Topological insulator Definition and 14 Discussions

A topological insulator is a material that behaves as an insulator in its interior but whose surface contains conducting states, meaning that electrons can only move along the surface of the material. Topological insulators have non-trivial symmetry-protected topological order; however, having a conducting surface is not unique to topological insulators, since ordinary band insulators can also support conductive surface states. What is special about topological insulators is that their surface states are symmetry-protected Dirac fermions by particle number conservation and time-reversal symmetry. In two-dimensional (2D) systems, this ordering is analogous to a conventional electron gas subject to a strong external magnetic field causing electronic excitation gap in the sample bulk and metallic conduction at the boundaries or surfaces.The distinction between 2D and 3D topological insulators is characterized by the Z-2 topological invariant, which defines the ground state. In 2D, there is a single Z-2 invariant distinguishing the insulator from the quantum spin-Hall phase, while in 3D, there are four Z-2 invariant that differentiate the insulator from “weak” and “strong” topological insulators.In the bulk of a non-interacting topological insulator, the electronic band structure resembles an ordinary band insulator, with the Fermi level falling between the conduction and valence bands. On the surface of a topological insulator there are special states that fall within the bulk energy gap and allow surface metallic conduction. Carriers in these surface states have their spin locked at a right-angle to their momentum (spin-momentum locking). At a given energy the only other available electronic states have different spin, so the "U"-turn scattering is strongly suppressed and conduction on the surface is highly metallic. Non-interacting topological insulators are characterized by an index (known as



{\displaystyle \mathbb {Z} _{2}}
topological invariants) similar to the genus in topology.As long as time-reversal symmetry is preserved (i.e., there is no magnetism), the



{\displaystyle \mathbb {Z} _{2}}
index cannot change by small perturbations and the conducting states at the surface are symmetry-protected. On the other hand, in the presence of magnetic impurities, the surface states will generically become insulating. Nevertheless, if certain crystalline symmetries like inversion are present, the



{\displaystyle \mathbb {Z} _{2}}
index is still well defined. These materials are known as magnetic topological insulators and their insulating surfaces exhibit a half-quantized surface anomalous Hall conductivity.
Photonic topological insulators are the classical-wave electromagnetic counterparts of (electronic) topological insulators, that provide unidirectional propagation of electromagnetic waves.

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  1. B

    Topological insulators and their optical properties

    I have tried to write down the boundary conditions in this case and looked into them. As conditions i) and ii) were trivial, i looked into iii) and iv) for information that I could use. But all I got was that for the transmitted wave to have an angle, the reflective wave should also have an...
  2. H

    A Is there a topological insulator without Spin Orbit Coupling (SOC)?

    There are some famous materials is determined as TI induced by SOC, like graphene and so on. But from some formula, for instance, Kane-Fu formula, they just need parities to get Z2 number. So I wonder if there is a known TI with weak soc.
  3. E

    A Surface states of 3D topological insulators

    I have a question (more like a curiosity) related to three-dimensional topological insulators, which support Dirac-like states at their surfaces. From the theory, it is well known that these states are immune to scattering from non-magnetic impurities, i.e. impurities that do not break...
  4. J

    Solid State Sources to learn about Topological Superconductivity

    Greetings. Does anyone know about any good places to learn about topological superconductivity from? Thanks in advance!
  5. J

    Solid State Books on Topological Insulators

    Hello. Do you know of any good material on topological insulators like books, review papers etc? I would prefer something more oriented towards theoretical physics(because I know that there are reviews out there that are purely experimental). Thank you!
  6. L

    A Why is the nearest hopping kept real in Haldane model?

    I am leaning the Haldane model : Haldane imaged threading magnetic flux though a graphene sheet, and the net flux of a unit cell is zero. He argued that since the loop integral ##\exp [ie/\hbar \oint {A \cdot dr} ]## along a path...
  7. DeathbyGreen

    A Pierels substitution integral approximation

    In the textbook "Topological Insulators and Topological Superconductors" by B. Andrei Bernevig and Taylor L. Hughes, there is a chapter titled "Hall conductance and Chern Numbers". In section 3.1.2 (page 17) they are discussing including an external field in a tight binding model, the Peierls...
  8. V

    A Question about Berry phase in 1D polyacetylene

    Hi. I'm taking a look at some lectures by Charles Kane, and he uses this simple model of polyacetylene (1D chain of atoms with alternating bonds which give alternating hopping amplitudes) [view attached image]. There are two types of polyacetylene topologically inequivalent. They both give the...
  9. T

    Mathematical theory for topological insulators

    I have been learning topological insulators recently, and I become more and more curious about the link between topological insulators and mathematical theory these days. I know topological insulators have something to do with fiber bundles and K-theory. I have a relatively good background of...
  10. M

    Topological insulator

    Hi every one, I face with a question on my works, As you know there in many articles Physicist introduce a material that has zero gap without spin-orbit coupling (SOC). By applying the SOC, a relatively small gap (0.1 eV) is opened and it becomes topological insulator. My question, Is that...
  11. T

    Z2 invariant and edge state

    I have been reading about Z2 topological invariant recently. However, after some literature survey, I still cannot understand Z2 invariant in language of time reversal polarization. Basically, my struggle includes the following two questions: As the ref paper says(see the picture below): On...
  12. SoumiGhosh

    Topological phase and spontaneous symmetry breaking coexist?

    As we know topological phases cannot be explained using spontaneous symmetry breaking and order parameter. But can they coexist? Suppose there is a system which is undergoing quantum phase transition to a anti-ferromagnetic phase from a disordered phase. So in the anti-ferromagnetic phase...
  13. M

    Effect of TRS potential on Topological insulator (QSH)

    Hi every body, I faced a paradox. The topological insulator is robust against a potential that does not breaks the TRS. But in the original work of Kane-Mele (PRL 95, 146802), the "staggered sublattice potential" that does not breaks the TRS,, makes zigzag ribbon trivial insulator (figure 1 in...
  14. T

    Relations between chern number and edge state

    I have been doing a literature survey about topological insulators for some time. What surprises me is the close relation between difference of chern number and number of edge states. However, I found most review or tutorial in topological insulator avoided direct proof of the relation. So can...