Spin-Orbit Coupled Bosons in One Dimension: Emergent Gauge Field and Lifshitz Transition

We describe a two-component Bose liquid in one-dimension (1D) in terms of scalar bosons coupled to an emergent dynamical gauge field. In the presence of strong spin-independent interactions and spin-orbit coupling, the system undergoes an interaction (or density) tuned quantum phase transition. Although the order parameter describes a broken Z₂ spin symmetry, the associated phase is qualitatively distinct from the Ising phase as it is accompanied by a non-vanishing momentum which is generated by the gauge fluctuations at the phase transition. This quantum phase transition is distinct from the conventional Ising transition and has a dynamical critical exponent z ≈ 2, typical of a Lifshitz transition. Our model describes 1D ultracold atoms with Raman-induced spin-orbit coupling, and provides a route to quantum emulation of a model dynamical gauge theory as well as exotic critical behavior.