A method to derive material-specific spin-bath model descriptions of materials displaying prevalent spin physics

TL;DR

The paper tackles the challenge of capturing low-energy spin physics in complex materials by introducing a two-stage approach: (i) identify spin-like orbital degrees of freedom via a local parity metric and parity optimization, and (ii) derive an effective spin-bath Hamiltonian by an extended Schrieffer-Wolff transformation that integrates out charge degrees of freedom. The authors reformulate the SW expansion as a linear system $L\,\vec{S}=\vec{V}$ in operator space and use an SVD-based separation to retain resonant terms while discarding gapped contributions, enabling application to generic Hamiltonians with four-index terms. They validate the method on model systems such as the single impurity Anderson model and the disordered Fermi-Hubbard chain, and on molecular chromium bromide, showing that the local parity of spin-like orbitals is a strong predictor of the quality of the resulting spin-bath description and that the transformed Hamiltonians reproduce low-energy spin spectra with high fidelity. The work provides a principled, first-principles pathway to map complex electronic-structure problems to spin-bath models, facilitating quantum simulation and scalable analysis of materials with dominant spin physics.

Abstract

Magnetism and spin physics are true quantum mechanical effects and their description usually requires multi reference methods and is often hidden in the standard description of molecules in quantum chemistry. In this work we present a twofold approach to the description of spin physics in molecules and solids. First, we present a method that identifies the single-particle basis in which a given subset of the orbitals is equivalent to spin degrees of freedom for models and materials which feature significant spin physics at low energies. We introduce a metric for the spin-like character of a basis orbital, of which the optimization yields the basis containing the optimum spin-like basis orbitals. Second, we demonstrate an extended Schrieffer-Wolff transformation method to derive the effective Hamiltonian acting on the subspace of the Hilbert space in which the charge degree of freedom of electron densities in the spin-like orbitals is integrated out. The method then yields an effective Hamiltonian describing spins coupled to a fermionic environment. This extended Schrieffer-Wolff transformation is applicable to a wide range of Hamiltonians and has been utilized in this work for model Hamiltonians as well as the Hamiltonian describing the active orbital space of molecular chromium bromide. This is achieved by reformulating the highly non-linear Schrieffer-Wolff equations into a linear set of equations corresponding to an operator basis.

Figures (9)

• Figure 1: Schematic representation of the lattice model described by the Hamiltonian (\ref{['eq:hamiltonian_toy_model']}). The lattice sites $A_i$ are shown in grey. On the orange lattice sites $B_i$ the fermions experience a strong repulsive Hubbard interaction $U$ which energetically discourages $n_{B_i} \neq 1$. We refer to the lattice sites $A_i$ and $B_i$ as the initial basis of the system. The hybridization between the lattice sites $A_i$ and $A_{i+1}$ is given by $t < 0$. We impose periodic boundary conditions via a hybridization $t$ between the lattice sites $A_{N=4}$ and $A_1$. There is a small hybridization $0 > V > t$ between lattice sites $A_i$ and $B_i$. The local Hubbard density-density interaction of strength $U \gg \vert t \vert$ is strongly repulsive. The connectivity and the coupling constants of the lattice models are chosen such that the lattice sites $B_i$ should be good realizations of spin-like orbitals.
• Figure 2: Ground state expectation values of the local parity $\langle P_q \rangle_0$ of the initial basis orbitals $A_i$ and $B_i$, as well as (a) the natural (spin) orbitals basis and (b) the parity optimized basis for the spin-bath chain model described by eq. (\ref{['eq:hamiltonian_toy_model']}). (a) The dark grey line shows the local parities of the original basis orbitals $A_i$ and $B_i$ and the dashed orange line indicates the local parities of the natural basis orbitals. The blue circles display the expectation values for the electron density $\langle n_q \rangle_0$ in the natural orbital basis. In the natural orbital basis we find $N_{\phi_q}=2$ orbitals $\phi_q$ for which $\langle n_q \rangle_0 = 1$, but the respective values of the local parities $\langle P_q \rangle_0 > 0$ highlight that $\vert \delta n_q\vert \gg 0$. (b) The dashed orange line now displays the ground state local parities of the parity optimized basis orbitals. The blue circles show the expectation value for electron density $\langle n_q \rangle_0$ in these optimized basis orbitals. In the optimized basis we find $N_{\phi_q} = 4$ orbitals $\phi_q$ with local parity $\langle P_q \rangle_0 \simeq -1$. We also observe $\langle n_q \rangle_0 = 1$ for each orbital. The orbitals $\phi_{q\leq 3}$ of the optimized basis are considered spin-like. They coincide with the lattice sites $B_i$, but the ordering has been shuffled in the optimization procedure.
• Figure 3: (a) Molecular structure of the closed configuration of the molecule para-benzyne ($\text{C}_6 \text{H}_4$). (b) Isosurface of one spin-like orbital $\phi_{q=1}$ of para-benzyne. (c) Isosurface of the other spin-like orbital $\phi_{q=2}$ of para-benzyne. The colors red and blue indicate the sign of the orbital wavefunction.
• Figure 4: Ground state local parities $\langle P_i\rangle_0$ of the basis orbitals of para-benzyne ($\text{C}_6 \text{H}_4$) in the initial CASSCF canonical molecular orbitals basis (dark grey line) and the two different parity optimized bases (dashed orange and blue). The local parities $\langle P_i \rangle_0$ of the original atomic basis orbitals each satisfy $\langle P_i \rangle_0 > 8.5\times 10^{-1}$, so the initial basis orbitals are not considered spin-like. The local parities $\langle P_q \rangle_0$ of the basis orbitals $\phi_q$ in the optimized basis, where the full set of $N=62$ orbitals has been optimized, are shown in orange. The resulting local parities $\langle P_q \rangle_0$, where the subset of $N=12$ basis orbitals $\phi_i$ with initial local parity $\langle P_i \rangle_0 < 9.5\times 10^{-1}$ has been optimized, are displayed in blue. The electron densities in the optimized basis orbitals is shown as green circles. The restricted set turns out to be equivalent to the active space of the CASSCF calculation used to obtain the reduced density matrices $\rho^{(1)}$ and $\rho^{(2)}$. The local parity $\langle P_q \rangle_0 \simeq -9.4\times 10^{-1}$ of two specific optimized basis orbitals $\phi_q$ is sufficiently small for the criteria $\langle n_q \rangle_0 \equiv 1$ and $\vert \delta n_q \vert \simeq 0$ to be simultaneously fulfilled. The isosurfaces of these spin-like orbitals are displayed in Figs. \ref{['fig:spin_like_1']} and \ref{['fig:spin_like_2']}.
• Figure 5: Rescaled ground state energy difference $U\times\Delta E_0/t=U\times\vert E_{0}^{ (\mathcal{P})\tilde{H}(\mathcal{P})} - E_{0}^{H}\vert /t$ between the ground state of the SIAM Hamiltonian and the transformed Hamiltonians $\tilde{H}$ and $\mathcal{P} \tilde{H} \mathcal{P}$ as a function of the interaction strength $U/t$. The dark grey line represents the scaled energy difference for the transformed Hamiltonian $\tilde{H}$. The dashed orange displays $U\times\Delta E_0/t$ for the effective Hamiltonian $\mathcal{P} \tilde{H} \mathcal{P}$ where the impurity site is restricted to single occupancy $n_d \equiv 1$. The solid blue line displays the local parity $\langle P_d \rangle_0$ of the impurity site in the true ground state for each value of the interaction strength $U/t$.