update docs

docs/make.jl

@@ -70,6 +70,7 @@ const PAGES = Any[
             "HEOMLS for Bosonic and Fermionic Bath" => "heom_matrix/M_Boson_Fermion.md",
             "HEOMLS for Master Equation" => "heom_matrix/master_eq.md",
         ],
+        "Parity Support" => "Parity.md",
         "Hierarchy Dictionary" => "hierarchy_dictionary.md",
         "Time Evolution" => "time_evolution.md",
         "Stationary State" => "stationary_state.md",

docs/src/Parity.md

@@ -0,0 +1,41 @@
+# [Parity Support](@id doc-Parity)
+## Introduction
+When the system Hamiltonian contains fermionic systems, the HEOMLS matrix ``\hat{\mathcal{M}}`` might be constructed into a different one depend on the parity of the input operator which ``\hat{\mathcal{M}}`` is acting on. This dependence intuitively originates from the properties of partial traces over composite fermionic spaces, where operators do not necessarily commute. 
+
+As an example, for an environment made out of a single fermion, the reduced matrix elements ``\langle{i}|\rho_\textrm{s}^p|{j}\rangle`` (in a basis  labeled by ``\langle i|`` and ``|{j}\rangle``) involve the perturbative sum of expressions of the form ``\langle i| (c \tilde{\rho}_\textrm{e} \tilde{\rho}_\textrm{s}^p c^\dagger+\tilde{\rho}_\textrm{e} \tilde{\rho}_\textrm{s}^p)|{j}\rangle`` (in terms of environmental operators ``\tilde{\rho}_{\textrm{e}}``, system operators ``\tilde{\rho}_\textrm{s}^p`` with parity ``p``, and the environment-annihilation operator ``c``). These quantities depend on the commutator between ``\tilde{\rho}_\textrm{s}^p`` and ``c``, which is trivial only for `EVEN`-parity (``p=+``). In the `ODD`-parity (``p=-``) case, the partial trace over the environment requires further anti-commutations, ultimately resulting in extra minus signs in the expression for the effective propagator describing the reduced dynamics. 
+
+It is important to explicitly note that, here, by `parity` we do not refer to the presence of an odd or even number of fermions in the system but, rather, to the number of fermionic (annihilation or creation) operators needed to represent ``\rho_\textrm{s}^p``. The reduced density matrix of the system should be an `EVEN`-parity operator and can be expressed as ``\rho_{\textrm{s}}^{p=+}(t)``. However, there are some situations (for example, [calculating density of states for fermionic systems](@ref doc-DOS)) where ``\hat{\mathcal{M}}`` is acting on `ODD`-parity ADOs, e.g., ``\rho_{\textrm{s}}^{p=-}(t)=d_{\textrm{s}}\rho_{\textrm{s}}^{+}(t)`` or ``\rho_{\textrm{s}}^{p=-}(t)=d_{\textrm{s}}^\dagger\rho_{\textrm{s}}^{+}(t)``, where ``d_{\textrm{s}}`` is an annihilation operator acting on fermionic systems.
+
+## Parity support for HEOMLS
+One can specify the parameter `parity::AbstractParity` in the function of constructing ``\hat{\mathcal{M}}`` which describes the dynamics of [`EVEN`](@ref)- or [`ODD`](@ref)-parity auxiliary density operators (ADOs). The default value of the parameter is `parity=EVEN`.
+```julia
+Hs::AbstractMatrix  # system Hamiltonian
+Bbath::BosonBath    # bosonic   bath object
+Fbath::FermionBath  # fermionic bath object
+Btier::Int          # bosonic   truncation level 
+Ftier::Int          # fermionic truncation level 
+
+# create HEOMLS matrix in EVEN or ODD parity
+M_even = M_S(Hs, EVEN)
+M_odd  = M_S(Hs, ODD)
+
+M_even = M_Boson(Hs, Btier, Bbath, EVEN) 
+M_odd  = M_Boson(Hs, Btier, Bbath, ODD) 
+
+M_even = M_Fermion(Hs, Ftier, Fbath, EVEN) 
+M_odd  = M_Fermion(Hs, Ftier, Fbath, ODD)
+
+M_even = M_Boson_Fermion(Hs, Btier, Ftier, Bbath, Fbath, EVEN) 
+M_odd  = M_Boson_Fermion(Hs, Btier, Ftier, Bbath, Fbath, ODD) 
+```
+
+## Base functions support
+### Multiplication between Parity labels
+```julia
+EVEN * EVEN # gives EVEN
+EVEN * ODD  # gives ODD
+ODD  * EVEN # gives ODD
+ODD  * ODD  # gives EVEN
+!EVEN       # gives ODD
+!ODD        # gives EVEN
+```

docs/src/heom_matrix/HEOMLS_intro.md

@@ -49,31 +49,6 @@ M = M_Boson_Fermion(Hs, Btier, Ftier, Bbath, Fbath; threshold=1e-7)
 !!! note "Default value of importance threshold"
     The full hierarchical equations can be recovered in the limiting case ``\mathcal{I}_\textrm{th}\rightarrow 0``, which is the default value of the parameter : `threshold=0.0`. This means that all of the ADOs will be taken into account by default.
 
-## [Parity Support for HEOMLS Matrices](@id doc-Parity)
-When the system Hamiltonian contains fermionic systems, the HEOMLS matrix ``\hat{\mathcal{M}}`` might be constructed into a different one depend on the parity of the input operator (HEOMLS) it is acting on. Usually, it is acting on the reduced density operator and [auxiliary density operators (ADOs)](@ref doc-ADOs), which are all in `EVEN`-parity. However, there are some situations (for example, [calculating density of states for fermionic systems](@ref doc-DOS)) where ``\hat{\mathcal{M}}`` is acting on ADOs with `ODD`-parity.
-
-One can specify the parameter `parity::AbstractParity` in the function of constructing ``\hat{\mathcal{M}}`` to be [`EVEN`](@ref) or [`ODD`](@ref). The default value of the parameter is `parity=EVEN`.
-```julia
-Hs::AbstractMatrix  # system Hamiltonian
-Bbath::BosonBath    # bosonic   bath object
-Fbath::FermionBath  # fermionic bath object
-Btier::Int          # bosonic   truncation level 
-Ftier::Int          # fermionic truncation level 
-
-# create HEOMLS matrix in EVEN or ODD parity
-M_even = M_S(Hs, EVEN)
-M_odd  = M_S(Hs, ODD)
-
-M_even = M_Boson(Hs, Btier, Bbath, EVEN) 
-M_odd  = M_Boson(Hs, Btier, Bbath, ODD) 
-
-M_even = M_Fermion(Hs, Ftier, Fbath, EVEN) 
-M_odd  = M_Fermion(Hs, Ftier, Fbath, ODD)
-
-M_even = M_Boson_Fermion(Hs, Btier, Ftier, Bbath, Fbath, EVEN) 
-M_odd  = M_Boson_Fermion(Hs, Btier, Ftier, Bbath, Fbath, ODD) 
-```
-
 ## Methods
 All of the HEOMLS matrices supports the following two `Base` Functions :
  - `size(M::AbstractHEOMLSMatrix)` : Returns the size of the HEOMLS matrix.