Finally, we briefly comment on the spectrum of the Hamiltonian (9.1) and the types of excitations it supports. Each of the terms in the Hamiltonian enforces one of the local relations of the vector space assigned to Σ byS. As such, the zero-energy ground space is just the vector space assigned to Σ byS. The deconfined anyonic excitations in the model correspond to violations of the plaquette and vertex terms. By construction, these are in one-to-one correspondence with the bounding idempotents of the tube category. This is simply because as discussed earlier, the Bp operator in the plaquette term is the projector Π1, which projects onto states containing no quasiparticles in the interior ofp. The fusion rules of the excitations can be computed with the tube category methods developed in earlier sections.
One can also consider the excitations corresponding to violations of the edge terms (note that these will only be present in theories with q-type objects). If a given state has a +1 eigenvalue under an edge term (1−De), then the fermions traveling across e pick up an additional minus sign relative to the background spin structure. This is illustrated in Figure
9.8, where the violated edge e is shown in red, and the two circles marked N denote the vortices created on either side of e. Diagrammatically we denote the additional minus sign with a branch cut (the dashed line in Figure 9.8). This implies that violating a single edge term nucleates a pair of vortices (the set of which are in one-to-one correspondence with the non-bounding idempotents of the tube category) on the plaquettes adjacent to the edge
e (recall that the plaquette terms are only non-zero in the ground space of the edge and vertex terms). This means that the vortex excitations can only be separated at the expense
Figure 9.8: A region of the graphG containing two vortex excitations. The two punctures hosting the vortex excitations are marked as circles, with the label N denoting their non- bounding spin structure. The dashed blue line is the spin structure defect connecting the two punctures. The edge colored red intersects the spin structure defect and is excited, violating the edge termDe in the Hamiltonian.
of a linear increase in energy, and so are linearly confined.
Alternatively, by modifying the Hamiltonian we can introduce vortices by hand. We remove the plaquette terms where we wish the vortices to reside, and require the corre- sponding plaquette boundaries to have a non-bounding spin structure. Relative to the un-modified Hamiltonian, the vortices will be connected by spin structure branch cuts. (The edge terms of the modified and unmodified Hamiltonians will differ for edges which intersect these branch cuts.) A ground state of the modified Hamiltonian will be an excited state of the unmodified Hamiltonian, whose energy depends on the choice of branch cut. To deconfine the vortices one needs to give the spin structure dynamics; we leave the study of this possibility to future work.
Chapter 10
Super pivotal state sums and
tensor networks
In this section we describe a version of the Turaev-Viro-Barrett-Westbury (TVBW) state sum [58, 59] for super pivotal fusion categories and a tensor network for the ground state wave function of the Hamiltonian constructed in Section 9. Related work was presented in [26], see also [41]. We first review the TVBW construction for bosonic spherical fusion categories. We then show how to write the state-sum as a tensor contraction on a tensor network. Next we detail the modifications needed for the fermionic versions of the state sum and tensor network. Lastly we use the state sum to write down an explicit wave function for the ground ground state of (9.1).
Before we begin, we need to establish some terminology regarding cell and handle de- compositions. Recall that a handle decomposition for a 3-manifoldM is built from a series of k-handles, with k = 0,1,2,3, each of which is identified with Dk×D3−k. Handle de- compositions can be obtained from cell decompositions by thickening each k-cell into a
k-handle. Conversely, each handle decomposition determines a cell decomposition by tak- ing the cores of the handles. (See Section 9.2 for more details.) We will often refer to a
k-cell and its associated k-handle with the same letter, since it will be convenient for us to be able to describe things in terms of both handle decompositions and their corresponding cell decompositions. We call Sk−1 ×D3−k the attaching region (or attaching boundary) of the k-handle, and Dk×S3−k−1 the non-attaching boundary. The attaching map of a
k-handle is a homeomorphism from the attaching region to a submanifold of the boundary of the union of the lower-dimensional handles. The topology ofM is encoded by the various attaching maps. The different types of k-handles are illustrated in Figure10.1.
Figure 10.1: The handles corresponding to a standard cubic cell decomposition (a). The 0-, 1- and 2-handles are shown in (b), (c) and (d), with colors green, blue and red. In (b) the blue disks on the 0-handle indicate where the 1-handles attach to the 0-handle, and the red rectangles indicate where the 2-handles attach to the 0-handle. In (c) the green disks indicate where the 0-handles attach to the 1-handle and the red rectangles indicate where the 2-handles attach to the 1-handle. Similarly, in (d) the blue and green rectangles indicate where the 2-handles attach to the 0- and 1-handles. We have omitted the 3-handles from the figure.