line integral along the closed path C into a surface in-tegral over the surface S where ∂S = C. Also, note that we assumed here a two dimensional parameter space such that we could use the rotation operator ap-plied to the Berry connection, i.e. ∇R × A(R). This expression is called the Berry curvature F , i.e. F(R) ≡ ∇R × A(R) (2.12)
which reads (in this two dimensional case) explicitly [7, 11] Fi j (R) = ∂Ai (R) − ∂Aj (R)
We saw previously that the Berry connection needs to be integrated to result a physical quantity, namely the Berry phase. That means, the Berry connection is as vector potential primarily a mathematical tool (in the same way as the electro-magnetic vector potential in electrodynamics [29]). On the other hand, the Berry curvature is a gauge-invariant local manifestation of the geometric properties of the wavefunctions in the parameter space [30]. Hence, we can make the analogy between the Berry curvature and the magnetic field in electrodynamics [29].
Having the notion of the Berry phase and the definition of the Berry curvature at hand, we can now define the first Chern number. In direct correspondance with the insight we gained for the Berry phase, cf. Eq. 2.10, we define the first Chern number as [7] C = 1 dS F (R) (2.14)
Here, the integral is defined over a closed surface S′ (i.e. without boundary). In practice, this surface will be either the sphere or a torus. For example in chapter 2.1.3, we assume the Brillouin to have periodic boundary conditions, such that it can be mapped on a torus. Furthermore, note that we can always make the connection to Eq. 2.11 where S was a surface with boundary, by cutting the closed surface S′ into two pieces, e.g. the sphere can be cut into two hemispheres.
With our previous discussion of the Berry phase, this formula seems well mo-tivated. From the physical side however, motivating the first Chern number can-not be done without mentioning the Integer Quantum Hall Effect (IQHE) [6]. Von Klitzing [6] realized the IQHE in a two-dimensional electron gas which was ex-posed to a homogenous magnetic field oriented perpendicular to the gas. Experi-mentally, it was found that the system is an insulator in the bulk, i.e. longitudinal elements of the conductivity tensor σ vanish, i.e. σx x = σyy = 0. On the other hand, the transverse element σxy was found to be quantized as [11] σxy = e2 C (2.16)
where C is an integer, and is indeed the first Chern number. The gas being a bulk insulator, the transverse conductivity must be connected to transport properties at the edges [31]. Indeed, chiral edge modes carry C units eh2 of conductance, where the sign of C determines the orientation of the edge transport. Mathe-matically, the connection with first Chern number was described by Thouless, Kohmoto, Nightingale, and de Nijs [32] who showed that the number C could in-deed by computed using Eq. 2.15. The first Chern number is therefore sometimes also called the TKNN invariant [31].
One important aspect of a system with a non-zero Chern number is its phe-nomenology in experimental realizations. The question is, how can it be that we obtain e.g. in a system that shows an IQHE a bulk insulator and a non-zero Hall conductivity which indicates transport taking place in the system [11]. The solu-tion to this question points to the occurence of edge modes, i.e. conductive modes that only occur at an interface of the system at hand with another system that has a different topological bulk invariant. While we will at this point only consider the following intuitive argument, we will in the following sections closer look at systems that exhibit such modes, namely the Haldane and Kane-Mele models. If we create an edge in a material hosting a bulk non-zero topological number, the material is interfaced with the vacuum which has trivial topological order. This induces a mismatch in terms of topological invariants at the edge which can only be resolved by the system through the creation of gapless edge or surface states [11] . This link between a topological invariant of a system and the emergence of surface states is the bulk-boundary correspondence

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Gauge independent numeric computation of the Berry curvatur and Chern number

The parameter space R introduced in previous sections, will in practice be for the momentum space in two dimensions. Explicitly, we discretize Eq. 2.14 and 2.15 in momentum space using k = 2π l (2.17)
where the Brillouin zone of the momentum space is divided into N discrete points and lx,y = 1, . . . , N − 1. Note that we assume periodic boundary conditions for the Brillouin zone such that it can effectively be mapped on a torus. The Chern number1 and Berry curvature, cf. Eq. 2.14 and 2.15, then read C = 1 ∑ dkx dky F (k) (2.18) 2π k∈BZ and F (k) = ∂Ax (k) − ∂Ay (k) . (2.19) ∂ky ∂kx
Although it might be tempting to just use these equations in order to compute the local Berry curvature or the global Chern number, it is unfortunately not that straight forward. The Berry connection A is not gauge independent. In prac-tice, this means that we get a different random phase when we diagonalize the Hamiltonian of the system one by one for each fixed wave vector k such that the resulting eigenstates are not smoothly connected. In order to compute the Berry curvature and Chern number in a gauge independent manner, we follow Ref. [33].
First, we note that we can change the gauge of an eigenstate |n(k)〉 at wave vector k according to U(1) gauge transformation |n(k)〉 → exp(iθ )|n(k)〉 (2.20) where θ is a smooth function.
On the level of the Berry connection, this results in the gauge transformation that we saw in Eq. 2.8. Then, we define U(1) link variables which capture relative phases of eigenstates on neighboring lattice sites as [33] Uµ (k) = 〈n(k)|n(k + µ)〉 (2.21) |〈n(k)|n(k + µ)〉|
where µ = x, y, µ is the unit vector in direction µ with length 2π/N, and the eigenstate |n〉 is the eigenstate corresponding to the n-th band.
Next, we define a lattice field strength by taking the product of all relative phases around the boundary of a plaquette (which consists of neighboring lattice sites at each vertex) [33]
F˜ (k) = log #Ux (k)Uy (k + x)Ux−1 (k + y)Uy−1 (k)$ (2.22)
where we select the default branch of the logarithm as < ˜ k / . In −π F ( ) i ≤ π fact, we have ˜ k k if it holds that < ˜ k / . If ˜ k is outside F ( ) = F ( ) −π F ( ) i ≤ π F ( )
this range, it means that we have vortex in the plaquette [8] (which as we shall see later on relates to a singularity in the corresponding wave functiion), and we can bring ˜ k back to , which means that we picked effectively up phase F( ) (−π π] factor of 2 . Thus, the field strength ˜ k counts the net number of vortices in π F ( ) the Brillouin zone and the sum C = 1 ∑ ˜ F (k) (2.23) 2π k∈BZ is the Chern number.
Finally, we refer to Fig. 2.1 which describes numerical method for the re-construction and visulzation of the Berry curvature in the Brillouin zone. This method will be used several times in this PhD thesis.

Table of contents :

1 Introduction 
2 Concepts of topological band theory 
2.1 Topology in condensed matter physics
2.1.1 The Berry Phase
2.1.2 Berry curvature, first Chern number and Hall conductivity
2.1.3 Gauge independent numeric computation of the Berry curvature and Chern number
2.2 Graphene
2.2.1 Hexagonal lattice structure
2.2.2 Graphene tight binding Hamiltonian
2.2.3 Symmetry protection of the Dirac cones
2.3 The Haldane honeycomb model
2.3.1 Quantum Hall state in graphene without external magnetic field
2.3.2 The Haldane Hamiltonian
2.3.3 Haldane phase diagram
2.3.4 Bulk and edge band structure of the Haldane model
2.3.5 Probing Chern numbers via the circular dichroism of light
3 Topological proximity effects in the Haldane-graphene model 
3.1 The Haldane-graphene model
3.2 Proximity effect in the Haldane-graphene model
3.3 Mathematical description of the Berry phase shift
3.3.1 Singularities in the eigenstates of the Haldane-graphene model
3.3.2 Lifting the singularities
3.4 Edge properties and strong coupling limit
3.4.1 Counter-propagating edge modes at different velocities
3.4.2 Strong coupling limit
3.5 Experimental realization
3.5.1 The Haldane honeycomb model in ultra cold atoms
3.5.2 The case of the bilayer system
3.5.3 The Haldane-Haldane model
3.6 Conclusion
4 Interaction effects in the Haldane honeycomb model 
4.1 Stochastic variables and Mean field theory
4.1.1 The model Hamiltonian
4.1.2 General remarks on the decoupling scheme
4.1.3 Decomposition of the quartic term
4.1.4 Hubbard-Stratonovich transformations
4.1.5 Self consistent mean field equations from a variational approach
4.1.6 Numeric solution to the self consistent mean field equations
4.2 Energetic analysis of the phase transition
4.3 Probing topology with light response
4.3.1 Circular dichroism of light at the Dirac points
4.3.2 Ground state circular dichroism
4.4 Stochastic Chern number
4.4.1 Stochastic Topological Number and Interpretation as a Disordered Situation
4.4.2 Light-Matter Response and Mott Transition
4.4.3 Energy distribution of excited quasi particles
4.4.4 Analogy with Temperature Effects
5 Analytical approach to the Kane-Mele-Hubbard model 
5.1 The Kane-Mele model
5.1.1 Model Hamiltonian
5.1.2 The Z2 topological invariant
5.2 The Kane-Mele-Hubbard model
5.3 The Kane-Mele-Hubbard model from a variational principle
5.3.1 Kane-Mele-Hamiltonian decoupling scheme
5.3.2 Hubbard-Stratonovich transformations
5.3.3 Interaction density matrix
5.3.4 Self-consistent mean field equations
5.3.5 Solution to the self-consistent mean field equations
5.4 Analytical approach to the Kane-Mele-Hubbard model
5.4.1 Decomposition and Hubbard-Stratonovich transformation
5.4.2 Transition line from saddle point conditions
5.5 Conclusion and comparison of the two methods
6 Conclusion 
Acknowledgments
Bibliography 

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