Topological Insulators
 a beginners guide 
This is an informal webresource to give
some background info useful to those attending the focus session on TI's at the
Physics@FOM Veldhoven meeting 2011.
Focus session 3
Location: Auditorium
Time: 11:0013:00, Tuesday 18th January
2011.
Chair: Mark Golden (UvA)
Speakers:
11:00  11:25 Laurens Molenkamp (Würzburg)
11:30  11:55 Carlo Beenakker (LEI)
12:00  12:25 Menno Veldhorst (UT)
12:30  12:55 Kareljan Schoutens (UvA)
I've borrowed content from numerous
webpages and pdfs of people's talks on the web to put this beginner's guide
together. Particular thanks in this regard to Joel Moore and ShouCheng Zhang.
For the more formally theoretically minded among you: I've very probably made
mistakes in trying to keep this short enough: apologies for that.
A more exhaustive overview can be found at http://rmp.aps.org/pdf/RMP/v82/i4/p3045_1.
Content
1. Topology
2. Order
3. Topological phases
4. Edge states in IQHE and 2D topological
insulators
5. Examples of 2D topological insulators
and their edge states
6. 3D Topological insulators
7. Why the excitement?
8. Outlook
1. Topology.
From
wiki:
Topology
(from the Greek τόπος, "place", and
λόγος, "study") is a major area of mathematics
concerned with spatial properties that are preserved under continuous
deformations of objects, for example, deformations that involve stretching, but
no tearing or gluing.
This means that an orange and a bowl are in
the same topological class, but this is a different class to a coffee mug or
doughnut: the latter two possessing a hole (they are homeomorphic).
The animated .gif shows how one can morph
between the doughnut and coffee mug without tearing or gluing.

A continuous deformation (homeomorphism) of a coffee cup into a doughnut (torus) and back. From en.wikipedia.org/wiki/Topology 
2. Order
In physics, an ordered state appears at low temperature when the system spontaneously loses one of the symmetries present at high temperature.
For example: crystals break translational and rotational symmetries of free space; ferromagnets break timereversal symmetry and superconductors break gauge symmetry.
Since the discovery of the (integer) quantum Hall effect in 1980, we know there are types of order that can occur without symmetry breaking. When a 2D electron gas is sufficiently cooled and placed in a strong enough magnetic field, the Hall conductance, σ_{xy}, exhibits plateaus with the quantised value n(e^{2}/h). This quantisation is accurate to 10^{9}, and is a result of topological order.
3. Topological phases
A topologically ordered phase will have a response function which is given by a topological invariant. The latter is a quantity that does not change under continuous deformation (like the number of holes in the coffee mug & doughnut). More precisely, many topological invariants come about from the integration of a geometric quantity. Both an orange and a doughnut are closed surfaces. The GaussBonnet theorem states that the integral of the curvature over the whole surface is a topological invariant.
k = (r_{1}r_{2})^{1}
The three surfaces on the left have  at their equators  negative, zero and positive Gaussian curvature 
GaussBonnet theorem says that for the closed surfaces below (sphere or torus):



In this equation g stands for the genus (the number of holes), being 0 for a sphere and 1 for the doughnut. 

In solid state systems like the integer quantum Hall effect, and topological insulators, the Brillouin zone plays the role of the surface and the Berry phase plays the role of the curvature.
The Berry (or geometric) phase is a phase
acquired over the course of a cycle, when the system is subjected to cyclic
adiabatic processes, resulting from the geometrical properties of the
parameter space of the Hamiltonian. For a crystalline solid, the parameter
space for the Berry phase is the crystal momentum, and the change of the
electron wavefunction within the unit cell gives a Berry connection (A) and a
Berry curvature (F):

The n in the Hall conductivity, is an integral of F, and is called the first Chern number.
In the integer quantum Hall effect, the Landau levels are either filled or empty, making it, in this sense like a band insulator (filled or empty bands with an energy gap in between). However, in the quantum Hall plateaus, there is finite (and very well defined) conductivity. How can this be?
The answer is that there are conducting edge states, forming the border of the topologically ordered phase (the 2D quantum Hall sample) and the 'regular' insulator  e.g. vacuum  surrounding the system.
4. Edge states in IQHE and 2D topological insulators
In fact, these edge states are chiral  acting as oneway conductance channels: the quantum spin Hall effect. The nice thing is that the topological invariant of the bulk 2D (topological) material tells us how many such 1D, chiral 'wires' there have to be at the surface of the system. These edge states are said to be topologically protected.
Going beyond quantum Hall systems, 2D topologically ordered, insulating phases can be found, in which the spinorbit coupling plays the role of the magnetic field in the QHE (in the former, timereversal symmetry is not broken, in the latter  through the magnetic field  it is).
It has been shown that at the boundary between a 2D topological insulator and an ordinary insulator (or vacuum), edge states occur in which spin up and spin down electrons are in IQHE like states, but each feels an opposite effective magnetic field, arising from the spinorbit coupling. See the pictures from Joel Moore.
The Chern number from IQHE, n, doesn’t survive in a system with only spinorbit interaction (and no external magnetic field), but Kane and Mele proposed a new topological invariant for fermionic systems with time reversal invariance (i.e. TRS is not broken).
The new topological invariant is not an integer, but rather a Chern parity  it can be either odd or even. It is called a Z2 invariant, and gets the letter
In a time reversal invariant electron system (i.e. no magnetic field, Efield is allowed), all energy eigenstates come in degenerate pairs (Kramers pairs).
Essentially, this Z2 invariant ν counts the number of Kramers pairs of edge modes, integrating over half of the Brillouin zone. If the overall Z2 sum of occupied bands is even, the system is a regular insulator, if the sum is odd, it is a topological insulator. For example the 2D system graphene possesses two Kramers pairs, has an even Z2 and thus is a 'trivial' system, whereas a material with one or three Kramers pairs would be a topological system.
5. Examples of 2D topological insulators and their edge states.
The first speaker of the focus session  Laurens Molenkamp  and his team in Würzburg were the first to realise 2D systems with the right sort of band structure and strong spinorbit coupling. Their experiments on Hg1xCdxTe quantum wells in which the electrical conductance of the quantum spin Hall edge states were measured were the first experimental confirmation of all the beautiful theory. In his talk Dirac fermions in HgTe quantum wells, we'll hear all about these exciting experiments. The theory and experiment combination of the quantum spin Hall effect was placed no. 6 in Science magazine's breakthroughs of the year, 2007
6. 3D Topological insulators
At the same time as Molenkamp et al. were publishing their pioneering experiments on 2D systems, theoreticians were predicting that just as 1D edge states demarcate the topological IQHE state and vacuum (the latter being a conventional insulator), there are 3D crystals whose electronic states possess topological properties and thus whose surfaces support conducting, 2D topologically protected surface states.
The picture below (from a News and Views article of Hari Manoharan's for Nature Nanotechnology), is shown schematically how the energy gap between the filled valence band and empty conduction band of a 3D insulator can be filled with an even (0,2) or odd (1) number of Kramers pairs. The Möbius strips symbolise the number of topological twists the band structure has got: only an odd number of twists gives the right geometric phase for topological behaviour.
Examples of crystals with the right band structure for the formation of topological surface states include Bi_{1x}Sb_{x} (a socalled first generation topological insulator) and Bi_{2}Se_{3} and Bi_{2}Te_{3} (both examples of second generation systems).
Angleresolved photoemission and scanning tunneling spectroscopy  being surface sensitive probes  have been the first experimental techniques to probe these 2D topological surface states, showing their Diraccone like dispersion relation and spinvortex like spin arrangement (the spin is always perpendicular to the momentum).
7. Why the excitement?
Other than the very basic satisfaction it gives condensed matter physicists to discover new quantum states of matter, the topological insulators would seem to offer experimental doorways to states that could be used in lowpower spintronic applications and as materials suitable for quantum information processing.
Majorana fermions in topological insulators. Majorana fermions are particles that are their own antiparticles and which obey nonAbelian statistics, protecting their quantum states from decoherence and making the clear link to quantum information processing. Menno Veldhorst's presentation will deal with transport experiments on TI systems, and also on Coupling superconductors to topological insulators, as his title puts it. Kareljan Schoutens' talk  on Topological quantum registers  will explain how braiding of excitations can be used to generate logical quantum gates, and will also introduce research into the generation of different kinds of anyons (particles with statistics between those of bosons and fermions).
A further source of excitement in the TI field are the possibilities for realising exotic entities such as dyons and magnetic monopoles, the latter predicted to exist in grand unification and string theories. The picture below from ShouCheng Zhang's 2009 Science paper illustrates the fractional statistics induced by image monopole effect when a magnetic film is placed on top of a TI surface state.
8. Outlook
We are starting to realise there is a sort of parallel universe out there with topological variants of the things we have come to know and cherish in the topologically trivial world. Many are convinced it will not be a very long wait until the first topological variants of strongly correlated or interacting states are able to be produced (for example a topological case of a fractional quantum Hall state, or a topological Mott insulator). With roots in both semiconductor and mesoscopic physics as well as in quantum field theory and the physics of strongly correlated systems, this field brings together new combinations of scientists (as does this focus session).
I hope this beginner's guide was of some use  sorry it got onto the webpage pretty last minute, and I hope you enjoy the focus session.
Mark Golden (UvA), focus session chair.