Numerical Tools to find Braiding Statistics of Quasiparticles

Question

While certain classes of systems that exhibit topological order can be solved exactly (such as the Toric Code, Abelian FQH Edges, etc.) there also exist systems (think of perturbed versions of the Toric Code or Abelian FQH Edges) that cannot be solved exactly.

What are the common numerical tools that are used to extract the (not necessarily abelian) braiding statistics of quasiparticles from these models? What is the main idea of these techniques? For which systems do they work well and for which systems do they fail?

Can one access the braiding statistics also from numerical exact diagonalization of a small system? (For say a perturbed Toric code I would guess so because at least in the unperturbed case the abelian braiding statistics do not depend on the system size)

I would be very happy if someone could provide a brief overview.

I am looking forward to your responses!

Answer 1

For a gapped system with topological order, the quasiparticle statistics and braidings can be extracted numerically from ground state entanglement of the system, see the paper by Yi Zhang et al.

The main idea is the following. Consider a topological phase on a torus. Using some (probably numerical) method, you obtain the degenerate ground states of the system on torus. You write down a most general linear combination of those degenerate ground states, say $\Psi=c_1\psi_1+c_1\psi_2+\cdots$, where $\psi_i$ are the ground states you obtained and $c_i$ the coefficients.

Then you partition the torus into two cylinders. After that, you can calculate the entanglement entropy between those two cylinders using the most general ground state. Then you minimize the entanglement entropy.

Since the entanglement entropy goes like $S=\alpha L -\gamma$ for a gapped 2D topological system, with $\gamma$ the topological entanglement entropy, and $\alpha$ some model-dependent parameter, minimizing the entanglement entropy means maximizing the topological entanglement entropy. Namely, you maximize the "topological information" that you can obtain.

Suppose you work on a system with ground state degeneracy $N$ on the torus. Upon the minimization process you will obtain one (or more) combination of $\{c_i\}$ which minimizes the entropy. Then you search for the states which have the minimal entanglement entropy in the state space orthogonal to the state(s) you have just found. Continue this process until you have found exactly $N$ states with minimal entanglement entropy. We call such $N$ ground states "minimally entangled states", which are actually the eigenstates of the nonlocal operators which distinguish the topologically degenerate ground states.

Just now we have partitioned the torus into two cylinders by cutting one of its uncontractible loops. Now we partition the torus into two other cylinders (or annuluses that can be deformed into cylinders) by cutting the other uncontractible loop of the torus. Similarly, you can calculate the entanglement entropy using the most general $\Psi$ and repeat the minimization procedure. You will find another group of minimally entangled states.

Then you calculate the overlap between these two groups of minimally entangled states. It's a matrix which we identify with the modular $S$ matrix characterizing the mutual statistics of quasiparticles. After all the modular $S$ transformation is just exchanging the two uncontractible loops of the torus.

For special models such as $Z_2$ spin liquid, one can also find the $T$ matrix from entanglement entropy, but there seems to be no general algorithm.

There have been at least two works using this method on FQHE in topological flat bands and Fibonacci & Ising models. They did perform exact diagonalization on small systems to find the ground states. I don't know about the newest works but I think those are enough for one to understand the procedures.

"For which systems it fails", I'm not sure but I think the above method works well for all gapped systems with topological order.

Answer 2

Abelian anyons were originally thought of as particles that interpolate between bosons and fermions. As such, their joint wavefunctions pick up a phase as one of them is exchanged with another one. This phase is generally not a multiple of $\pi$, and one says that the particles obey fractional exchange statistics, which is the same as what you call braiding statistics. Let us call this phase $\alpha$.

Now, if one wants to push the interpolation picture further, one may further entertain the thought that anyons may also obey fractional exclusion statistics, i.e., that only a certain number of anyons is allowed in a certain number of quantum-mechanical states. For fermions, this amounts to the Pauli exclusion principle (1 fermion per 1 quantum-mechanical state) and for bosons it reduces to no restriction at all. Let us associate another parameter $\beta$ to the rate of change of the Fock space of a certain species of anyons upon increasing the system size.

If the interpolation analogy is correct, then $\alpha$ and $\beta$ should be uniquely related to each other. Therefore, one can infer $\alpha$ from the value of $\beta$.

Why would one want to evaluate $\alpha$ indirectly through $\beta$ instead of getting $\alpha$ directly? The short answer is that $\beta$ is much easier to obtain in finite-system calculations of topologically ordered, correlated particle systems. It turns out that this approach is viable and yields correct results in exact-diagonalization calculations.

I am not aware of any other approach that can yield similar information generically for correlated systems. Perhaps another answer will be more enlightening in this respect.