What is Topological insulators: Definition and 24 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. D

    Solid State Texts on Topological Effects/Phases in Materials

    I am looking to learn about these topological effects or phases in solids. More specifically, I'm trying to find a set of lecture notes or a textbook or some other text that do not shy away from discussing homotopy classes and the application algebraic topology to describe these materials. I...
  2. 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...
  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. J

    Solid State Books: Weyl semimetals, Topological Insulators

    Hello! What are some good sources(preferably textbooks) to learn about Weyl semimetals? I also want some sources to learn about topological insulators and anything containing the Integer Quantum Hall effect would be great. As an aside, if you have any good book on theoretical condensed matter...
  7. D

    I Why and how Dirac cones are "tilted"?

    Given a Weyl Hamiltonian, at rest, \begin{align} H = \vec \sigma \cdot \vec{p} \end{align} A Lorentz boost in the x-direction returns \begin{align} H = \vec\sigma\cdot\vec{p} - \gamma\sigma_0 p_x \end{align} The second term gives rise to a tilt in the "light" cone of graphene. My doubts...
  8. T

    A Why is the Brillouin zone a torus?

    Recently, topological concepts are popular in solid state physics, and berry connection and berry curvature are introduced in band theory. The integration of berry curvature, i.e. chern number, is quantized because Brillouin zone is a torus. However, I cannot justify the argument that...
  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. E

    Properties of the Dirac point and Topological Insulators

    I understand that the centring of the Fermi energy at the Dirac point is a highly sought after property in Topological Insulators but I'm unsure as to exactly why? I see that the state at the conical intercept will be unique but I'm not sure of what is theorized to happen to the electrons...
  11. S

    Scattering in Topological Insulators

    It is well known that back-scattering of surface electrons in topological insulators is prohibited due to Kramer's degeneracy theorem as long as Time Reversal Symmetry is not broken by magnetic field or magnetic impurities. I would like to know what effect this has on scattering length and...
  12. C

    Questions on Topological Insulators, confused on some terminologies

    Hi all! I am currently reading stuff related to quantum hall effect and topological insulators, and have a couple of questions. 1. I read about that band insulators can be classified into two types: topological trivial insulators and topological non-trivial insulators. And there is a...
  13. E

    Fabrication of topological insulators

    Hi I have been searching some papers online to find how practically we can approach for the fabrication of topological insulators. Can somebody please help me regarding this by providing some web links or some insight on the fabrication of topolopgical insulators...
  14. G

    Topological Insulators and Inversion Symmetry

    Hi, I was curious if specific symmetries (or lack thereof) in crystal structure are necessary for the formation of topological insulators. Specifically, do we require that inversion symmetry (or inversion asymmetry) be present in the lattice in order to form the TI state? Thanks, Goalie33
  15. K

    Counterexamples to the bulk-boundary correspondence (topological insulators)

    In the literature on topological insulators and superconductors the 'bulk-boundary correspondence' features quite heavily. One version of this conjecture says roughly: "At an interface between two materials belonging to the same symmetry class with bulk invariants n and m, precisely |n-m|...
  16. R

    Topological Insulators Explained: Quantum Hall Effect & More

    I'm sorry if this is in the incorrect section, but can someone please explain what topological insulators are, the quantum hall effect, how you make a topological insulator and anything else that is relevant to the topic. Thanks.
  17. K

    Kitaev's Periodic Table (of Topological Insulators & SCs)

    Hi PF, I'm trying to come to grips with the work of Alexei Kitaev on applying notions from (topological) K-theory to the task of classifying phases of topological insulators and superconductors (paper here: http://arxiv.org/pdf/0901.2686v2.pdf). Despite having plenty of citations, I've yet to...
  18. M

    Time reversal symmetry in Topological insulators of HgTe quantum Wells

    Hi everyone, While reading about the BHZ model used to describe HgTe quantum well topological insulators, I read at many places that the effective Hamiltonian (which is a 4 x 4 matrix) can be written in block diagonal form and the lower 2x2 block can be derived from upper 2x2 block as...
  19. N

    What do topological insulators have to do with hall currents?

    Hello, So a topological insulator can induce a magnetic field when an electric charge is near to it (I can give a reference if necessary), but the thing is, the paper interprets the origin of this magnetic field as being the hall currents on the surface of the topological insulator. Now I...
  20. U

    Why topological insulators are topological?

    Hi, this is my first post here! I've been studying about topological insulators, but still I can't understand why this materials are called topological, I've read about topological analogy between the donut and the coffee mug and the smooth changes on the Hamiltonian, but I can't get the full...
  21. T

    Studying How start studying Topological Insulators

    Hi all, I'am starting a Phd In Theoretical Condensed Matter Physics, and I would like to produce a thesis on the Topological Insulators topic. Unfortunately I don't have a background in Consensed Matter Physics (in my curriculum there are exams about General Relativity, Quantum Field Theory...
  22. O

    Pointers to start understanding topological insulators

    I've recently started learning about topological insulators. I've read a considerable amount of (review) papers on the subject, yet I still only have a phenomenological understanding of what a topological insulator is. I know for example, that the gapless surface states have to be there because...
  23. H

    Is the topological insulators a result of boundary conditions with SO coupling ?

    Hi, these days I have been trying to understand the essentials of the so-called topological insulators (TBI), such as Bi2Te3, which seem very hot in current research. As i understand, these materials should possesses at the same time gapped bulk bands but gapless surface bands, and spin-orbit...
  24. T

    Why are Topological Insulators Called Topological?

    why are topological insulators called TOPOLOGICAL insulators? what factor of topology apperas in the phenomenon