Evidence for spontaneous breaking of a continuous symmetry at a non-conformal quantum critical point in one dimension

Abstract

In this work, we present numerical evidence for the spontaneous breaking of a continuous $U(1)$ symmetry in a nearest-neighbor interacting spin-1 chain at a quantum critical point separating two XY quasi-long-range ordered phases distinguished by a spontaneously broken $\mathbb{Z}_2$ symmetry. Remarkably, the continuous symmetry breaking emerges precisely at the critical point of the discrete order parameter, suggesting a novel mechanism beyond currently established scenarios. At criticality, the XY correlations develop true long-range order, accompanied by a finite perpendicular magnetization, a zero-frequency Bragg peak in the transverse dynamical structure factor, and sharp gapless collective excitations. From complementary static and dynamical observables, we quantitatively determine the critical exponents, obtaining a dynamical exponent $z=1.50\pm0.04$ and an anomalous dimension $η=1.04\pm0.03$. Remarkably, the value of $z$ coincides with the one-dimensional Kardar--Parisi--Zhang (KPZ) exponent despite the system being an equilibrium quantum many-body system. We further show that the non-interacting continuum limit is equivalent to the recently introduced transverse quantum fluid, displaying their off-diagonal long-ranged order in one dimension. Complementing the numerical study, we derive the renormalization-group flow equations of the continuum theory to second order in the $\varepsilon$ expansion. We identify an interacting fixed point whose critical behavior differs from the Ising universality class already at two-loop order, although the perturbative exponents remain far from the numerical values.