5.5 Time-Reversal Invariant Topological Insulators with Cold Atoms
6.1.2 Topological Edge States and the Bulk Boundary Correspondence
In the previous section, we identified the quantum Hall effect as a topological feature of an infinite system.
To drive a quantum Hall system, or more general, a topological non-trivial system, into a topologically trivial system, the topology of the system must be changed. In general, changing the topology is possible by allowing bulk bands, which were formerly well separated by a bulk gap such that the Chern number of both bands was well defined, touch each other at certain points in the Brillouin zone. When two bands touch each other, the Chern number, i.e. the topology, of the individual bands is no longer well defined.
Further deformation of the system will lead to splitting of the bands, such that the Chern number is well defined again but may have changed during the process of deformation.
Now imagine a finite topologically non-trivial system. The boundary of the system can be seen as an interface between a topologically non-trivial system to a topologically trivial system, namely the vacuum.
Somewhere on the way from the center to the boundary of the system, the energy gap has to vanish, since this is required for changing the topological invariant from a finite integer to zero. Therefore, there will exist low energy states, which are bound to the region where the energy gap vanishes. From a topological point of view, these low energy states must also be very robust against external perturbations since, as with the Chern number of the infinite system, they exist due to purely topological arguments. As first shown by Halperin [77], these low energy states are located close to the boundary of the system, decaying exponentially into the bulk and are therefore referred to as (topological) edge states. Moreover, the edge states are chiral, which means they propagate in one direction only along a given edge of the system, and therefore are responsible for the strictly quantized Hall conductance in finite systems. The chirality of the edge states is also the reason for their robustness against external perturbations or interactions, as backscattering processes can only occur when two or more states are counterpropagating. Without backscattering, hybridization or localization of the states is impossible.
Since the introduction of edge states by Halperin [77] and the introduction of the TKNN invariant for the bulk system [182], many publications have proven that the Hall conductance obtained from the infinite system via TKNN coincides with the one obtained from the analysis of the edge states in finite system, which is known as the bulk-boundary correspondence [79, 71, 44].
We will now derive a very simple expression for the quantized Hall conductance in terms of edge states.
For this purpose, we imagine a two-dimensional lattice system on a cylindrical geometry, which consists ofMy lattice sites and lattice spacingay in they-direction andMxlattice sites and lattice spacingaxin thex-direction. While the system shall be translationally invariant in they-direction, it is confined in the x-direction2. The confining potential shall not be specified, the only restriction we make for this potential is that it still allows for a clear distinction between bulk bands and bulk gaps, while the latter may now
2This is realized by periodic boundary conditions in they-direction and choosing the Landau gauge for the vector potentialA.
contain gapless edge states which connect the different bulk bands of the system and are located close to the boundary of the system, i.e. states from different boundaries have no real space overlap. Because the system is translationally invariant in they-direction, the quasi-momentumky remains a good quantum number, such that the eigenstates and eigenenergies of the system can be labeled as|ψα(ky)i, α(ky), respectively, whereαis an additional label but not a band index. Fig. 6.1 shows the integrated spectral density
ρ(ky, ω) =X
α
hψα(ky)| 1
ω−H+i0+|ψα(ky)i (6.10)
of possible systems, where one can clearly distinguish between bulk band regions and bulk gap regions, which are now traversed by gapless edge states. The particles in the system are considered to be electrons,
Figure 6.1: Schematic illustration of the band structure of a finite system for different topological invari-antsN = 0,1,2(from left to right). The red region indicates a completely filled band, while the green region indicates a empty band in between the two bands, there is a bulk gap, filled with single edge modes, which are located at the left (red) and right (blue) boundary of the system. The Fermi energy is located in the bulk gap, such that the system finds itself in a normal insulator forN = 0, in quantum Hall state for N= 1and in a quantum Hall state with increased Hall conductance forN = 2. The topological invariant Nis obtained according to Eq. (6.18).
with the elementary chargeq=−eand a Fermi energy located in a bulk gap. Now, a voltageVxis applied in thex-direction, which is sufficiently small, such that the energyEV =|eVx|<< Γis much smaller than the bulk gap. This voltage leads to a linearly shifted chemical potential along thex-direction, i.e.
µ(x) =F−eVx
x Mx
, (6.11)
such that the difference in the chemical potential between the left and the right boundary of the system is∆µ = eVx. Since the bulk of the system is gapped, the change in the chemical potential does not influence the bulk at all. On the other hand, there are gapless states at the edges of the system, which will change their occupation according to the changed chemical potential. Therefore, since the chemical potential is larger at the left edge, there will be more states populated on the left edge than on the right edge, leading to an asymmetry in the current on both sides and therefore to a net current densityjyin the whole system. This current density is determined by
jy= Iy
Myay
=− e Myay
X
α0
vα0(ky) =− e
~Myay
X
α0
∂α0(ky)
∂ky
, (6.12)
whereIyis the total current in they-direction,vα(ky) =∂kyα(ky)/~is the velocity associated with the state|ψα(ky)iand the sum runs over all statesα0 that are occupied on the left side but unoccupied on the right side of the system. In the limit of a weak perturbation, this can be simplified to
jy =− e
~Myay
∂F(ky)
∂ky
∆n, (6.13)
whereF(ky)is the energy of the edge states at the Fermi surface and∆nis the number of additionally occupied states. The chemical potential on the left boundary of the system has changed by∆µ=eVx, due
6.1. The Quantum Hall Effect 93
to the applied voltage. The number of additionally occupied states in the energy interval[F, F +eVx] shall now be expressed in terms of the voltage. Therefore, we rewrite the change in the chemical potential in terms of thek-space volume∆kythat is available due to the energy change
eVx= ∆µ=
∂F(ky)
∂ky
∆ky. (6.14)
In the last equation, the absolute value of the derivative of the dispersion is taken because both thek-space volume and the energy shift are defined as being positive. The number of states in thisk-space volume is straight-forward to determine. In a lattice ofMysites, with lattice spacingay, the quasi-momentumkyis distributed overMyequally distributed values in the interval[−π/ay, π/ay]and therefore
∆ky= 2π∆n
Myay. (6.15)
Combining equations (6.13)-(6.15) leads to the Hall current jy=− e2
2π~
∂F(ky)
∂ky
∂F(ky)
∂ky
−1
Vx=−e2 hsgn
∂F(ky)
∂ky
Vx (6.16)
and therefore to the Hall conductance σxy= jy
Vx
=−e2 hsgn
∂F(ky)
∂ky
(6.17) which is evidently quantized in terms ofe2/h. This derivation for the Hall conductance was performed for the case where a single edge state is present on the left edge of the system. For the case of multiple edge modes, this derivation can easily be extended to represent the contribution of every single edge mode to the Hall conductance as long as they are well separated ink-space. Therefore, the combined Hall conductance for an arbitrary number of edge states is given by the sum
σxy=−e2 h
X
α
sgn
∂α(ky)
∂ky
, (6.18)
whereαlabels the edge modes crossing the Fermi edge andα(ky)is the dispersion of theα-mode at the Fermi edge.
The derivation of Eq. (6.18) presented in this thesis is very simple and does not directly rely on any topological arguments, however, the presence of edge states, which is required for a non-zero value of σxy, is a result of the topology of the system, as argued above. However, the integer
ν =X
α
sgn
∂α(ky)
∂ky
(6.19)
can also be interpreted in terms of topology, as pointed out by Hatsugai [79]. In his analysis, he found that (6.19) is the winding number of the edge states on a Riemann surface, which is obtained by analyt-ically continuing the energy onto the complex plane. Additionally, Hatsugai showed that this number is identical to the total Chern number of the occupied bands and therefore proved the bulk-boundary cor-respondence. Whereas Eq. (6.18) is independent of the confining potential, the formulas for the Hall con-ductance in [79] are derived only for the case of open boundaries but without a trapping potential. While we present a generalized formulation of the Hall conductance for finite systems (6.18), which is formally equivalent to the one obtained in [79] for the open system, no proof exists for the bulk-boundary corre-spondence to hold for arbitrary confining potentials. For a certain confining potential, the bulk-boundary correspondence has then to be validated individually by either numerical or analytical calculations (see section 6.3 for the case of potentials relevant in optical lattice experiments).