Torus

The ground-state degeneracy of the Levin-Wen model on a torus, for every category it was run on, with its checks and timings. Its stages follow this description; the checks are defined at the bottom of the page.

The Levin-Wen string-net model on a $2\times2$ honeycomb lattice on the torus, with a vertex projector $A_v$ on every vertex and a plaquette projector $B_p$ on every hexagon,

H = -\sum_{v} A_v - \sum_{p} B_p,\qquad B_p = \frac{1}{\dim(\mathcal{C})}\sum_{s} d_s\, B_p^{s} .

The vertex projectors are imposed exactly by restricting to the physical basis of fusion-allowed edge labellings; the plaquette operators are eq. 20 of PhysRevB.102.115154, with block inverses $F^{-1}$ at the inverse moves. Every ground state has energy $E_0 = -(|V| + |P|)$ , and the dimension of the ground space is the number of simple objects of the center:

\operatorname{GSD}(T^2) = \operatorname{rank}(\mathcal{Z}(\mathcal{C})) .

The $B_p$ are commuting projectors, so their product $P = \prod_p B_p$ projects onto the joint $B_p = 1$ eigenspace, which is the ground space, and the degeneracy is its trace, $\operatorname{GSD} = \operatorname{Tr}P$ . Each $B_p$ has between 6 and a few hundred nonzeros per column, so $P$ is never formed: the trace is a sum over columns of three sparse matrix-vector products against a row of $B_4$ , and the cost is linear in the basis dimension. This works for unitary and non-unitary input alike, and it is the same for both.

Input	Method	Delivers
any category	the trace of the sparse commuting projector $P = \prod_p B_p$ ; $P^2 = P$ is checked on a random vector and the trace must be an integer.	rank
unitary, physical basis of at most 3000 states	as a cross-check, $H$ is also assembled densely and its lowest eigenspace counted; the count must equal the trace, and this is where $E_0 = -(\|V\|+\|P\|)$ is read off directly.	agrees with the trace
the four B_p would not fit in 16 GB, or the basis exceeds 2 million states	skipped, not estimated, with the reason recorded (fill is sampled on 2000 columns before anything is built).	nothing
non-unitary, ill-conditioned at 16 figures	the projector check fails ( $\operatorname{Tr}P$ off an integer, $P^2 \neq P$ ) and the category is refused rather than rounded.	not computed
blocks, dimensions, fusion, braiding of $\mathcal{Z}(\mathcal{C})$	a trace on one torus carries none of this; it is a count.	not a torus quantity

Stages

Stage	Builds	Checks
1	physical basis: the fusion-allowed edge labellings of the $2\times2$ honeycomb torus, so every $A_v$ is imposed exactly	its size: above 2 million states the category is skipped, with the reason recorded
2	plaquette operators $B_p$ , eq. 20 of PhysRevB.102.115154, sparse, with block inverses $F^{-1}$ at the inverse moves	their fill, sampled on 2000 columns before anything is built: if the four $B_p$ would not fit in 16 GB the category is skipped
3	$\operatorname{GSD} = \operatorname{Tr}P$ , $P = \prod_p B_p$ , as a sum over columns of three sparse matrix-vector products against a row of $B_4$	$\\|P^2 v - P v\\| / \\|P v\\|$ on a random vector and the distance of $\operatorname{Tr}P$ from an integer, both within $10^{-6}$ , or the category is refused
4	for unitary input with a basis of at most 3000 states, $H$ assembled densely and its lowest eigenspace counted	the count equals the trace; $E_0 = -(\|V\|+\|P\|)$

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