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An anomalous energy splitting of spin triplet states at zero magnetic fieldhas recently been measured in germanium quantum dots. This zero-field splittingcould crucially alter the coupling between tunnel-coupled quantum dots, thebasic building blocks of state-of-the-art spin-based quantum processors, withprofound implications for semiconducting quantum computers. We develop ananalytical model linking the zero-field splitting to spin-orbit interactionsthat are cubic in momentum. Such interactions naturally emerge in holenanostructures, where they can also be tuned by external electric fields, andwe find them to be particularly large in silicon and germanium, resulting in asignificant zero-field splitting in the $\mu$eV range. We confirm ouranalytical theory by numerical simulations of different quantum dots, alsoincluding other possible sources of zero-field splitting. Our findings areapplicable to a broad range of current architectures encoding spin qubits andprovide a deeper understanding of these materials, paving the way towards thenext generation of semiconducting quantum processors.

Anomalous zero-field splitting for hole spin qubits in Si and Ge quantum dots