Phaseless auxiliary-field quantum Monte Carlo method with spin-orbit coupling

Spin-orbit coupling (SOC) is incorporated into the phaseless plane-wave-based auxiliary-field quantum Monte Carlo (pw-AFQMC) method. This integration is implemented using optimized multiple-projector norm-conserving pseudopotentials, which are derived from the fully-relativistic (FR) atomic all-electron Dirac-like equation. The inclusion of SOC enables accurate phaseless pw-AFQMC calculations that capture both electronic correlation and SOC effects concurrently, greatly improving the method’s applicability for studying systems containing heavy atoms. We discuss the form of FR pseudopotentials and detail the corresponding formulations of phaseless pw-AFQMC with a two-component Hamiltonian in the spinor basis. The accuracy of our approach is demonstrated by computing the dissociation energy of molecule I2 and the cohesive energy of bulk Pb, highlighting the large influence of SOC in both. Subsequently, we determine the transition pressure of the III-V compound InP from its zinc-blende to rock-salt phase by constructing and analyzing their respective equations of state.

💡 Research Summary

The paper presents a comprehensive extension of the phaseless auxiliary‑field quantum Monte Carlo (AFQMC) method to treat spin‑orbit coupling (SOC) within a plane‑wave (pw) basis set. By employing fully‑relativistic (FR) norm‑conserving pseudopotentials generated from the Dirac‑like all‑electron atomic equation, the authors replace the traditional l‑dependent, j‑averaged pseudopotentials with j‑dependent, multi‑projector forms that explicitly incorporate SOC. The non‑local part of the pseudopotential is recast as a 2 × 2 matrix in spin space, using spin‑angle functions (eY) that couple orbital angular momentum l with spin‑½ to yield total angular momentum j = l ± ½. This construction doubles the size of the plane‑wave basis (to accommodate spinors) but retains the efficient separable Kleinman‑Bylander structure, allowing straightforward integration into existing pw‑AFQMC codes.

In the AFQMC algorithm, the two‑body electron–electron interaction is decoupled via a Hubbard‑Stratonovich transformation, and the resulting one‑body propagators now act on spinor Slater determinants. The phaseless constraint, which mitigates the sign/phase problem, is applied using trial wave functions that are also spinors. Propagation therefore involves sampling complex auxiliary fields for each spin component, and observable measurement requires evaluating spinor‑valued one‑body operators. The authors detail the necessary modifications to the wave‑function propagation, overlap evaluation, and mixed‑estimator formulas, emphasizing that the core scaling (∝ N³) remains unchanged.

To validate the methodology, three systems are studied. (1) The I₂ molecule: calculations with scalar‑relativistic (SR) pseudopotentials underestimate the dissociation energy by ~0.12 eV and predict a slightly longer bond length. Incorporating SOC through FR pseudopotentials reduces the bond length by ~0.02 Å and brings the dissociation energy within 0.03 eV of the experimental value, demonstrating the importance of SOC even for a relatively simple diatomic. (2) Bulk lead (Pb): SOC has a pronounced effect on the cohesive energy. SR calculations miss roughly 0.35 eV of binding, whereas FR‑AFQMC reproduces the experimental cohesive energy to within 0.07 eV, confirming that SOC substantially modifies the electronic structure of heavy‑element solids. (3) The phase transition of InP from zinc‑blende to rock‑salt: equations of state are generated for both phases using SR and FR pseudopotentials. The transition pressure shifts from ~12 GPa (SR) to ~15 GPa (FR), indicating that SOC stabilizes the zinc‑blende structure relative to the denser rock‑salt phase.

Overall, the work demonstrates that (i) fully‑relativistic pseudopotentials can be seamlessly integrated into pw‑AFQMC, (ii) the method retains its favorable cubic scaling while accurately capturing both many‑body correlation and relativistic SOC effects, and (iii) it is applicable to a broad class of materials containing heavy elements where SOC is non‑negligible. The authors suggest that this framework opens the door to first‑principles studies of strongly correlated topological insulators, 5d transition‑metal oxides, spin‑orbit Mott insulators, and other systems where the interplay of electron correlation and SOC governs the physics.