Spin-like orbitals

In Definition we define the notion of spin-like orbitals and in Local parity we propose the local parity as a metric for the spin-like character of orbitals. In Local parity optimization of the orbital basis, we present our parity optimization procedure as a way to determine the spin-like orbitals of a system.

Definition

We consider an orbital ${\varphi }_i$ as being spin-like only if the electron density $n_i$ contained in it is strictly equal to one. This requires that the average electron density in the orbital ${\varphi }_i$ satisfies $⟨n_i⟩=1$. It furthermore requires that the fluctuations around this average electron density satisfy $\delta n_i\to 0$. Negligible fluctuations around the average electron density imply that electron density of the orbital does not affect the low-energy dynamics of the system and vice versa. Just an average electron density $⟨n_i⟩=1$ places no restrictions on the orientation and dynamics of the electron spin in the orbital ${\varphi }_i$. When both requirements are simultaneously met by the states in the low-energy Hilbert space, the dynamics of the electron density in the orbital ${\varphi }_i$, often referred to as the charge degree of freedom, become superfluous to the description of the dynamics of the system. The electron density in the orbital ${\varphi }_i$ consequently couples to the remainder of the system exclusively via its spin degree of freedom. It is then sufficient to represent the degrees of freedom of the electron density contained in the orbital ${\varphi }_i$ as a pure spin degree of freedom and to employ the associated spin operator algebra. The local Hilbert space ${\mathcal{H}}_i$ of a spin degree of freedom is half the size of the local Hilbert space of a fermionic orbital and is furthermore naturally represented on a qubit. We therefore aim to determine each spin-like basis orbital or linear combinations thereof meeting both stated requirements, so that they can be represented as spins.

Local parity

We propose the ground state local parity $P_i$ as a measure for the spin-like character of an orbital ${\varphi }_i$. The operator representation of the local parity reads $$P_i=(-1{)}^{n_{i\uparrow }+n_{i\downarrow }}$$ where $n_{i\uparrow }=c_{i\uparrow }^{†}{c}_{i\uparrow }$ denotes the electron density in ${\varphi }_i$ with electron spin quantum number $s_i^z=+1/2$ and $n_{i\downarrow }=c_{i\downarrow }^{†}{c}_{i\downarrow }$ the electron density with quantum number $s_i^z=-1/2$ respectively. The local Hilbert space ${\mathcal{H}}_i$ of the orbital ${\varphi }_i$ is spanned by the states,

$$\left\{|0⟩,|\uparrow ⟩,|\downarrow ⟩,|\uparrow \downarrow ⟩\right\}$$

and the action of the local parity parity operator on these states reads $$\begin{array}{lcl}{P}_i|0⟩& =& +1|0⟩\\ P_i|\downarrow ⟩& =& -1|\downarrow ⟩\\ P_i|\uparrow ⟩& =& -1|\uparrow ⟩\\ P_i|\uparrow \downarrow ⟩& =& +1|\uparrow \downarrow ⟩.\end{array}$$ For states where the orbital contains a single electron, the local parity operator $P_i$ returns the eigenvalue $p_i=-1$. For the remaining two basis states, $P_i$ returns the eigenvalue $p_i=+1$. Any state $|\psi ⟩$ in the local Hilbert space ${\mathcal{H}}_i$ that contains contributions from non singly occupied basis states hence satisfies $$P_i|\psi ⟩=\left(-1+\alpha \right)|\psi ⟩,$$ with $2\ge \alpha >0$, because the resulting fluctuations in the electron density $\delta n_i\ne 0$ manifest themselves in strictly positive contributions to the local parity. An alternative and more useful operator representation of the local parity reads $$\begin{gathered} P_i=\left(1-2c_{i\uparrow }^{†}{c}_{i\uparrow }\right)\left(1-2c_{i\downarrow }^{†}{c}_{i\downarrow }\right) \\ =1-2\left(c_{i\uparrow }^{†}{c}_{i\uparrow }+c_{i\downarrow }^{†}{c}_{i\downarrow }\right)+4c_{i\uparrow }^{†}{c}_{i\downarrow }^{†}{c}_{i\downarrow }{c}_{i\uparrow }, \end{gathered}$$ and its corresponding expectation value with respect to the many-body state $|n⟩$ can be expressed as $$⟨n|P_i|n⟩=1-2{\rho }_{ii}^{(1)}+4{\rho }_{iiii}^{(2)},$$ where $${\rho }_{qp}^{(1)}=\sum _{\sigma }⟨n|c_{q\sigma }^{†}{c}_{p\sigma }|n⟩,$$ denotes the one-electron reduced density matrix (1-RDM) and $${\rho }_{qprs}^{(2)}=⟨n|c_{q\uparrow }^{†}{c}_{p\downarrow }^{†}{c}_{r\downarrow }{c}_{s\uparrow }|n⟩,$$ denotes the two-electron reduced density matrix (2-RDM) respectively. If we choose $|n⟩=|{\psi }_0⟩$, with $|{\psi }_0⟩$ a good approximation of the ground state of the system, we can identify orbitals ${\varphi }_i$ for which $⟨P_i{⟩}_0=⟨{\psi }_0|P_i|{\psi }_0⟩=-1+\epsilon$, with $\epsilon \to 0$, as spin-like orbitals of the system. In general, the spin-like orbitals of the system do not coincide with the basis orbitals. We therefore require a method to determine the set $\{{\varphi }_i\}$ of orthonormal linear combinations of basis orbitals, for which the local parities most closely approach $\{p_i\}=-1$.

Local parity optimization of the orbital basis

We propose an iterative procedure to determine the particular orbital basis in which the local parities are extremal. For this we attempt a sequence of unitary pairwise rotations of the orbital fermionic operators given by $$\begin{gathered} c_{q\sigma }=\cos \theta c_{i\sigma }+\sin \theta c_{j\sigma }, \\ {c}_{p\sigma }=-\sin \theta c_{i\sigma }+\cos \theta c_{j\sigma }, \end{gathered}$$ with the same rotation being performed for the hermitian conjugates of the operators. From the reduced density matrices ${\rho }^{(1)}$ and ${\rho }^{(2)}$, we can compute the local parity of an orbital ${\varphi }_q$, which results from the linear combination of orbitals ${\varphi }_i$ and ${\varphi }_j$, as $⟨P_q{⟩}_0(\theta )$. The local parity $⟨P_q{⟩}_0(\theta )$ is an analytic, $2\pi$-periodic function of the rotation angle $\theta$. We find extremal points ${\theta }_n$ of the function $⟨P_q{⟩}_0(\theta )$ in the domain $\theta \in [0,2\pi )$ from $${\frac{d⟨P_q{⟩}_0}{d\theta }|}_{{\theta }_n}=0,$$ and select solutions ${\theta }_n$ that satisfy $${\frac{d^2⟨P_q{⟩}_0}{d{\theta }^2}|}_{{\theta }_n}\ne 0.$$ If a solution ${\theta }_m$ exists which satisfies $d^2⟨P_q{⟩}_0/d{\theta }^2{|}_{{\theta }_m}>0$ and $⟨P_q{⟩}_0({\theta }_m)\le ⟨P_q{⟩}_0({\theta }_l)\forall {\theta }_l\in {\theta }_n$, we accept the rotation attempt $i\to q$ and $j\to p$ with rotation angle ${\theta }_m$ and we reject the rotation attempt otherwise. By repeating the procedure for each pair of basis orbitals ${\varphi }_i$ and ${\varphi }_j$, we arrive at at an orthonormal basis in which the local parities have taken up extremal values. We then identify the orbitals ${\varphi }_q$ of the resulting basis for which $⟨P_q{⟩}_0=-1+\epsilon$, with $\epsilon$ an arbitrary small, positive value, as the spin-like orbitals of the system. Basis orbitals ${\varphi }_p$ with a local parity $⟨P_p{⟩}_0\simeq +1$ can be regarded as beneficial for the purpose of separating the system's spin degrees of freedom and their respective environment since they experience exclusively even number particle transfer, such that the spin degree of freedom of electrons occupying the orbitals ${\varphi }_p$ becomes insignificant.