Lineas de la mano

Having discussed the algebraic theory describing topological excitations in a gapped quan- tum liquid, we will now turn to braiding.

Imagine we adiabatically tune Vtrap such that the simple type a1 at location x1is rotated around itself by 2π. The states in V(a1, b2, . . . ) can potentially acquire a geometric phase19 eiθa_{. The phase θ}

a is called the topological spin of anyon type a. We can implement this in our graphical notation by giving the lines a framing and think of them as ribbons, thus whenever we undo a twist in the ribbon the corresponding state acquires a phase eiθa_.

It turns out that the right way to define the data for braiding is by introducing the so-called R-symbol, defined graphically by

b

a

c

b

a

c

The R-matrix cannot be arbitrary and must satisfy certain non-trivial constraints which comes from consistency, one can go from one diagram to another in multiple ways giving us multiple mappings between these states and consistency requires them to be equal. Starting with three anyons, similar to the pentagon equation, we can derive a consistency relation of the schematic form RF R = PF RF that the R and F symbols must satisfy. This is called the hexagon equation. Again the McLane Coherence theorem ensures that no other consistency relations are needed.

Any solution of the pentagon and hexagon equations, will give us a consistent anyon model and correspond to a particular topological order in 2+1D. If start with the graphical notation (1.17), we can braid the anyons in any arbitrary way. By using the F and R- symbols, eq. (1.19) and (1.20), we can turn this braid diagram back into (1.17). This gives us a map on the fusion space that is nothing but the wanted Braid group representations discussed earlier.

The mathematical structure behind this is called a modular tensor category (MTC),20

and 2 + 1D topological orders can be classified by classifying MTCs. It naively appears that the solution space of the pentagon and hexagon equations form a continuous algebraic variety, but it turns out that they also containg gauge freedoms and after a gauge fixing, the solution space becomes discrete. We say that MTCs are rigid, meaning that no contin- uos deformations are possible. The classification of MTCs is an active research direction [58].

These equations first appeared in the work of Moore and Seiberg on 1 + 1D Rational Conformal Field theories [59,60] which in turn are very closely related to topological order in 2 + 1D [61,62]. We will however not go into this very deep and interesting direction.

Important for this thesis are two crucial pieces of universal data in a MTC called the 20_{Stricly speaking, we have only presented part of the definition of a MTC here.}

modular S and T matrices21 Sab =

a

_b

These matrices generate a projective representation of SL(2, Z) and satisfy the relations, (ST )3 = e2πi8 c−C, S2 = C, C2 = 1 (1.23)

where c− is called the chiral central charge and is another universal quantity. Note that knowing S and T , gives us c− mod 8, we will explain why when discussing boundary physics. For abelian topological orders with N anyons these reduce to Sab = _N11/2

eiθa+b eiθa_eiθb, Tab = δabeiθa. Here the T matrix is just the self-statistics while the S matrix gives us the mutual particle statistics (up a normalization). Thus S and T matrices contain informa- tion about statistics of particle excitations. Despite being properties of excitations, they can be computed from the ground states by the universal wavefunction overlap (1.9) as representations of the mapping class group of the torus.

It turns out that from these matrices one can compute the fusion rules, given by the Verlinde formula [63]. Knowing the fusion rules, one can solve the pentagon and hexagon equations and potentially reconstruct the whole catagory theory. Thus until recently it was believed that the S and T matrices can be used to compute any universal quantity in the MTC, but counter examples were recently found [64,65,66]. New invariants were proposed, but they are also in principle computable as mapping class group representations.22

For more details about modular tensor categories and their application to describe anyon models, see [67,56, 68].

A final note before we end this section. We have learned that topological excitations of gapped quantum liquids form degenerate fusion spaces that are stored non-locally and thus completely protected against local perturbations as long as the anyons are well separated. Not only that, we can act on this space using braiding of particles which is a non-local 21_{This diagramatic notation can be defined by introducing inner products to our current construction.} In order to save space, we will not go through the details here.

22_{The smallest known examples are several rank 49 categories corresponding to discrete gauge theories} G = Z11o Z5 with certain group cohomology twists [ω] ∈ H3(Z11o Z5, U (1)). The group is of order 55, and there are 5 inequivalent twists that produces a rank 49 category.

operation and thus also protected against local perturbations. These important properties has given rise to the proposal of using non-abelian anyons to do quantum computation, an idea going under the name topological quantum computation [69, 70, 71]. Thus topological order is a physical mechanism for fault tolerant error correction, rendering millions of redundant qubits, as needed in conventional error correction codes, unnecessary.