Theory behind HQS Spin Mapper

In Spin-like orbitals 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. We discuss the basic concept of the Schrieffer-Wolff transformation in Perturbative similarity transformation. We detail our proposed extended Schrieffer-Wolff transformation method in sections Symmetry specification of block-offdiagonal operators and Schrieffer-Wolff transformation as a system of linear equations for unique block-offdiagonal operators . In Full workflow we show how the described methods are combined into our workflow for deriving effective spin-bath model Hamiltonians for materials with relevant low-energy spin physics.

Spin-like orbitals

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.

Perturbative similarity transformation

In principle, any Hamiltonian can be completely diagonalized by means of a particular unitary transformation $U$. In practice, finding the particular unitary transformation $U$ often requires a complete diagonalization of the Hamiltonian to begin with. Here we recap how a perturbative similarity transformation, namely the Schrieffer-Wolff transformation, can be used to determine an approximate transformation operator $U$, or the generator $S$ thereof, which does not diagonalize the Hamiltonian fully, but yields a block-diagonal Hamiltonian instead. These blocks consist of the orthonormal states in the Hilbert space $\mathcal{H}$ that share the a given choice of characteristics, e.g. the local particle quantum number $n_i$. We will denote the set of terms in the Hamiltonian that are already block-diagonal in the initial basis as $H_0$. The remaining terms connect different blocks, i.e are block-offdiagonal, and are denoted $V$. The complete Hamiltonian thus reads $$H=H_0+V.$$ A unitary similarity transformation of the Hamiltonian is given by $$H=U^{†}HU={\text{e}}^{-S}\left(H_0+V\right){\text{e}}^S={\text{e}}^{-S}{H}_0{\text{e}}^S+{\text{e}}^{-S}V{\text{e}}^S,$$ where $S$ is an anti-hermitian operator. One refers to it as the generator of the Schrieffer-Wolff transformation. The key problem of the Schrieffer-Wolff transformation is finding the generator $S$ such that the transformed Hamiltonian $H$ becomes entirely block-diagonal. In order to arrive at an equation for $S$ one makes use of the Campbell-Baker-Hausdorff formula to expand $$H={\mathrm{e}}^{S}H{\mathrm{e}}^{-S}=\sum _{m=0}^{\infty }\frac{1}{m!}[S,H{]}_m=H+[S,H]+\frac{1}{2}\left[S,[S,H]\right]+\dots ,$$ where for generators $S$ satisfying $\Vert S\Vert \ll \Vert H_0\Vert$, with $\Vert \cdot \Vert$ a suitable norm, one can approximate the expression as $$H=H_0+V+\left[S,H_0\right]+\left[S,V\right]+\mathcal{O}(S^2).$$ Considering that commutators of pairs of block-diagonal operators or pairs of block-offdiagonal operators respectively generally become block-diagonal, while the commutators of block-diagonal operators with block-offdiagonal operators become block-offdiagonal, one chooses the equation $$[S,H_0]=-V\iff [H_0,S]=V,$$ by which one can determine the generator $S$ which removes the block-offdiagonal terms $V$ of the Hamiltonian. If a solution $S$ exists, one can use $$\left[S,\left[S,H_0\right]\right]=\left[S,-V\right]=-\left[S,V\right]\notin \mathcal{O}(S^2),$$ to simplify the expression for the transformed Hamiltonian $$H={\mathrm{e}}^{-S}\left(H_0+V\right){\mathrm{e}}^S\simeq H_0+\frac{1}{2}\left[S,V\right]+\mathcal{O}(S^2),$$ where the terms originating from $[S,V]/2$ contain the perturbative corrections arising from the consecutive application of two block-offdiagonal operators. A subsequent projection to the subspaces $${\mathcal{P}}_n=\sum _{p\in {\mathcal{P}}_n}|p⟩⟨p|,$$ yields the block-diagonal Hamiltonian $$H_{\mathrm{block}-\mathrm{diagonal}}=\sum _n{\mathcal{P}}_n\tilde{H}{\mathcal{P}}_n,$$ where $n$ denotes the distinct blocks of the Hilbert space.

Symmetry specification of block-offdiagonal operators

The block-offdiagonal part of the Hamiltonian $V$ consists of a sum of block-offdiagonal terms. These in turn comprise products of individually block-offdiagonal operators $x$. In the following we detail a method to decompose generic block-offdiagonal operators $x$ into distinct components. Each of the components exclusively connects two distinct blocks of the Hilbert space $\mathcal{H}$, often associated with distinct quantum numbers of a symmetry of the system. A given block-offdiagonal operator $x:\mathcal{H}\to \mathcal{H}$ satisfies $$\left[H_0,x\right]=\epsilon z\ne 0,$$ where $\epsilon$ denotes an arbitrary scalar and $z:\mathcal{H}\to \mathcal{H}$ an arbitrary operator. Let $A$ be a diagonal operator in the initial basis. It can be identical to the symmetry operator differentiating the blocks of the Hilbert space, but it is not required to be. One can use the spectrum of $A$ to expand the operator $x$ as $$x=\sum _q{x}_q=\sum _q{\beta }_q\prod _{i\ne q}\left(A-a_i\right)x,$$ where the different $x_q$ couple the target subspace associated with the eigenvalue $a_q$ of $A$ to other subspaces of the Hilbert space. If the operator $A$ satisfies $$[x,A]=0,$$ then both subspaces, initial and final, coupled via $x_q$ are specified by the eigenvalue $a_q$. This is possible for the fermionic creation and annihilation operators $c_{i\sigma }^{†}$ and $c_{i\sigma }$. The coefficients ${\beta }_q$ are solutions to the equation $${\beta }_q\prod _{i\ne q}\left(a_q-a_i\right)=1.$$ The symmetry-specified block-offdiagonal operators $x_q$ satisfy $$\left[x_q,x_{q^{\prime }}^{†}\right]\propto {\delta }_{q{q}^{\prime }},$$ where the operator $x_q^{†}$ denotes the hermitian conjugate of the operator $x_q$.

Schrieffer-Wolff transformation as a system of linear equations for unique block-offdiagonal operators

Following the procedure outlined in the section Perturbative similarity transformation we separate the Hamiltonian of the system into its block-diagonal and block-offdiagonal contributions as $$H=H_0+V.$$ The block-offdiagonal contribution $V$ comprises each block-offdiagonal term $v$ $$\begin{gathered} V=\sum _{\left\{v\right\}}{\alpha }_v\left[\left(\prod _j{o}^j\prod _i{x}^i\right)+{\left(\prod _j{o}^j\prod _i{x}^i\right)}^{†}\right] \\ =\sum _{\left\{v\right\}}{\alpha }_v\left[\left(\prod _{i,j}{o}^j\sum _{q(i)}{x}_{q(i)}^i\right)+{\left(\prod _{i,j}{o}^j\sum _{q(i)}{x}_{q(i)}^i\right)}^{†}\right], \end{gathered}$$ where ${\prod }_i{x}^i$denotes sequences of individually block-offdiagonal operators, ${\prod }_j{o}^j$ denotes sequences of individually block-diagonal operators, and $x_{q(i)}^i$ denotes the symmetry-specified components of the operator $x^i$. We introduce a vector spaces ${\mathcal{V}}_0^h$ and ${\mathcal{V}}_0^a$, for which each unique pair of hermitian, or anti-hermitian respectively, symmetry-specified operator sequences in $V$ corresponds to a unique orthonormal basis vector $${\hat{e}}_v=\{\begin{array}{ll}\left({\prod }_j{o}^j{\prod }_i{x}_{q(i)}^i\right)+{\left({\prod }_j{o}^j{\prod }_i{x}_{q(i)}^i\right)}^{†}& \in {\mathcal{V}}_0^h\\ \left({\prod }_j{o}^j{\prod }_i{x}_{q(i)}^i\right)-{\left({\prod }_j{o}^j{\prod }_i{x}_{q(i)}^i\right)}^{†}& \in {\mathcal{V}}_0^a\end{array},$$ where ${\mathcal{V}}^h$ denotes the hermitian vector space and ${\mathcal{V}}^a$ the anti-hermitian vector space. We define the linear map $$L:\begin{array}{l}{\mathcal{V}}_0^h\to {\mathcal{V}}_1^a\\ {\mathcal{V}}_0^a\to {\mathcal{V}}_1^h\end{array},$$ where the action of the linear map on a vector ${\hat{e}}_v$ is given by $$L{\hat{e}}_v=\left[H_0,{\hat{e}}_v\right],$$ where $$\begin{array}{lcl}(L{\hat{e}}_v)\in {\mathcal{V}}_1^a& \mathrm{if}& {\hat{e}}_v\in V_0^h\\ (L{\hat{e}}_v)\in {\mathcal{V}}_1^h& \mathrm{if}& {\hat{e}}_v\in V_0^a\end{array},$$ with $$\dim \left({\mathcal{V}}_1\right)\ge \dim \left({\mathcal{V}}_0\right),$$ since ${\mathcal{V}}_1$ includes additional unique operator sequences generated by $[H_0,{\hat{e}}_v]$. In the vector spaces $\mathcal{V}$, we can translate the equation for the generator $S$ of the Schrieffer-Wolff transformation to $$L\vec{S}=\vec{V}\iff \left[H_0,S\right]=V,$$ with $\vec{S}\in {\mathcal{V}}_0^a$ and $\vec{V}\in {\mathcal{V}}_1^h$. Determining the generator $S$ of the Schrieffer-Wolff transformation becomes equivalent to solving the set of linear equations. In general one finds $$\mathrm{rank}(L)\le \dim \left({\mathcal{V}}_0\right)\le \dim \left({\mathcal{V}}_1\right),$$ and there is consequently no unique solution $\vec{S}$ to the set of equations. It is intuitive that the terms of $V$ approximately bring about transitions between distinct eigenstates of $H_0$ belonging to different blocks of the Hilbert space. This is reflected by $$\left[H_0,{\hat{e}}_v^h\right]=\left(\Delta E_{0,v}\prod _l{o}^l\right){\hat{e}}_v^a,$$ where $\Delta E_{0,v}$ denotes the eigenvalue difference between the two eigenstates of $H_0$ and ${\prod }_l{o}^l$ denotes an arbitrary sequence of individually block-diagonal operators. From the previous equations we can approximate $$\Vert \vec{S}\Vert \approx \sqrt{\sum _{\{v\}}{\left(\frac{{\alpha }_v}{\Delta E_{0,v}}\right)}^2},$$ which highlights the necessity for a significant energy gap $\Delta E_{0,v}$ between the separate subspaces of the Hilbert space coupled by $V$ in order for the series expansion of $H$ to $\mathcal{O}(S^2)$ to be considered a good approximation. We consider terms $v$ for which $$\frac{{\alpha }_v^2}{|\Delta E_{0,v}|}\ge 1,$$ to be the resonant terms of $V$ which should be retained in the transformed Hamiltonian $H$. To identify these resonant terms of $V$, we employ a singular value decomposition (SVD) of the linear map $L$ and arrive at $$L=U\Sigma W^{†}=U\left({\Sigma }_{>}+{\Sigma }_{<}\right)W^{†}=L_{\mathrm{gapped}}+L_{\mathrm{resonant}},$$ where ${\Sigma }_{>}$ comprises the singular values $${\sigma }_i>\frac{1}{N_v}\sum _{\{v\}}\sqrt{{\alpha }_v^2},$$ where $N_v$ denotes the number of terms $v$. We define the Moore-Penrose pseudoinverse of $L$ that acts exclusively on the gapped, i.e. non-resonant, terms in $V$ as $$L_{\mathrm{gapped}}^{+}=\left(W{\Sigma }_{>}^{+}{U}^{†}\right),$$ where ${\Sigma }_{>}^{+}$ denotes the pseudoinverse of ${\Sigma }_{>}$. In the absence of a unique solution to $\left[H_0,S\right]=V$, the closest approximate solution is given by $$\vec{S}=\left(L_{\mathrm{gapped}}^{+}\vec{V}\right)\in {\mathcal{V}}_0^a,$$ and the resulting transformed Hamiltonian reads $$\tilde{H}=H_0+V_{\mathrm{resonant}}+\left[S,V_{\mathrm{resonant}}\right]+\frac{1}{2}\left[S,V_{\mathrm{gapped}}\right],$$ where $V_{\mathrm{gapped}}=[H_0,S]$ and $V_{\mathrm{resonant}}=(V-V_{\mathrm{gapped}})$.

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.