Fermions

Struqture can be used to represent Fermion operators, hamiltonians and open systems, such as:

\[ \hat{O} = \sum_{j=0}^N \alpha_j \left( \prod_{k=0}^N f(j, k) \right) \left( \prod_{l=0}^N g(j, l) \right) \] with \[ f(j, k) = \begin{cases} c_k^{\dagger} \\ \mathbb{1} \end{cases} , \] \[ g(j, l) = \begin{cases} c_l \\ \mathbb{1} \end{cases} , \] and \[ \dot{\rho} = \mathcal{L}(\rho) = -i [\hat{H}, \rho] + \sum_{j,k} \Gamma_{j,k} \left( L_{j}\rho L_{k}^{\dagger} - \frac{1}{2} \{ L_k^{\dagger} L_j, \rho \} \right) \]

The simplest way that the user can interact with these matrices is by using symbolic representation: "c0a0" represents a \( c^{\dagger}_0\ c_0 \) term. Note: Here c is used in the equations to represent any fermion operator while b was used to represent a boson operator. However, in string serialisation strquture uses the convention that c always represents a creation operator, whether in the fermionic or bosonic degrees of freedom and a always represents an annihilation operator. This is a very scalable approach, as indices not mentioned in this string representation are assumed to be acted on by the identity operator: "c7a25" represents a \( c^{\dagger}_7 c_{25} \) term, where all other terms (0 to 6 and 8 to 24) are acted on by \(I\).

However, for more fine-grain control over the operators, we invite the user to look into the FermionProduct and HermitianFermionProduct classes, in the Building blocks section. If not, please proceed to the coherent or decoherent dynamics section.