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remove fermions (broken work in progress)
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‎docs/src/man/sectors.md‎

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@@ -5,71 +5,77 @@ using TensorKit
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```
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Symmetries in a physical system often result in tensors which are invariant under the action
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of the symmetry group, where this group acts as a tensor product of group actions on every tensor
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index separately. The group action on a single index, or thus, on the corresponding vector space,
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can be decomposed into irreducible representations (irreps). Here, we restrict to unitary
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representations, such that the corresponding vector spaces also have a natural Euclidean inner
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product. In particular, the Euclidean inner product between two vectors is invariant under the
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group action, i.e. the scalar field corresponds to the trivial representation of the group.
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The corresponding vector spaces will be canonically represented as ``V = ⨁_a ℂ^{n_a} ⊗ R_{a}``,
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where ``a`` labels the different irreps, ``n_a`` is the number of times irrep ``a`` appears and
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``R_a`` is the vector space associated with irrep ``a``. Irreps are also known as spin sectors
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(in the case of ``\mathsf{SU}_2``) or charge sectors (in the case of `\mathsf{U}_1`), and we
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henceforth refer to `a` as a sector. As is briefly discussed below, the approach we follow does
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in fact go beyond the case of irreps of groups, and sectors would more generally correspond to
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simple objects in a (ribbon) fusion category. Nonetheless, every step can be appreciated by using
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the representation theory of ``\mathsf{SU}_2`` or ``\mathsf{SU}_3`` as example. The vector space
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``V`` is completely specified by the values of `n_a`.
8+
of the symmetry group, where this group acts as a tensor product of group actions on every
9+
tensor index separately. The group action on a single index, or thus, on the corresponding
10+
vector space, can be decomposed into irreducible representations (irreps). Here, we
11+
restrict to unitary representations, such that the corresponding vector spaces also have a
12+
natural Euclidean inner product. In particular, the Euclidean inner product between two
13+
vectors is invariant under the group action, i.e. the scalar field corresponds to the
14+
trivial representation of the group.
15+
16+
The corresponding vector spaces will be canonically represented as
17+
``V = ⨁_a ℂ^{n_a} ⊗ R_{a}``, where ``a`` labels the different irreps, ``n_a`` is the number
18+
of times irrep ``a`` appears and ``R_a`` is the vector space associated with irrep ``a``.
19+
Irreps are also known as spin sectors (in the case of ``\mathsf{SU}_2``) or charge sectors
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(in the case of `\mathsf{U}_1`), and we henceforth refer to `a` as a sector. As is briefly
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discussed below, the approach we follow doesin fact go beyond the case of irreps of groups,
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and sectors would more generally correspond to simple objects in a (ribbon) fusion
23+
category. Nonetheless, every step can be appreciated by using the representation theory of
24+
``\mathsf{SU}_2`` or ``\mathsf{SU}_3`` as example. The vector space ``V`` is completely
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specified by the values of ``n_a``.
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The gain in efficiency (both in memory occupation and computation time) obtained from using
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symmetric tensor maps is that, by Schur's lemma, they are block diagonal in the basis of
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coupled sectors. To exploit this block diagonal form, it is however essential that we know
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the basis transform from the individual (uncoupled) sectors appearing in the tensor product
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form of the domain and codomain, to the totally coupled sectors that label the different blocks.
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We refer to the latter as block sectors. The transformation from the uncoupled sectors in the
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domain (or codomain) of the tensor map to the block sector is encoded in a fusion tree (or splitting
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tree). Essentially, it is a sequential application of pairwise fusion as described by the group's
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form of the domain and codomain, to the totally coupled sectors that label the different
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blocks. We refer to the latter as block sectors. The transformation from the uncoupled
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sectors in the domain (or codomain) of the tensor map to the block sector is encoded in a
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fusion tree (or splitting tree). Essentially, it is a sequential application of pairwise
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fusion as described by the group's
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[Clebsch-Gordan (CG) coefficients](https://en.wikipedia.org/wiki/Clebsch–Gordan_coefficients).
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However, it turns out that we do not need the actual CG coefficients, but only how they transform
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under transformations such as interchanging the order of the incoming irreps or interchanging
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incoming and outgoing irreps. This information is known as the topological data of the group, i.e.
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mainly the F-symbols, which are also known as recoupling coefficients or [6j-symbols](https://en.wikipedia.org/wiki/6-j_symbol)
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(more accurately, it's actually [Racah's W-coefficients](https://en.wikipedia.org/wiki/Racah_W-coefficient))
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However, it turns out that we do not need the actual CG coefficients, but only how they
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transform under transformations such as interchanging the order of the incoming irreps or
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interchanging incoming and outgoing irreps. This information is known as the topological
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data of the group, i.e. mainly the F-symbols, which are also known as recoupling
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coefficients or [6j-symbols](https://en.wikipedia.org/wiki/6-j_symbol) (more accurately,
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it's actually [Racah's W-coefficients](https://en.wikipedia.org/wiki/Racah_W-coefficient))
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in the case of ``\mathsf{SU}_2``.
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Below, we describe how to specify a certain type of sector what information about them
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needs to be implemented. Then, we describe how to build a space `V` composed of a direct sum
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of different sectors. In the last section, we explain the details of fusion trees, i.e. their
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construction and manipulation. But first, we provide a quick theoretical overview of the required
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data of the representation theory of a group.
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of different sectors. In the last section, we explain the details of fusion trees, i.e.
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their construction and manipulation. But first, we provide a quick theoretical overview of
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the required data of the representation theory of a group.
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## Representation theory and unitary fusion categories
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Let the different irreps or sectors be labeled as ``a``, ``b``, ``c``, … First and foremost,
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we need to specify the *fusion rules* ``a ⊗ b = ⨁ N_{a,b}^{c} c`` with `N_{a,b}^c` some non-negative
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integers. There should always exists a unique trivial sector ``u`` such that ``a ⊗ u = a = u ⊗ a``.
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Furthermore, there should exist a unique sector ``\overline{a}`` such that ``N_{a,\overline{a}}^{u} = 1``,
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whereas for all ``b ≂̸ \overline{a}``, ``N_{a,b}^{u} = 0``. For example, for the representations
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of ``\mathsf{SU}_2``, all irreps are self-dual (i.e. ``a = \overline{a}``) and the trivial
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sector corresponds to spin zero.
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The meaning of the fusion rules is that the space of transformations ``R_a ⊗ R_b → R_c`` (or vice
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versa) has dimension ``N_{a,b}^c``. In particular, we assume the existence of a basis consisting of
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unitary tensor maps ``X_{a,b}^{c,μ} : R_c → R_a ⊗ R_b`` with `μ = 1, \ldots, N_{a,b}^c` such that
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we need to specify the *fusion rules* ``a ⊗ b = ⨁ N_{a,b}^{c} c`` with `N_{a,b}^c` some
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non-negative integers. There should always exists a unique trivial sector ``u`` such that
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``a ⊗ u = a = u ⊗ a``. Furthermore, there should exist a unique sector ``\overline{a}``
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such that ``N_{a,\overline{a}}^{u} = 1``, whereas for all ``b ≂̸ \overline{a}``,
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``N_{a,b}^{u} = 0``. For example, for the representations of ``\mathsf{SU}_2``, all irreps
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are self-dual (i.e. ``a = \overline{a}``) and the trivial sector corresponds to spin zero.
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The meaning of the fusion rules is that the space of transformations ``R_a ⊗ R_b → R_c``
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(or vice versa) has dimension ``N_{a,b}^c``. In particular, we assume the existence of a
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basis consisting of unitary tensor maps ``X_{a,b}^{c,μ} : R_c → R_a ⊗ R_b`` with
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``μ = 1, \ldots, N_{a,b}^c`` such that
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``(X_{a,b}^{c,μ})^† X_{a,b}^{c,μ} = \mathrm{id}_{R_c}``
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and
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``\sum_{c} \sum_{μ = 1}^{N_{a,b}^c} X_{a,b}^{c,μ} (X_{a,b}^{c,μ})^\dagger = \mathrm{id}_{R_a ⊗ R_b}``
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The tensors ``X_{a,b}^{c,μ}`` are the splitting tensors, their hermitian conjugate are the fusion
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tensors. They are only determined up to a unitary basis transform within the space, i.e. acting
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on the multiplicity label ``μ``. For ``\mathsf{SU}_2``, where ``N_{a,b}^c`` is zero or one and the
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multiplicity labels are absent, the entries of ``X_{a,b}^{c}`` are precisely given by the CG
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coefficients. The point is that we do not need to know the tensors ``X_{a,b}^{c,μ}``, the topological
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data of (the representation category of) the group describes the following transformation:
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The tensors ``X_{a,b}^{c,μ}`` are the splitting tensors, their hermitian conjugate are the
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fusion tensors. They are only determined up to a unitary basis transform within the space,
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i.e. acting on the multiplicity label ``μ``. For ``\mathsf{SU}_2``, where ``N_{a,b}^c`` is
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zero or one and the multiplicity labels are absent, the entries of ``X_{a,b}^{c}`` are
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precisely given by the CG coefficients. The point is that we do not need to know the
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tensors ``X_{a,b}^{c,μ}``, the topological data of (the representation category of) the
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group describes the following transformation:
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* F-move or recoupling: the transformation between ``(a ⊗ b) ⊗ c`` to ``a ⊗ (b ⊗ c)``:
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@@ -79,25 +85,27 @@ data of (the representation category of) the group describes the following trans
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``σ_{R_a,R_b} ∘ X_{a,b}^{c,μ} = ∑_{ν} [R_{a,b}^c]^μ_ν X_{b,a}^{c,ν}
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The dimensions of the spaces ``R_a`` on which representation ``a`` acts are denoted as ``d_a``
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and referred to as quantum dimensions. In particular ``d_u = 1`` and ``d_a = d_{\overline{a}}``.
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This information is also encoded in the F-symbol as ``d_a = | [F^{a \overline{a} a}_a]^u_u |^{-1}``.
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Note that there are no multiplicity labels in that particular F-symbol as `N_{a,\overline{a}}^u = 1`.
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The dimensions of the spaces ``R_a`` on which representation ``a`` acts are denoted as
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``d_a`` and referred to as quantum dimensions. In particular ``d_u = 1`` and
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``d_a = d_{\overline{a}}``. This information is also encoded in the F-symbol as
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``d_a = | [F^{a \overline{a} a}_a]^u_u |^{-1}``. Note that there are no multiplicity labels
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in that particular F-symbol as `N_{a,\overline{a}}^u = 1`.
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If, for every ``a`` and ``b``, there is a unique ``c`` such that ``a ⊗ b = c``
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(i.e. ``N_{a,b}^{c} = 1`` and ``N_{a,b}^{c′} = 0`` for all other ``c′``), the category is abelian.
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Indeed, the representations of a group have this property if and only if the group multiplication
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law is commutative. In that case, all spaces ``R_{a}`` associated with the representation are
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one-dimensional and thus trivial. In all other cases, the category is nonabelian. We find it
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useful to further finegrain between categories which have all ``N_{a,b}^c`` equal to zero or
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one (such that no multiplicity labels are needed), e.g. the representations of ``\mathsf{SU}_2``,
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and those where some ``N_{a,b}^c`` are larger than one, e.g. the representations of ``\mathsf{SU}_3``.
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If, for every ``a`` and ``b``, there is a unique ``c`` such that ``a ⊗ b = c`` (i.e.
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``N_{a,b}^{c} = 1`` and ``N_{a,b}^{c′} = 0`` for all other ``c′``), the category is abelian.
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Indeed, the representations of a group have this property if and only if the group
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multiplication law is commutative. In that case, all spaces ``R_{a}`` associated with the
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representation are one-dimensional and thus trivial. In all other cases, the category is
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nonabelian. We find it useful to further finegrain between categories which have all
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``N_{a,b}^c`` equal to zero or one (such that no multiplicity labels are needed), e.g. the
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representations of ``\mathsf{SU}_2``, and those where some ``N_{a,b}^c`` are larger than
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one, e.g. the representations of ``\mathsf{SU}_3``.
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Using ``R_u ≂ ℂ``, ``λ_{R_} ``η_{R_{\overline{a}}}``
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Using ``R_u ≂ ℂ``, ``λ_{R_}`` ``η_{R_{\overline{a}}}``
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If, for every ``a`` and ``b``, there is a unique ``c`` such that ``a ⊗ b = c``
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(i.e. ``N_{a,b}^{c} = 1`` and ``N_{a,b}^{c′} = 0`` for all other ``c′``), the category is Abelian.
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Indeed, the representations of a group have this property if and only if the group multiplication
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law is commutative. In that case, all spaces ``R_{a}`` associated with the representation are
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(i.e. ``N_{a,b}^{c} = 1`` and ``N_{a,b}^{c′} = 0`` for all other ``c′``), the category is Abelian. Indeed, the representations of a group have this property if and only if the group multiplication law is commutative. In that case, all spaces ``R_{a}`` associated with the representation are
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one-dimensional and thus trivial.
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‎src/sectors/fermions.jl‎

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# Sectors giving rise to super vector spaces, which have at least a Z2 grading
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#==============================================================================#
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abstract type AbstractFermion end
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struct Fermion{G<:Sector,F} <: Sector end
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sector::G
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parity::F
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end
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z2fermionparity(c::ℤ₂) = isodd(c.n)
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u1fermionparity(c::U₁) = isodd(c.charge)
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const FermionParity = Fermion{ℤ₂,typeof(z2fermionparity)}
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Base.@pure BraidingStyle(::Type{<:Fermion}) = Fermionic()
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# FermionParity: only total fermion parity is conserved
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struct FermionParity <: Fermion
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parity::Bool
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end
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FermionParity(p::Int) = FermionParity(isodd(p))
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Base.one(::Type{FermionParity}) = FermionParity(false)
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Base.conj(p::FermionParity) = p
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⊗(p1::FermionParity, p2::FermionParity) = (FermionParity(p1.parity != p2.parity),)
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Base.@pure FusionStyle(::Type{FermionParity}) = Abelian()
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const fℤ₂ = FermionParity
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Base.show(io::IO, ::Type{FermionParity}) = print(io, "fℤ₂")
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Base.show(io::IO, p::FermionParity) = get(io, :compact, false) ? print(io, Int(p.parity)) : print(io, "fℤ₂(", Int(p.parity), ")")
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# FermionNumber: total number of fermions is conserved -> U1 symmetry
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struct FermionNumber <: Fermion
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num::Int
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end
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Base.one(::Type{FermionNumber}) = FermionNumber(0)
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Base.conj(n::FermionNumber) = FermionNumber(n.num)
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⊗(n1::FermionNumber, n2::FermionNumber) = (FermionNumber(n1.num+n2.num),)
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Base.@pure FusionStyle(::Type{FermionNumber}) = Abelian()
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const fU₁ = FermionNumber
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Base.show(io::IO, ::Type{FermionNumber}) = print(io, "fU₁")
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Base.show(io::IO, n::FermionNumber) = get(io, :compact, false) ? print(io, n.num) : print(io, "fU₁(", n.num, ")")
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const AbelianFermion = Union{FermionParity,FermionNumber}
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Nsymbol(a::G, b::G, c::G) where {G<:AbelianFermion} = c == first(a ⊗ b)
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Fsymbol(a::G, b::G, c::G, d::G, e::G, f::G) where {G<:AbelianFermion} =
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Int(Nsymbol(a,b,e)*Nsymbol(e,c,d)*Nsymbol(b,c,f)*Nsymbol(a,f,d))
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Rsymbol(a::G, b::G, c::G) where {G<:AbelianFermion} = fermionparity(a) && fermionparity(b) ? -Int(Nsymbol(a, b, c)) : Int(Nsymbol(a, b, c))
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"""
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function fermionparity(s::Fermion) -> Bool
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Returns the fermion parity of a sector `s` that is a subtype of `Fermion`, as a
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`Bool` being true if odd and false if even.
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"""
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fermionparity(p::FermionParity) = p.parity
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fermionparity(n::FermionNumber) = isodd(n.num)
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struct FermionSpin <: Fermion
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dim::Int
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FermionSpin(j::Int) = j >= 0 ? new(2*j+1) : throw(SU2IrrepException)
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function FermionSpin(j::Rational{Int})
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if j.den == 2
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new(j.num+1)
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elseif j.den == 1
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new(2*j.num+1)
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else
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throw(SU2IrrepException)
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end
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end
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end
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_getj(s::FermionSpin) = (s.dim-1)//2
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Base.one(::Type{FermionSpin}) = FermionSpin(0)
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Base.conj(s::FermionSpin) = s
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⊗(s1::FermionSpin, s2::FermionSpin) = SectorSet{FermionSpin}( abs(_getj(s1)-_getj(s2)):(_getj(s1)+_getj(s2)) )
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Base.@pure FusionStyle(::Type{FermionSpin}) = SimpleNonAbelian()
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Nsymbol(s1::FermionSpin, s2::FermionSpin, s::FermionSpin) = (abs(s1.dim-s2.dim)+1) <= s.dim <= (s1.dim+s2.dim-1) && isodd(s1.dim+s2.dim - s.dim)
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# TODO: Fsymbol -> 6j-symbols of SU(2), compute, cache, ... -> Recycle those of SU2Irrep
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const fSU₂ = FermionSpin
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Base.show(io::IO, ::Type{FermionSpin}) = print(io, "fSU₂")
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Base.show(io::IO, s::FermionSpin) = get(io, :compact, false) ? print(io, _getj(s)) : print(io, "fSU₂(", _getj(s), ")")
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fermionparity(s::FermionSpin) = iseven(s.dim)
1+
# TODO

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