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We analyze the exact ground state of XXZ zigzag spin chain with appliedmagnetic field and find the quantum critical surface. Using the theorem ofpositive semi-definite matrix, we can prove that the ground states for aspecific region, are fully polarized state and one magnon states. With Betheansatz, we argue that this is the quantum critical surface in all cases. Afirst in the literature, we derive the analytical expression of quantumcritical surface for superconducting quantum dots array in the presence of gatevoltage.

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