Microscopic Theory Revealing Ising Criticality with Distinct Sublattice Orders in Pseudospin-1/2 Chain

The one-dimensional transverse Ising model is a paradigmatic example of quantum criticality. In spin-orbit coupled systems, however, effective Ising interactions arise alongside bond-dependent couplings such as Kitaev ($K$) and $\Gamma$ terms in addition to the Heisenberg ($J$) interaction, leading to complex magnetic orders beyond the pure Ising limit. We first explore how the generic $J-K-\Gamma$ model with four-fold screw symmetry in a spin-orbit-entangled pseudospin-1/2 chain manifests via sublattice order, and then test whether field-driven transitions retain Ising universality. We find sublattice-dependent magnetic order below the critical field and a distinct sublattice pattern persisting above it. Despite these complex magnetic order structures, the transition remains in the Ising universality class with central charge $c=1/2$. Our work provides a route to the microscopic Hamiltonian and emergent Ising criticality while allowing microscopic physics of the sublattice orders to manifest at low and high fields. Application to antiferromagnetic Ising materials such as BaCo$_2$V$_2$O$_8$ is also discussed.