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In short, what makes a superconductor topological is the nontrivial band structure of the Bogoliubov quasiparticles. Generally one can classify non-interacting gapped fermion systems based on single-p …

Topological order can not be described in Ginzburg-Landau symmetry breaking paradigm. It is actually fair to say that topological order are more or less the properties of (gapped) quantum phases that …

You did not include the proper infinitesimal imaginary part for frequency in the first equation for the Green's function, which would have given the correct pole structures. So it is not clear whether …

Chiral $p$-wave superconductor and He A phase can be considered being equivalent phases of matter, for the following reason: The fact that the superconductor is charged does not make too much differen …

One first needs to define what it means for a state to be topological when the system is gapless. This is not an easy question! I don't think there is a unified answer at the moment. I'll give some …

Your statement itself is not quite right. What is not conserved is the chiral current, namely the current of fermions at one of the Weyl nodes. The physics can be understood essentially in one-dimensi …

There is certainly a way to preserve the time-reversal symmetry with open boundary conditions: just put the two edge spins into a singlet state. You may say that this is cheating, since essentially we …

Well it depends on what you mean by "exactly solvable". Many people would consider any quadratic Hamiltonian(i.e. non-interacting) exactly solvable, so the BCS Hamiltonian of a p+ip superconductor alr …

In the Ising anyon model, there are three topological charges, $1,\sigma,\psi$. $\sigma$ can be thought as carrying a Majorana zero mode, and $\psi$ is an ordinary fermionic excitation. This can also …

A very good question. First of all, topological order strictly speaking is only defined for gapped states. But to some extent it can coexist with gapless degrees of freedom. A rather trivial example i …

This is why people start to abandon calling the zero modes in vortices "Majorana fermions" because they are NOT fermions. $\gamma$ is a Majorana zero mode, which means it always has to pair up with an …

The equation follows from the a general result called Hellman-Feynman theorem: suppose one has a Hamiltonian $H(\lambda)$ depending on a parameter $\lambda$ (e.g. your $q$), with eigenstates $|n(\lamb …

For one thing, toric codes (and other error-correcting codes) are really about ways to store quantum information(logical qubits) in several physical qubits, so there is not much point in asking for a …

The definition in terms of LU transformations is more fundamental. In the general case, we believe that topological orders are described by modular tensor categories (MTC), and the equivalence of topo …

Q1: Consider the example of a Haldane phase protected by time-reversal symmetry. There is a spin-1/2 on the edge, and to polarize it (so the system is gapped and the ground state is unique) one must b …