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

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 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 plus or minus0.04 and an anomalous dimension η=1.04 plus or minus0.03. 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 epsilon 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.