Variational Monte Carlo Method in the Presence of Spin-Orbit Interaction and Its Application to the Kitaev and Kitaev-Heisenberg Model

TL;DR

This paper introduces an advanced variational Monte Carlo method capable of accurately modeling systems with strong spin-orbit interactions, demonstrated on the Kitaev and Kitaev-Heisenberg models, capturing complex quantum states including spin liquids.

Contribution

The paper develops a generalized variational wave function framework with complex parameters and quantum number projections, enabling accurate simulations of strongly correlated spin-orbit coupled systems.

Findings

Successfully reproduces ground state degeneracy of Kitaev spin liquids.

Accurately estimates ground state energies for Kitaev and Kitaev-Heisenberg models.

Demonstrates the method's capability to handle complex quantum states with strong spin-orbit interactions.

Abstract

We propose an accurate variational Monte Carlo method applicable in the presence of the strong spin-orbit interaction. Our variational wave functions consist of generalized Pfaffian-Slater wave functions that involve mixtures of singlet and triplet Cooper pairs, Jastrow-Gutzwiller-type pro- jections, and quantum number projections. The generalized wave functions allow any symmetry breaking states, ranging from magnetic and/or charge ordered states to superconducting states and their fluctuations, on equal footing without any ad hoc ansatz for variational wave functions. We de- tail our optimization scheme for the generalized Pfaffian-Slater wave functions with complex-number variational parameters. Generalized quantum number projections are also introduced, which im- poses the conservation of not only spin and momentum quantum numbers but also Wilson loops. As a demonstration of the capability of the present variational Monte Carlo method, the accuracy and efficiency of the present method is tested for the Kitaev and Kitaev-Heisenberg models. The Kitaev model serves as a critical benchmark of the present method: The exact ground state of the model is a typical gapless quantum spin liquid far beyond the applicability range of simple mean-field wave functions. The newly introduced quantum number projections precisely reproduce the ground state degeneracy of the Kitaev spin liquids, in addition to their ground state energy. Our framework offers accurate solutions for the systems where strong electron correlation and spin-orbit interaction coexist.