  Continuous symmetry breaking (CSB) in low-dimensional systems, forbidden by
the Mermin-Wagner theorem for short-range interactions, may take place in the
presence of slowly decaying long-range interactions. Nevertheless, there is no
stringent bound on how slowly interactions should decay to give rise to CSB in
1D quantum systems at zero temperature. Here, we study a long-range interacting
spin chain with $U(1)$ symmetry and power-law interactions
$V(r)\sim1/r^\alpha$, directly relevant to ion-trap experiments. Using
bosonization and renormalization group theory, we find CSB for $\alpha$ smaller
than a critical exponent $\alpha_c(\le 3)$ depending on the microscopic
parameters of the model. Furthermore, the transition from the gapless XY phase
to the gapless CSB phase is mediated by the breaking of conformal symmetry due
to long-range interactions, and is described by a new universality class akin
to the Berezinskii-Kosterlitz-Thouless transition. Our analytical findings are
in good agreement with a numerical calculation. Signatures of the CSB phase
should be accessible in existing trapped-ion experiments.
