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You can find an excellent description of what a topological insulator is in this brief presentation from the Yazdani Group at Princeton: Topological Insulators. To answer your question on the meaning of negative effective mass: The effective mass is actually determined by the behavior of the energy levels $E({\bf k})$ as functions of the crystal wave ...

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The tricky part is to explain the plateaus in $\sigma_{xy}$ and why they coincide with vanishings of $\sigma_{xx}$. To make this more evident let us suppose first you had a system without disorder, say you have an interacting Galilean that is fully translationally invariant. Ideally this would be realized e.g. in a Torus. Then you can show, following Kohn's ...

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Disorder is necessary to observe the integer quantum Hall effect (IQHE), but it is not necessarily true that current is carried only by the edge states. Disorder gives rise to localized bulk states between Landau levels. It is the existence of these localized states that gives us the broad plateaus in resistivity at integer filling fractions: increasing the ...

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In condensed matter "bulk" does not refer to the dimensionality of the problem but the location in the material. It refers to the volume of the crystal, as opposed to, e.g., surface effects. Many organic conductors behave as 1D systems, yet you can talk about bulk properties. Copper oxide superconductors have a 2D physics. However, often you will find ...

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It is not true that if $\mathbf{k}\neq -\mathbf{k}$ the matrix element $\langle u_i(\mathbf{k}|T|u_j(\mathbf{k})\rangle$ vanishes. Remember that $u(\mathbf{k})$ are Bloch wavefunctions, which are eigenvectors of the momentum space Hamiltonian $H(\mathbf{k})$ (e.g. $H(\mathbf{k})$ in Kane-Mele model is just a $4\times 4$ matrix at a given $\mathbf{k}$). ...

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