Topological insulators

Topological insulator is a novel quantum state of electrons with a vanishing electrical conductance at zero temperature. Unlike conventional insulators, the topological ones are characterized by quantum entanglement and properties which cannot be detected with any local probe.
To put this in broader context, we classify phases of matter by their symmetries. For example, positions of atoms in a crystal, averaged over a long time, are generally changed when we translate the system by a vector, so we say that a crystal is not invariant under translations. On the other hand, a liquid is invariant under translations because all of its atoms are in constant and almost free motion. The laws of physics which lead to localization of atoms at their lattice sites in a crystal are fundamentally invariant under translations, there is no preferred position in the universe for any particular atom in the crystal. Therefore, we say that the crystal is a result of spontaneous symmetry breaking. The groundstates of almost all known quantum manybody systems break some symmetries spontaneously, and it typically takes thermal fluctuations to restore them, such as in a liquid. States with and without broken symmetries can be distinguished by their local properties. For example, take a Scanning Tunneling Microscope and keep looking for an atom in a microscopically small patch of space. If you can find it and it keeps siting there, you are dealing with a crystal. If the image looks blurred and you cannot isolate an atom, you are dealing with a liquid.
A closely related characterization of phases rests on the excitation spectrum. An insulator is an electronic quantum system in which all chargecarrying excitations cost a finite energy. Therefore, in order to make current flow, one has to invest a huge amount of energy, a fixed amount per electron. A metal, on the other hand, has no energy gap for electrons, but Fermi statistics and Pauli exclusion principle require that the amount of invested energy be proportional to the amount of current. Also, electron scattering leads to dissipation, described by Ohm's law. Finally, superconductors are also gapless, but their elementary chargecarrying excitations are bosons, so called Cooper pairs of two electrons glued by some attractive interaction. It takes a negligible amount of energy to excite a finite current, and this current is not dissipated.
Hightemperature superconductors (cuprates) are a family of materials which pose an enormous challenge in condensed matter physics for more than twenty years since their discovery. Their large critical temperature for superconductivity (up to about 150K) gives hope that it might be possible to engineer one day a roomtemperature superconductor and revolutionize our lives. However, we still do not understand the microscopic mechanism for superconductivity. Under normal conditions, these materials are insulators with broken symmetries. There are many theories which try to explain how they become so good superconductors after sufficient doping. The crucial question is what creates the energy gap for charged excitations.
There are two generic and well understood kinds of insulators, band and Mott. Band insulators have gapped fermionic excitations by the virtue of having a completely filled electron band below the Fermi energy. This kind of insulator can be obtained from noninteracting electrons in a periodic lattice potential and does not (normally) involve symmetry breaking. Mott insulators have gapped bosonic charged excitations and occur at densities which correspond to a partially filled electron band. The charge gap is created by Coulomb interactions between electrons. This kind of insulator typically involves spontaneous symmetry breaking in the form of a magnetic or chargedensitywave order.
A number of theories propose that underdoped cuprates could be topological insulators as an explanation of their highly unusual properties in the so called pseudogap state. These are very strange states of matter because they do not break any symmetries, yet the spectrum of all charge excitations is gapped due to interactions. This situation is very difficult to find in electronic systems, and until recently we had only one known example achieved in laboratories: quantum Hall effect. Recently, however, physicists have predicted and engineered a new kind of topological insulator, which will be the first subject of our sessions.
Our great fortune is that the new kind of topological insulators can be understood largely from the perspective of singlebody quantum mechanics. Therefore, not much training in quantum theory is required to get into this subject, which makes it ideal to gently introduce students to condensed matter physics.
Experimental realization of topological insulators in HgTe/CdTe quantum wells
Atomically thin films of HgTe semiconductors embedded in the bulk CdTe semiconductor have been theoretically predicted and then experimentally showed to have a topologically insulating ground state when the film thickness is small enough. The topological state in question is an integer spin quantum Hall state (SQHS), which is a timereversal invariant cousin of the well understood integer quantum Hall state (QHS).
Semiconductor heterostructures like HgTe/CdTe are very useful for creating twodimensional electron gases. A proper choice of materials creates a quantum well potential which completely quantizes motion in the direction perpendicular to the layer. At sufficiently low temperatures the electron gas is physically twodimensional. If strong magnetic field is applied perpendicular to the layer, the twodimensional electron states acquire a spectrum of Landau levels. Classically, electrons move in cyclotron orbits, which implies that magnetic field localizes them in the plane. Quantummechanically, these localized states have discrete energy levels, which also must be macroscopically degenerate because a cyclotron orbit can be localized anywhere in the plane without any extra energy cost. Such macroscopically degenerate discrete energy levels are called Landau levels.
A topological insulator in HgTe/CdTe quantum wells can be thought of as a twocomponent quantum Hall state. Electrons come in two distinct states of spins: the spin projection on an arbitrary axis can be either "up" or "down". Spin is physically a magnetic moment attached to an electron. The motion of an electron through the crystal lattice can be studied in the reference frame attached to the electron. In this reference frame it appears that the lattice is moving. The lattice ions exert electric field on the electron, which appears to alternate in time in the electron's reference frame. According to Faraday's law, this alternating electric field induces a magnetic field, which then couples to the magnetic moment of the electron spin. This gives rise to the so called spinorbit interaction. The mathematical structure of spinorbit interaction resembles very much that of the magnetic field on a charged particle and may lead to Landau level quantization of electron spectra. However, the "magnetic field" which an electron sees depends on the spin, and changes sign when the spin is flipped from "up" to "down".
The intricate properties of electronic spectra in HgTe and CdTe semiconductors (bandstructures) and their quantum mechanical mixing by the presence of the quantum well gives rise to SQHS. To understand this one needs to consider energy levels in the vicinity of the Fermi energy (which determine the low temperature dynamics) and their properties under various symmetry transformations. Symmetries are a very powerful tool in physics to elucidate properties of quantum systems: even if we can't solve a quantum problem, much can be learned about the solution based on its symmetries. Such a symmetry analysis points to the picture of SQHS given in the previous paragraph, and a simple introductory discussion can be found in this paper.
Integer quantum Hall states are topological in the sense that there is a charge gap, but a different one than found in ordinary band insulators. This is best revealed at the system boundaries where chiral edge states appear. An electron occupying an edge state can propagate freely along the edge in only one direction, but cannot be detached from the edge. If the quantum well were shaped as a disk, electron motion along the edge could be sustained only in a direction given by the "righthand rule" with respect to the external magnetic field. There are gapless edge states in a quantum Hall insulator, so that electrical current can flow along the edge even though the bulk system is insulating.
A topological insulator is very similar, but edge states come in pairs, one for each spin projection. The "righthand rule" makes electrons of opposite spins propagate in opposite directions along the same edge. Therefore, edges cannot conduct electrical current, but they can conduct spin current (magnetic dipole current). Various edge perturbations (impurities, interactions) can generally introduce quantum mechanical mixing between the edge eigenstates of the ideal Hamiltonian, and open up a gap in the edge energy spectrum. If this happens, the insulator becomes nontopological and cannot conduct (spin) current neither through the bulk nor along edges. However, it can be showed that nonmagnetic perturbations (which do not flip spin) can open a gap only if there is an even number of edge state pairs. Therefore, a topological insulator with an oddnumber of edge state pairs is a stable state of matter which takes large perturbations to destroy. This kind of state is indeed found in HgTe/CdTe quantum wells.
Characterization of topological bandinsulators
Dr. Kai Sun from the Joint Quantum Institute (JQI) at the University of Maryland gave a seminar and taught us how to characterize topological insulators in a variety of noninteracting or weakly interacting fermion models. Here is his Power Point presentation.
Research articles
A nice introductory article by Kane
Another nice introductory article by Kane and Mele
Science article by B. Andrei Bernevig et.al.
The first article by Kane and Mele on topological insulators
PRL article by Kane and Mele
Phys.Rev B article by Fu and Kane
Phys. Rev. A article by Tudor Stanescu et.al.