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Mathematicians like to write tensor product, since in many cases (or maybe in all cases) anyon types (simple objects) are indeed irreducible representations of some algebraic object (e.g. Hopf algebra, quantum groups), and irreducible representations of finite groups provide a large family of examples for fusion categories, where $\otimes$ and $\oplus$ ...

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In simplest terms, the presence of sub-gap zero energy localized modes (Majorana modes) makes a superconductor topological. A superconducting ground state is just a bunch of Cooper pairs and the BdG Hamiltonian describes excitations above the ground state. If the excitation spectrum has these localized modes then it is a topological superconductor otherwise ...

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I can provide an example for bosonic models. \begin{eqnarray} \mathcal{H} & = & \mathcal{K} + \mathcal{T}_\text{soc} +\frac{U}{2}\sum_{i\tau} \hat n_{i\tau}( \hat n_{i\tau}-1) \nonumber \\ & & + U^{\prime} \sum_i \hat n_{i\uparrow} \hat n_{i\downarrow} + V\sum_{i\tau} \hat{n}_{i\tau}\hat{n}_{i+1\tau} \nonumber ...

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Gauge symmetry imposes local conservation laws, which are called Ward Identities in QED and Slavnov-Taylor identities for non-Abelian gauge theories. Those identities relate amplitudes or limit them. An example of those constraints imposed by gauge symmetry is the transversality of the vacuum polarization. To be more precise, gauge symmetry does not allow ...

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The Green function you gave above is not the Green function for an insulator, which should generically has no pole if there is no Fermi surface. The Volovik argument is a bit circular, since you know from the beginning that $p=p_{F}$ is the Fermi momentum, defining the Fermi surface. When $p\approx p_{F}$ you can infer the form of the Green function as you ...

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I assume this is the formula for Chern number in a Chern insulator. The physical reason that such a formula exists is that this is exactly what Kubo formula gives you for the Hall conductance, which holds for interacting systems too.

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