Interacting quantum criticality with U(1) symmetry breaking in 1+1 dimensions

The Hohenberg-Mermin-Wagner theorem motivates the expectation that continuous symmetries remain unbroken in one-dimensional quantum ground states. Here, contrary to this expectation, we demonstrate the spontaneous breaking of U(1) spin-rotation symmetry and the emergence of an interacting quantum critical point in lattice spin-$1\mathrm{/}2$ and spin-$1$ chains. Using subsystem Binder scaling, we locate the critical point and determine the correlation-length critical exponent as $\nu =0.60\pm 0.02$, distinct from the Gaussian value $0.5$ but consistent with the perturbative renormalization-group prediction. We further analyze the finite-size scaling of the lowest neutral excitation gap and obtain the dynamical critical exponent $z=1.986\pm 0.005$. The result $z<2$ is incompatible with microscopic realizations of this critical theory by frustration-free models, implying a mechanism for continuous symmetry breaking in one-dimensional quantum ground states distinct from frustration freeness.