Quantum critical surface of the zigzag spin chain under magnetic field: Application to superconducting quantum dots

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 states. With Bethe ansatz, we argue that this is the quantum critical surface in all cases. A first in the literature, we derive the analytical expression of quantum critical surface for superconducting quantum dots array in the presence of gate voltage.