Quantum critical surface of the XXZ zigzag spin chain with applied magnetic field and its application

We analyze the exact ground state of XXZ zigzag spin chain with applied magnetic field and find the quantum critical surface. Using the theorem of positive semi-definite matrix, we can prove that the ground states for a specific region are fully polarized state and one-magnon state. We also argue that this is the quantum critical surface in all cases. Applied to the superconducting quantum dots array, our result gives the analytical expression of quantum critical surface for the system in the presence of gate voltage.