Diagnosing weakly first-order phase transitions by coupling to order parameters

The hunt for exotic quantum phase transitions described by emergent fractionalized de-grees of freedom coupled to gauge fields requires a precise determination of the fixed point structure from the field theoretical side, and an extreme sensitivity to weak first -order transitions from the numerical side. Addressing the latter, we revive the classic definition of the order parameter in the limit of a vanishing external field at the transi-tion. We demonstrate that this widely understood, yet so far unused approach provides a diagnostic test for first-order versus continuous behavior that is distinctly more sensi-tive than current methods. We first apply it to the family of Q-state Potts models, where the nature of the transition is continuous for Q & LE; 4 and turns (weakly) first order for Q > 4, using an infinite system matrix product state implementation. We then employ this new approach to address the unsettled question of deconfined quantum criticality in the S = 1/2 Neel to valence bond solid transition in two dimensions, focusing on the square lattice J -Q model. Our quantum Monte Carlo simulations reveal that both order parameters remain finite at the transition, directly confirming a first-order scenario with wide reaching implications in condensed matter and quantum field theory.