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Copy file name to clipboardExpand all lines: docs/src/man/Implementation.md
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explain the f-symbol and n-symbol storage system
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# MultiTensorKit implementation: $\mathsf{Rep_{A_4}}$ as an example
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# MultiTensorKit implementation: $\mathsf{Rep(A_4)}$ as an example
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This tutorial is dedicated to explaining how MultiTensorKit was implemented to be compatible with with TensorKit and MPSKit for matrix product state simulations. In particular, we will be making a generalised anyonic spin chain. We will demonstrate how to reproduce the entanglement spectra found in [Lootens_2024](@cite). The model considered there is a spin-1 Heisenberg model with additional terms to break the usual $\mathsf{U_1}$ symmetry to $\mathsf{Rep(A_4)}$, while having a non-trivial phase diagram and relatively easy Hamiltonian to write down.
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This will be done with the `A4Object = BimoduleSector{A4}``Sector`, which is the multifusion category which contains the structure of the module categories over $\mathsf{Rep_{A_4}}$. Since there are 7 module categories, `A4Object` is a $r=7$ multifusion category. There are 3 fusion categories up to equivalence:
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- $\mathsf{Vec A_4}$: the category of $\mathsf{A_4}$-graded vector spaces. The group $\mathsf{A}_4$ is order $4!/2 = 12$. It has thus 12 objects.
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- $\mathsf{Rep_{A_4}}$: the irreducible representations of the group $\mathsf{A}_4$, of which there are 4. One is the trivial representation, two are one-dimension non-trivial and the last is three-dimensional.
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- $\mathsf{Rep H}$: the representation category of some Hopf algebra which does not have a name. It has 6 simple objects.
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- $\mathsf{Vec_{A_4}}$, the category of $\mathsf{A_4}$-graded vector spaces. The group $\mathsf{A}_4$ is order $4!/2 = 12$. It has thus 12 objects.
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- $\mathsf{Rep(A_4)}$: the irreducible representations of the group $\mathsf{A}_4$, of which there are 4. One is the trivial representation, two are one-dimensional non-trivial and the last is three-dimensional.
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- $\mathsf{Rep(H)}$: the representation category of some Hopf algebra which does not have a name. It has 6 simple objects.
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````julia
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using TensorKit, MultiTensorKit, MPSKit, MPSKitModels
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## Constructing the Hamiltonian and matrix product state
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TensorKit has been made compatible with the multifusion structure by keeping track of the relevant units in the fusion tree manipulations. With this, we can make `GradedSpace`s whose objects are in `A4Object`:
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````julia
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D1 =A4Object(6, 6, 1) # unit in this case
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D2 =A4Object(6, 6, 2) # non-trivial 1d irrep
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D3 =A4Object(6, 6, 3) # non-trivial 1d irrep
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D4 =A4Object(6, 6, 4) # 3d irrep
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````
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Since we want to replicate a spin-1 Heisenberg model, it makes sense to use the 3-dimensional irrep to grade the physical space, and thus construct our Hamiltonian. We don't illustrate here how to derive the considered Hamiltonian in a $\mathsf{Rep(A_4)}$ basis, but simply give it.
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Since we want to replicate a spin-1 Heisenberg model, it makes sense to use the 3-dimensional irrep to grade the physical space, and thus construct our Hamiltonian. We don't illustrate here how to derive the considered Hamiltonian in a $\mathsf{Rep(A_4)}$ basis, but simply give it. For now, we construct a finite-size spin chain; below we will repeat the calculation for an infinite system.
H1 =@mpohamsum(-2* h1{i,j} for (i, j) innearest_neighbours(lattice))
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H2 =@mpohamsum(h2{i,j} for (i, j) innearest_neighbours(lattice))
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H = H1 + J1 * H2 + J3 * H3
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````
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## Constructing the matrix product state
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For now, we will select $\mathsf{Vec}$ as the module category:
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For the matrix product state, we will select $\mathsf{Vec}$ as the module category for now:
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````julia
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M =A4Object(1, 6, 1) # Vec
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````
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Afterwards, we build the physical and virtual space of the matrix product state:
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and construct the finite MPS:
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````julia
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D =40# bond dimension
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V = Vect[A4Object](M => D)
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Vb = Vect[A4Object](M =>1) # non-degenerate boundary virtual space
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init_mps =FiniteMPS(L, P, V; left=Vb, right=Vb)
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````
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> [!IMPORTANT]
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> We must pass on a left and right virtual space to the keyword arguments `left` and `right` of the `FiniteMPS` constructor, since these would by default try to place a trivial space of the `Sector`, which does not exist for `BimoduleSector` due to the semisimple unit.
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> We must pass on a left and right virtual space to the keyword arguments `left` and `right` of the `FiniteMPS` constructor, since these would by default try to place a trivial space of the `Sector`, which does not exist for any `BimoduleSector` due to the semisimple unit.
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## DMRG2 and the entanglement spectrum
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We can now look to find the ground state of the Hamiltonian with two-site DMRG. We use this instead of the "usual" one-site DMRG because the two-site one will smartly fill up the blocks of the local tensor during the sweep, allowing one to initialise as a product state in one block and more likely avoid local minima, a common occurence in symmetric tensor network simulations.
The truncation scheme keyword argument is mandatory when calling `DMRG2` in MPSKit. Here, we choose to truncate such that all singular values are larger than $10^{-4}$, while setting the default tolerance for convergence to $10^{-8}$. More information on this can be found in the [MPSKit](https://github.com/QuantumKitHub/MPSKit.jl) documentation.
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Now that we've found the ground state, we can compute the entanglement spectrum in the middle of the chain.
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````julia
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spec =entanglement_spectrum(ψ, round(Int, L/2))
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````
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This returns a dictionary which maps the objects grading the virtual space to the singular values. In this case, there is one key corresponding to $\mathsf{Vec}$. We can also immediately return a plot of this data by the following:
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````julia
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using Plots # !
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entanglementplot(ψ;site=round(Int, L/2))
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````
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This plot will show the singular values per object, as well as include the "effective" bond dimension, which is simply the dimension of the virtual space where we cut the system. #TODO: actually include the plot (or run everything as ipynb)
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## Search for the correct dual model
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Consider a quantum lattice model with its symmetries determing the phase diagram. For every phase in the phase diagram, the dual model for which the ground state maximally breaks all symmetries spontaneously is the one where the entanglement is minimised and the tensor network is represented most efficiently [Lootens_2024](@cite). Let us confirm this result, starting with the $\mathsf{Rep(A_4)}$ spontaneous symmetry breaking phase.
Copy file name to clipboardExpand all lines: docs/src/man/multifusioncats.md
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The most general case is the top left figure, where all three regions have a different coloring. The top middle region having the same coloring from the top left and top right strands follow from the delta function in the tensor product definition. However, as will be explained more in detail later, this most general trivalent junction with three colorings will never be needed. In short, we will always be considering a single bimodule category $\mathcal{C}_{ij}$ at a time, and the only other non-diagonal subcategory which fuses with this is its opposite $\mathcal{C}_{ji}$. This is displayed in the top middle and right. Similarly, two colorings are required when considering the fusion between a fusion and module strand, shown in the bottom left and middle figure. The simplest trivalent junctions boil down to fusions within fusion categories, which is obviously drawn with just one color. This is shown in the bottom right.
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With this coloring system, we can specify which associator must be called to perform a particular F-move. SHOW THE COLORS
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With this coloring system, we can specify which associator must be called to perform a particular F-move. Such an F-move would look like
where by $\mathbb{1}_a$ we mean the right unit of $a$ (the left unit we would denote $^{}_a \mathbb{1}$).
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where by $\mathbb{1}_j$ we mean the unit of $\mathcal{C}_j$.
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### More on the topological data: gauge choices and distilling properties of the subcategories
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The most generic F-move contains 4 colors. For that reason, MultiTensorKit will require the F-symbol data to provided as some data file (currently .txt) with 4 + 6 + 4 + 2 = 16 columns. The first 4 refer to the colors, the following 6 label the simple objects of the corresponding subcategories, the next 4 label multiplicities, and the final 2 provide the real and imaginary value of the F-symbol itself.
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In a similar manner, the N-symbols contain maximally three colors, so these data must provide 3 columns labeling the colors, 3 columns labeling the simple objects and a final column with the dimension of the corresponding vector space.
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Besides the B-move (and closely related A-move, which we do not illustrate), we can also see how the quantum dimension and Frobenius-Schur indicator expressions get modified. We already know that an F-move of the form $F^{a \bar{a} a}_{a}$ needs to be evaluated for these topological data. Graphically, we find that
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Besides the B-move (and closely related A-move, which we do not illustrate), we can also see how the quantum dimension and Frobenius-Schur indicator expressions get modified. We already know that an F-move of the form $F^{c \bar{c} c}_{c}$ needs to be evaluated for these topological data. Graphically, we find that
need to show other changed expressions like A-move, dimension, frobenius-schur indicator, what else outside of TensorKitSectors in terms of fusion tree manipulations?
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An important property of the F-symbols is that they must satisfy the **triangle identities**. In fusion category theory, this states that isomorphisms between (simple) objects $a$ and the tensor product between $a$ and the unit $\mathbb{1}$ exists, and that
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$$(a \otimes \mathbb{1}) \otimes b \cong a \otimes (\mathbb{1} \otimes b)$$
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for $b$ in the same fusion category. This can be straightforwardly generalised to multifusion categories. This requires a particular gauging of these trivalent vertices.
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Besides the triangle identities, the (multi)fusion category must also fulfill the **pentagon equations**. These encapsulate the two identical manners to evaluating the fusion of four objects in the (multi)fusion category. Every fusion category's F-symbols must satisfy these individually, but also the (bi)module associators between bimodule and fusion categories. One can check that, for every pair of fusion categories, their bimodule category and opposite bimodule category, there are 32 pentagon equations to be satisfied. In the multifusion notation, they can be represented by
no figures up till now with arrows, will this even be necessary? maybe if we show B-moves with M->Mop
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We briefly mentioned earlier how unitary F-symbols are required within TensorKit, and thus necessarily MultiTensorKit.
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### Braiding
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A very important aspect of MultiTensorKit is that all `BimoduleSector`s are defined to *not* support braiding: `TensorKitSectors.BraidingStyle(::Type{<:BimoduleSector}) = NoBraiding()`. We do this for two reasons. On the one hand, there is no natural 1-categorical way of defining braidings between the components of the multifusion category. It is possible that the diagonal fusion categories themselves are braided, but a "componentwise" braiding is unwise to support. On the other hand, it is entirely possible to write matrix product state manipulations in a planar manner (which has been done in [MPSKit](https://github.com/QuantumKitHub/MPSKit.jl)), thus avoiding the need of a braiding tensor.
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