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We propose an accurate variational Monte Carlo method applicable in thepresence of the strong spin-orbit interaction. Our variational wave functionsconsist of generalized Pfaffian-Slater wave functions that involve mixtures ofsinglet and triplet Cooper pairs, Jastrow-Gutzwiller-type pro- jections, andquantum number projections. The generalized wave functions allow any symmetrybreaking states, ranging from magnetic and/or charge ordered states tosuperconducting states and their fluctuations, on equal footing without any adhoc ansatz for variational wave functions. We de- tail our optimization schemefor the generalized Pfaffian-Slater wave functions with complex-numbervariational parameters. Generalized quantum number projections are alsointroduced, which im- poses the conservation of not only spin and momentumquantum numbers but also Wilson loops. As a demonstration of the capability ofthe present variational Monte Carlo method, the accuracy and efficiency of thepresent method is tested for the Kitaev and Kitaev-Heisenberg models. TheKitaev model serves as a critical benchmark of the present method: The exactground state of the model is a typical gapless quantum spin liquid far beyondthe applicability range of simple mean-field wave functions. The newlyintroduced quantum number projections precisely reproduce the ground statedegeneracy of the Kitaev spin liquids, in addition to their ground stateenergy. Our framework offers accurate solutions for the systems where strongelectron correlation and spin-orbit interaction coexist.

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