We perform a detailed study on the consequences of the nonsymmorphic symmetries in the Luttinger phase of the one-dimensional (1D) spin-$\frac{1}{2}$ Kitaev-Heisenberg-Gamma model with an antiferromagnetic Kitaev interaction. Although the low-energy Hamiltonian has an emergent U(1) symmetry, the bosonization formulas for the spin operators contain ten nonuniversal coefficients which only respect the discrete nonsymmorphic symmetries and can be determined by our density matrix renormalization group simulations to a high degree of accuracy. Using the nonsymmorphic bosonization formulas, the response to weak magnetic field is analyzed, and the zigzag phase in the 2D Kitaev-Heisenberg-Gamma model is recovered by weakly coupling an infinite number of 1D chains. Furthermore, we find a line of critical points with an emergent $\mathrm{SU}{\left(2\right)}_1$ conformal symmetry located on the boundary of the Luttinger liquid phase. Our work reveals the rich physics related to nonsymmorphic symmetries in strongly correlated quantum magnetic models, and demonstrates the usefulness of 1D analysis for understanding the 2D Kitaev physics.

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• Frustrated magnetism
• Magnetism
• Bosonization
• Density matrix renormalization group
• Discrete symmetries in condensed matter
• Effective field theory
• Kitaev model
• Spin chains