Negative exchange interaction in Si quantum dot arrays via valley-phase induced $\mathbb{Z}_2$ gauge field

The exchange interaction $J$ offers a powerful tool for quantum computation based on semiconductor spin qubits. However, the exchange interaction in two-electron systems in the absence of a magnetic field is usually constrained to be non-negative $J \geq 0$, which inhibits the construction of various dynamically corrected exchange-based gates. In this work, we show that negative exchange $J < 0$ can be realized in two-electron Si quantum dot arrays in the absence of a magnetic field due to the presence of the valley degree of freedom. Here, valley phase differences between dots produce a non-trivial $\mathbb{Z}_2$ gauge field in the low-energy effective theory, which in turn can lead to a negative exchange interaction. In addition, we show that this $\mathbb{Z}_2$ gauge field can break Nagaoka ferromagnetism and be engineered by altering the occupancy of the dot array. Therefore, our work uncovers new tools for exchange-based quantum computing and a novel setting for studying quantum magnetism.