[Paper Review] Phaseless auxiliary-field quantum Monte Carlo method with spin-orbit coupling

Zheng Liu, Shiwei Zhang+1 more|arXiv (Cornell University)|Feb 12, 2026

Advanced NMR Techniques and Applications0 citations

TL;DR

The paper integrates spin-orbit coupling into the phaseless plane-wave AFQMC using optimized norm-conserving FR pseudopotentials, enabling simultaneous treatment of electron correlation and SOC in heavy-element systems and validating with I2, Pb, and InP.

ABSTRACT

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.

Motivation & Objective

• Motivate the need for accurate many-body treatments including spin-orbit coupling in heavy-element systems.
• Develop a formalism to include spin-orbit coupling in phaseless pw-AFQMC using fully-relativistic pseudopotentials.
• Showcase the method on molecular and solid-state systems to quantify SOC impact on energies and phase transitions.
• Demonstrate compatibility with two-component spinor formalism and two-body propagation within the AFQMC framework.

Proposed method

• Formulate the Hamiltonian in a plane-wave basis with local and non-local pseudopotentials, including spin-orbit coupling through j-dependent projectors.
• Adopt fully-relativistic (FR) pseudopotentials and reformulate non-local terms to operate in a spinor (two-component) basis.
• Apply the phaseless AFQMC framework with a second-order Trotter–Suzuki decomposition to separate one- and two-body parts.
• Use Hubbard-Stratonovich transformations to linearize the two-body propagator and propagate Slater determinants in a spinor basis.
• Implement measurement via mixed estimators and Green’s functions that account for spinor structure and SOC-induced spin mixing.
• Leverage FFT-based efficiency and a reduced FE representation for non-local operators to manage computational cost in SOC-enabled pw-AFQMC.

Experimental results

Research questions

• RQ1 How can spin-orbit coupling be consistently incorporated into phaseless pw-AFQMC with plane-wave basis and ONCV FR pseudopotentials?
• RQ2 What is the impact of SOC on computed dissociation energies, cohesive energies, and phase-transition pressures in heavy-element systems?
• RQ3 Can the two-component spinor AFQMC formulation accurately capture SOC effects alongside electronic correlation in real materials and molecules?
• RQ4 How do FR pseudopotentials and SOC-aligned non-local operators influence computational scaling and accuracy of pw-AFQMC?

Key findings

• SOC-inclusive phaseless pw-AFQMC yields accurate energetics for systems with heavy elements by treating correlation and SOC concurrently.
• Demonstrations include the dissociation energy of I2 and the cohesive energy of bulk Pb, illustrating the large influence of SOC on these properties.
• The approach enables calculation of structural transitions, exemplified by the InP zinc-blende to rock-salt phase transition via equations of state.
• The framework shows how to implement two-component Hamiltonians and spinor-based observables within pw-AFQMC, expanding applicability to SOC-rich materials.