Theoretical background

For users interested in the theoretical background behind the HQS Spin Mapper, we refer them to the sections Spin-like orbitals and Local Parity and Transformation of the many-body Hamiltonian. These sections are however not required reading for usage of the package.

In section Spin-like orbitals and Local Parity we

• define the notion of spin-like orbitals,
• propose the local parity as a metric for the spin-like character of orbitals,
• and present our parity optimiziation procedure as a way to determine the spin-like orbitals of a system.

The section Transformation of the many-body Hamiltonian describes

• the basic concept of the Schrieffer-Wolff transformation,
• our proposed extended Schrieffer-Wolff transformation method,
• and Schrieffer-Wolff transformation as a system of linear equations for unique block-offdiagonal operators.

Here, in section Full workflow we describe how the methods are combined into our workflow for deriving effective spin-bath model Hamiltonians for materials with relevant low-energy spin physics.

Full workflow

In the following, we outline the steps of the workflow that we use to derive an effective spin-bath model Hamiltonian from a first principles description of a material.

Computation of the required system information

We start with an ab-initio electronic structure calculation of the material to determine its ground state. The electronic structure method needs to be a post-Hartree-Fock or related method - this excludes density functional theory - to capture the effect of correlations in the two-particle reduced density matrix ${\rho }^{(2)}$. From the electronic structure calculation we obtain the basis orbitals in which the Hamiltonian of the system is formulated, and the one-electron and two-electron integrals which specify the Hamiltonian description. For the ground state of the calculation we compute the one-particle and two-particle reduced density matrices ${\rho }^{(1)}$ and ${\rho }^{(2)}$.

Determination of spin-like basis orbitals

We utilize the reduced density matrices to assign a local parity $⟨P_i{⟩}_0$ to the basis orbitals i${\varphi }_i$. We then perform pairwise rotations of the basis orbitals to determine the basis in which the local parities of the basis orbitals are extremized. If there exist optimized basis orbitals ${\varphi }_q$ with $(⟨P_q{⟩}_0+1)<\epsilon$, where we typically choose $\epsilon \le 10^{-1}$, we proceed with the subsequent steps of the workflow. If no spin-like orbitals are found, we terminate the workflow.

Schrieffer-Wolff transformation of the Hamiltonian

We use our Schrieffer-Wolff transformation approach to integrate out the valid terms of the Hamiltonian that modify the electron density in the spin-like orbitals, which leads to renormalized couplings of the electron spins in the spin-like orbitals to the environment. The valid terms are the ones that couple subspaces of the Hilbert space between which there exists a significant energy gap.

Construction of the effective spin-bath Hamiltonian

The transformed Hamiltonian $H$ is projected to the particular subspace of the Hilbert space where the electron density of spin-like orbitals ${\varphi }_q$ is fixed to $n_q\equiv 1$. Utilizing the identity $n_q=n_{q\uparrow }+n_{q\downarrow }\equiv 1$ fermionic operators acting on the spin-like operators are substituted with the corresponding spin operators. The resulting Hamiltonian $\mathcal{P}H\mathcal{P}$ is the effective spin-bath representation of the material.

Representation on a device (optional)

The effective spin-bath model Hamiltonian $\mathcal{P}\tilde{H}\mathcal{P}$ is re-expressed in terms of the spin operators that are realized on the specific device.