  We investigate the quantum-critical behavior between the rung-singlet phase
with hidden string order and the N\'eel phase with broken $SU(2)$-symmetry on
quantum spin ladders with algebraically decaying unfrustrated long-range
Heisenberg interactions. Combining perturbative continuous unitary
transformations (pCUT) with a white-graph expansion and Monte Carlo simulations
yields high-order series expansions of energies and observables in the
thermodynamic limit about the isolated rung-dimer limit. The breakdown of the
rung-singlet phase allows to determine the critical line and the entire set of
critical exponents as a function of the decay exponent of the long-range
interaction. A non-trivial regime of continuously varying critical exponents as
well as long-range mean-field behavior is demonstrated reminiscent of the
long-range transverse-field Ising model.

Strongly Correlated Electrons;Quantum Physics