Theory of criticality-enabled $U(1)$ symmetry breaking in a class of 1+1D systems

We present an analytic theory of the recently proposed phenomenon of spontaneous $U(1)$ symmetry breaking at 1+1D Lifshitz quantum critical points. The low-energy field theory contains two periodic scalars $\theta$ and $\phi$, with a Berry phase coupling that makes $\theta$ canonically conjugate to the $U(1)$ charge density ${\mathrm{\partial }}_x\phi$. Within a controlled large-$N$ limit, we demonstrate that the $U(1)$-charged phase vertex $e^{i\beta \theta }$ develops long-range order, while the conjugate vertex $e^{i\beta \phi }$ decays as a stretched exponential, $\log ⟨e^{i\beta \phi (x)}{e}^{-i\beta \phi (0)}⟩\propto -\mathrm{\mid }\beta {\mathrm{\mid }}^{4\mathrm{/}3}\mathrm{\mid }x{\mathrm{\mid }}^{2\mathrm{/}3}$. Going beyond the large-$N$ limit, we give an analytic argument that the Lifshitz field theory supports $U(1)$ long-range order provided its dynamical exponent satisfies $z\mathrm{\ne }1$, a condition strongly supported by existing calculations. We test these predictions using finite-size and infinite-system DMRG in an itinerant-fermion chain with $U(1)⋊{\mathbb{Z}}_2$ symmetry. At the ${\mathbb{Z}}_2$ ferromagnetic transition, the spin sector of the chain maps to the Lifshitz field theory. Consistent with analytic predictions, the $U(1)$-charged spin-nematic bond operator $S_i^{+}{S}_{i+1}^{+}$ exhibits long-range order, while the electron Green’s function shows stretched-exponential decay. Together, these results elucidate the mechanism and consequences of criticality-enabled $U(1)$ symmetry breaking in 1+1D.