We start by constructing the PEPS representation of the ground state N Y
p
Bp(⊗e|1ie) , (6.1)
whereN is a normalization factor. Graphically, (a patch of) this state can be represented as
= N X
{µ}
dµ1dµ2· · ·
µ1 µ2
µ3 µ4 , (6.2)
where qudits in the |1i state are represented by the gray edges. To find the state from this graphical notation, one has to resolve the loops appearing in Eq. (6.2) into the lattice.
This is done in two steps. First we fuse the neighboring loops along each edge using Eq. (2.33):
µ ν
=X
k
vk vµvνδµνk
µ ν µ
ν
k .
Then, the resulting configuration at every vertex is resolved into the lattice using Eq. (2.52):
i
j λ k
µ
ν =vλvµvνGijkλµν i
j k
. (6.3)
The PEPS tensor is obtained by splitting the factor in Eq. (6.3) evenly between the two adjacent vertices. The result is
ν0 µ
µ0 ν
λ0 λ
i
j k
i0 j0
k0
=δii0δjj0δkk0δλλ0δµµ0δνν0√
vivjvkGijkλµν, (6.4)
wherei0 ,j0 and k0 represent the physical degrees of freedom associated to the qudits on the 3 edges. Note that the physical degrees of freedom are doubled, since each of them is appearing at the two vertices connected to a given edge. The diagonal structure of Eq. (6.4), ensures that the values of the two physical indices representing the same qudit are always equal. We impose the convention that for every closed loop with label µ on the virtual level, a factor dµ is implied. This convention automatically takes care of the
factorsdµi appearing in Eq. (6.2), which can then be represented as
. (6.5)
To simplify the notation, we will follow the convention that a pair of legs crossing through a tensor represents a Kronecker delta between the indices on these legs, indicat-ing that the tensor is diagonal in this pair of indices.
The tensor network state constructed above corresponds specifically to the ground state Eq. (6.1). It can, however, be modified to represent any string-net ground state. To do this, first note that we can obtain any ground state by acting with the total projector B = QpBp on some state |φi ∈ Hs.n. corresponding to a configurations of strings on the lattice. The tensor network state for B|φi, can then be obtained by modifying our original construction, to account for additional strings running between the vacuum loops in Eq. (6.2). Locally, such a string-configuration can take two forms: a single string, or two strings fusing to a third one.
The first configuration looks as follows:
X
where we have pulled the strings into the different loops along its path using Eq. (2.33).
UsingF-moves, we can again pull ribbons from neighboring plaquettes into each edges:
µ0
The first two factors on the right hand side can be recognized as the symmetrized contri-bution to each vertex from Eq. (6.3). The first factor on the right hand side of Eq. (6.7) will therefore contribute to the vertex to the left of the edge, while the second factor will contribute to the right vertex, ensuring that we can use the PEPS tensor Eq. (6.4) on both vertices. The third and fourth factors on the right hand side will contribute to the upper and lower plaquette of the edge in Eq. (6.7), respectively. When combined with the factors in Eq. (6.6), these contributions result in a total factor of the form vµvvµ0
s ·vsvµvµ0 =dµdµ0 for each plaquette separately [instead of a single quantum dimension factor like in Eq. (6.2)].
If we now place the crossing tensor s
k ν
µ0 µ
ν0
=Gµµν0νk0s, (6.8)
in the PEPS at every crossing of the ribbon with labelswith a lattice edge, this gives the correct superposition of qudit states. Note that the quantum dimension factors in each plaquette are again taken care of by the convention for closed loops at the virtual level.
Eq. (6.6) can then be represented with the following tensor network state:
s
, (6.9)
where the additional loops in the plaquette correspond to the second summation on the right hand side of Eq. (6.6). Note that in the tensor network representation above, one should ensure that the string-label is fixed to the correct string-labels. In case one were to use the tensors above to represent a closed string, an additional Kronecker-delta tensor must be included to avoid summing over all string-labels.
The tensor network representation of a ground state obtained from a string-configuration containing fusing strings, is derived in a similar fashion. We again start by pulling the
strings onto the various loops using Eqs. (2.33) and (2.52):
The edge crossings can again be resolved as in Eq. (6.7), giving exactly the same situation as before for the upper plaquette. For the lower plaquette, the combined contribution from of the last factor in Eq. (6.7) for the three edge crossings combines with the right hand side of Eq. (6.10) to a total factor of the form
√vsvµ0vµ00√
vtvµvµ00√
vuvµvµ0vµvµ0vµ00Gsµµut00µ0 =dµdµ0dµ00√
vsvtvuGsµµut00µ0. (6.11) We can now define the tensor
µ to represent the fusion of strings on the virtual level. This tensor, along with the closed loop convention at the virtual level, takes care of all remaining factors in Eq. (6.11), and gives the correct superposition of qudit states. A ground state obtained from a initial configuration containing fusing strings can then be represented as:
s
t u
, (6.13)
where one must again be cautious to fix the string-labels to the correct values when “closing off” this tensor network on the virtual level.
The tensors in Eqs. (6.4), (6.8) and (6.12) can be used to construct the tensor network representation of any ground state of the Levin-Wen model. Ground states of the extend Levin-Wen model can be constructed by adding the following tensor on ever vertical edge:
x µ ν
y
x0
y0 t =δxx0δyy0δx0y0δt1. (6.14)
This tensor simply includes a trivial tail qudit and replaces the qudit on the vertical edge by two identical ones. Since its action is trivial, we deem it unnecessary to draw it explicitly. In the tensor network diagrams below, its pretense is implied in any plaquette where the tail qudit is not specified explicitly.