A basic notion when studying non-abelian anyons is that the system's groundstate is degenerate. Not only that, but exchanging two anyons' position rotates the state in this degenerate subspace. I'm having difficulty wrapping my head around this last idea. Quoting from the Anyon page in wikipedia:

"[...] so that multiple distinct states of the system have the same configuration of particles."

This seem extremely weird. I can understand having two distinct states with the same configuration of particles as supressing an actual difference in particle configuration inside what you call the vacuum of the system (which is actually a 2D condensed matter system, for example). But how moving particles around and putting them back in positions indistinguishable from the initial stage moves the vaccum around state ia beyond me. My question is: How is this possible? What is happening in terms of the underlying condesed matter system?


Anyons are quantum statistic induced by topology. The topological role can be seen with the projective space. In $R^3$ we can form $\{\mathbb R^3 - pt\}/\mathbb Z_2 = \mathbb RP^2$ and the first homotopy is $\pi_1(\mathbb RP^2) = \mathbb Z_2$. This is similar to the idea in my last paragraph. This result is closely connected to why quantum statistics has $[\phi, \phi^†]_\pm = 1$, - for bosons and + for fermion, statistics. Interestingly for $R^2$ we have $\{\mathbb R^2 - pt\}/\mathbb Z_2 = \mathbb RP^1$ and $\pi_1(\mathbb RP^1) = \mathbb Z$, which is why there are anyons in 2 dimensions. For this reason we may apply braid rules to anyonic statistics.

The Kitaev lattice in two dimensions describes spins on the edgelinks, and in general we consider $\sigma_x$ and $\sigma_z$. We form $A_v$ as a product of $\sigma_x$ at a vertex $v$ and we form $B_p$ as the product of $\sigma_z$ around a plaquette or square face bounded by edge links. This mean we have an elementary topological Poincare duality between spins defined at a vertex, a point, and those identified with an area. We then have a Hamiltonian of the sort $$ H=J_e\sum_vA_v-J_mB_p, $$ where the spins at a vertex define an “electric field” and those with the plaquette a “magnetic field.” Now using $\{\sigma_i,~\sigma_j\}=2\delta_{ij}$ it is possible to show that the $A_v$ and $B_p$ commute and are simultaneously diagonalizable. The product of a $\sigma_x$ with $\sigma_z$ gives a $\sigma_y$ and this leads to an elementary algebra of the electric and magnetic monopole fields $E$ and $M$ with the excitations $X$ so $$ E\times M = X, X\times E=M, M\times X=E. $$ which gives the abelian anyon. This may be written according to braid rules for anyons.

The products of operators $B=\prod_p\sigma_z$ and $B'=\prod_{p'}\sigma_z$ for orthogonal sequences of plaquettes, think of walking by stepping on a set of floor tiles in one direction for $B$ and $B'$ for stepping on floor tiles heading in the other direction. These operators in the two orthogonal directions define a four-fold vacuum degeneracy $$ |B,B'\rangle=|1,1\rangle,|-1,1\rangle,|1,-1\rangle,|-1,-1\rangle. $$ Because the algebra is abelian a path around will close up with commutation. This means these four ground states are degenerate.

For nonabelian anyons the above algebra, which can be thought of as a transport around the lattice does not close. The transformation of a state $|X_i\rangle\rightarrow |X,_i\rangle$ $=\sum_jU_{ij}|X_j\rangle$ is such that the operator $U_{ij}$ rotates a state into one that is a superposition of the field fluxes. The above algebra is then $x_a\times y_b=N_{abc}z_c$. We no longer have closure of a path around a loop, for now a state or field is mixed with other fields. Further, this means the vacuum degeneracy is broken.

All of this obeys braid rules for anyons. That is a subject a bit beyond this post. The abelian anyons are in a sense a global symmetry, while nonabelian anyons have local symmetries and the vacuum degeneracy is broken.

  • $\begingroup$ Sorry, but what does it mean for a path around a loop not to have closure, physically? Does it mean that moving the anyon around and back to the start changed something inside the lattice? $\endgroup$ – Lucas Baldo Sep 5 '19 at 0:23
  • $\begingroup$ The loop is closed, but field information does not return onto itself. $\endgroup$ – Lawrence B. Crowell Sep 5 '19 at 21:55
  • $\begingroup$ Field information? Sorry, I'm just beggining my studies on this, and it all seems a lot of info. Do you have a reccomendation for sources on non-abelian anyons basics? $\endgroup$ – Lucas Baldo Sep 6 '19 at 16:02
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    $\begingroup$ Look at NONABELIONS IN THE FRACTIONAL QUANTUM HALL EFFECT by Reed and Moore, physics.rutgers.edu/~gmoore/MooreReadNonabelions.pdf . This gets into Laughlin wave functions as well, which is a detail I avoided. $\endgroup$ – Lawrence B. Crowell Sep 7 '19 at 21:51

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