Kagome loop

[Chenqitrg]

By performing a step of contraction, a 2D tensor network on honeycomb graph can be transformed to one on a kagome graph, where the LoopTNR can be designed.

This PR realizes LoopTNR on kagome graph. It can be applied to the honeycomb graph as well. Most functions in LoopTNR can be used here. The only subtlety is the entanglement filtering and the final contraction.
Convention.pdf

Further test is still needed.

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[Chenqitrg]

@soonyotta

[Adwait-Naravane]

This is very interesting. I would like to know more about how coarse graining is performed here.

[Chenqitrg]

The diagram for this coarse graining is attached in the file.

[Chenqitrg]

An observation: since system size is increased by three times after each RG step, the error accumulates much faster than that on the square lattice. The leading order perturbation can be estimated to be $x \times 3^n$, where $x$ is the initial error and $n$ is the number of RG steps. Therefore if $x \sim 10^{-7}$ and $n \sim 15$ when the perturbation is $O(1)$. The current test shows the RG step is stable when $\chi = 16$ and $n < 20$.

[Chenqitrg]

A little strange: tests can pass locally but it reports cft_data! not defined.

[VictorVanthilt]

As of #122 the cft data functions should not mutate the tensors its calculating with.

[Chenqitrg]

@VictorVanthilt OK, will try to correct those parts.

[Chenqitrg]

Summary of the modification:

1. Added the Loop-TNR algorithm on the kagome lattice and its corresponding CFT data calculation approach (the GSD calculation is a special case of the CFT data when the only conformal dimensions are 0's)
2. Defined subtypes LinearLoopScheme{E, S} <: LoopScheme{E, S} <: TNRScheme{E, S}. Added KagomeLoopTNR{E, S, TT} <: LinearLoopScheme{E, S} and LoopTNR{E, S, TT} <: LinearLoopScheme{E, S}. And slightly modified SLoopTNR{E, S, TT} <: LoopScheme{E, S}, because they are all based on the Loop optimization. Also, the run! method can be used universally by the KagomeLoopTNR and the usual LoopTNR.
3. Added the classical Ising model on the dual graph of the triangular lattice, including the one protected by the $Z_2$ symmetry and the one not protected by the $Z_2$ symmetry
4. Tested the CFT data using the dual triangular Ising model, including the central charge and scaling dimensions
5. Refined the previous cft_data method based on the Krylov method: the returned dictionary now stores only conformal dimensions, and is constructed by a certain type rather than Any. The central charge is just another number and is not stored in the same dictionary as scaling dimensions. By introducing methods _action_assignmentor_no_approximation and _action_assignmentor_approximation, the cft_data_solver can be used universally for all shapes including ones on the Kagome lattice.

[Chenqitrg]

The current one is not very compatible with NNR-TNR. I am preparing to isolate the cft_data calculation first and try to fix the Dict{Any, Any} problem. The Kagome Loop will be left to the future.