# What is parafermion in condensed matter physics?

Recently, parafermion becomes hot in condensed matter physics (1:Nature Communications, 4, 1348 (2013),[2]:Phys. Rev. X, 2, 041002 (2012), [3]:Phys. Rev. B, 86, 195126 (2012),[4]:Phys. Rev. B,87, 035132, (2013)).

But I have little knowledge about parafermion (fractionalizing Majorana fermion). So I have a few questions:

1. Please give a pedagogical introduction to the parafermion. The more, the better.
2. The common and different characteristic comparing to Majorana fermion.
3. The relation to Ising anyon, Fibonacci anyon, and so on.
4. The differentiation of non-Abelian statistics with parastatistics, fractional statistics, and so on.
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1. The $\mathbb{Z}_k$ para fermions arise in several statistical mechanics models. They are both interesting and subtle because their exchange statistics depend on their positions (in one-dimension). It is most intuitive to understand them as excitations arising from a $\mathbb{Z}_k$ quantum clock model. In the $\mathbb{Z}_k$ clock model we have local operators at each site, $\sigma$ and $\tau$ which obey $\sigma^k =\tau ^k = 1$ and $\sigma \tau = \omega \tau \sigma$ with $\omega = e^{2 \pi i /k}$. From these operators we can define a Hamiltonian, $$H_{clock} = \sum_n (-J\sigma_n^{\dagger} \sigma_{n+1}+h.c.) -h(\tau_n +\tau_n^\dagger)$$ This then takes the familiar form of a 1D transverse field Ising model (k=2). The transverse field Ising model allows for a Jordan-wigner transformation which takes it to free fermions. The generalization of this transformation takes the clock model to one of (not-free) parafermions (I'll elaborate on this). The transformation is: \begin{eqnarray} \alpha_j &=& \sigma_j \prod_{i<j} \tau_i \\ \beta_j &=& \sigma_j \tau_j \prod_{i<j} \tau_i. \end{eqnarray} With these transformations in place we may verify that the Hamiltonian in terms of the parafermion operators takes the form: $$H_{clock} = \sum_n -J\omega\alpha_{n+1}^\dagger \beta_n - h\omega \beta_n^{\dagger} \alpha_n +h.c.$$ With the parafermions satisfying the commutation relations $\alpha_j \alpha_{j^{\prime}} = \omega^{sgn(j^\prime - j)} \alpha_{j^{\prime}} \alpha_j$ and similarly for the others. Hence the site dependent commutation relations. The introduction to http://arxiv.org/abs/1209.0472 provides more details in a very readable manner.
2. How do these parafermions compare to majorana modes in condensed matter physics? Well it is clear that they are a direct generalization of the Majorana operators found from the Jordan-Wigner transformation (or k=2 case here). But they are entirely different beasts when it comes to zero modes. This is most easily seen by trying to solve for the theories spectrum. For $k=2$ we have Majorana's, we can compute the spectrum by solving for ladder operators that satisfy $[H,\gamma_k] = E_k \gamma_k$. This is a relatively straight forward excercise and results in plane wave solutions with a spectrum that looks like $E_k = \pm2\sqrt{h^2(\cos{k}-1)^2 + J^2\sin^2{k}}$ (this should be check). If one tries the same methodology with the parafermions it becomes quite clear that commuting something linear in parafermions yields bilinears in parafermions. This signals to us that our theory is no longer free. This is what I would think to be the largest difference.
3. How do they relate to Ising anyons and Fibonacci anyons? Ising anyons are intimately related to majorana zero modes - they satisfy the same non-abelian statistics up to an overall $U(1)$ phase. I do know of one connection which someone else could perhaps ellaborate on. It is known that these models are all self-dual and have a critical point at $h = J$ and here we consider $h$ and $J$ real. These critical points are described by parafermion conformal field theories (CFT) in the thermodynamic limit. The CFT governing the k=3 theory has a field whos operator product satisfies the fibonacci fusion rules.
4. In two-dimensions funny things can happen when we exchange particles. Rather than the $\pm$ sign distinguishing bosonic like particles from fermionic like particles in 3 or more dimensions in two dimensions particles may pick up an arbitrary phase $e^{i \theta}$. This would be characteristic of an Abelian anyon. A non-Abelian anyon can also pick up such a phase but, even more strangely rather than just an overall phase - it's entire (degenerate) ground state may undergo a unitary transformation. An excellent review on the subject is given here: http://journals.aps.org/rmp/abstract/10.1103/RevModPhys.80.1083.