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 $θ$ and $ϕ$, with a Berry phase coupling that makes $θ$ canonically conjugate to the $U(1)$ charge density $\partial_x ϕ$. Within a controlled large-$N$ limit, we demonstrate that the $U(1)$-charged phase vertex $e^{iβθ}$ develops long-range order, while the conjugate vertex $e^{iβϕ}$ decays as a stretched exponential, $\log\langle e^{iβϕ(x)} e^{-iβϕ(0)}\rangle\propto-|β|^{4/3}|x|^{2/3}$. 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 $z\neq1$, a condition strongly supported by existing calculations. We test these predictions using finite-size and infinite-system DMRG in an itinerant-fermion chain with $U(1) \rtimes \mathbb{Z}_2$ 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.